ࡱ>   kqt` Pbjbjss 2K...B8 *B,: }~}~}~      $h#pDy. S*}~  D ޳޳޳  @ . ޳ ޳޳RN. P^0H <.P}~޳g }~}~}~DD }~}~}~    BBBBBBBBB Rotational Dynamics of Rigid Solids  George Kapp 2002 Rev 2008 Table of Contents Page 1. Introduction.........................................................................2 2. Coordinate System...................................................................4 3. Circular to Angular Transformations........................................4 4. The Cause Variable..................................................................6 5. Mathematical Interlude. ..........................................................8 6. Angular Relationships; The Strategy. .....................................10 7. Cause and Effect....................................................................10 8. Moment of Inertia, a Macro Quantity......................................12 9. Newtons Second Law for Rotation. ........................................12 10. Angular Impulse and Momentum. .........................................14 11. Energy Relationships; The Strategy. ......................................15 13. Work; An Alternative View. ....................................................16 14. Parallel Axis Theorem.............................................................16 15. Rotational Equilibrium...........................................................17 16. Examples...............................................................................19 17. Appendix I, Moment of Inertia Computations.........................27 18. Appendix II. The Combination Method for determining the Total Kinetic Energy.37 Rotational Dynamics of Rigid Solids 1 Introduction. It is this authors intent to start with the laws of Newton, applied to a collection of particles, and deduce all laws of rotation. To provide a crystal clear path from one to the next. In consideration of this journey, the student must be aware that the basis for true understanding and clarity is as much a mater of perspective and interpretation, as it is an exercise in mathematics and the application of fundamental laws of nature.   Consider the above situation: We have a rigid block. We give it a 180 degree rotation about each of the two axes shown, the center of mass axis, and the Y axis which is parallel to the cm axis. For the cm axis, the result is,   and for the Y axis, the result is,     We compare the result of each operation. Many observers would consider the results as dissimilar truly visual inspection would agree. The cm rotation produces no translation of the center of mass of the box whereas the Y rotation produces considerable displacement of the center of mass of the box. And yet, BOTH were 180 degree rotations. A mater of perspective and interpretation. We require a distinction between position, and orientation. While the change in position of the box is different for each event, both events show the same change in orientation of the box. The orientation has changed by 180 degrees. We will designate the objects cm position with the usual position vector Rcm, and define the objects orientation as its angular position q. It can also be seen from the example that a change in angular position (a rotation) about a given axis is equivalent to a change in angular position (a rotation) about any other parallel axis. It is important to create a clear understanding of the above paragraph here and now. We often use words such as rotate and spin; and in most cases our mind associates an axis through the body of the object. When we say rotate or spin, we must THINK change in orientation change in angular position. We must also be cautioned to guard against our minds selection of axis. Consider the following question: Is a ball rolling along on a flat table spinning about an axis which passes thru the point of contact?  The answer is of course, Yes! We now define the quantities of angular motion. The angular position variable (orientation) is defined as q. The angular velocity variable (the rate of change of angular position) is defined as w. The angular acceleration variable (the rate of change of angular velocity) is defined as a. The defining relationships are:  EMBED Equation.3   EMBED Equation.3  Notice the similarity to the translational variables, velocity and acceleration. Consider a new question. We observe a point mass particle moving through space. We observe it for only an infinitesimally short time. Is the particles velocity translational or angular? It would seem that the question is impossible to answer. If we were able to observe the particle at other times, before and after the given observation, we may feel we are in a better position to answer the question. The answer is however a mater of perspective and interpretation! We have the ability to describe the motion of the particle as either translational, or angular. It is a mater of choice. We often make this choice as a result of other information, its history and future, however the choice can not be wrong, only inconvenient. It is exactly for this reason, convenience, that the angular quantities are conceived. We have provided the mind set for the discussions to follow, we next present the tools. 2. Coordinate system. Of the many coordinate systems available, mathematics and engineering tend to favor the right hand coordinate system. Positive angular quantities are found by placing the thumb in the direction of positive on the translational axis (right hand) and observing the direction of the fingers. To represent the angular quantity as a vector, we think thumb, not fingers. This preserves the vector icon of the arrow.         It must also be remembered that the choice of coordinate system is ultimately that of the problem solver. The above coordinate system is not mandatory, it will however provide a basis of communication. 3. Circular to Angular Transformations. Three circular to angular transformations will be used to make the connection between the translational view and the angular view. Their basis lies in the fact that we can easily view the circular motion of a particle as both translational and angular. Consider a particle which is displaced along a circular path of radius r, by an amount Ds.    This displacement could also be interpreted as an angular displacement of Dq about an axis thru the center of that circle directed out from the page. Using the formula for the length of a circular sector from geometry, we have  EMBED Equation.3  where the angular displacement MUST be measured in radians. We now consider the velocity of our particle. By differentiating the above relationship with time, for constant radius, we have  EMBED Equation.3  We recognize the term on the left of the equal sign, ds/dt, as the speed of the particle along the path, the tangential velocity, vt. The derivative on the right side of the equal sign, dq/dt, is also recognized as the angular velocity of the particle about any  out of the page axis, w. This yields the second transform,  EMBED Equation.3  The third and final transform is obtained from the above in like manner,  EMBED Equation.3   EMBED Equation.3  where at is the acceleration component tangent to the circle of motion, and a is the angular acceleration of the particle about any axis perpendicular to the plane of the circle.  EMBED Equation.3  All angular quantities require radian angle measure. Vt and at are measured in the "r=0" frame of reference. The Vector aspects of these equations will be discussed later. 4. The Cause Variable. We start with the question, why do objects angularly accelerate? To answer this question, let us consider a simple see saw ( a rigid board balanced on a fulcrum).   We consider the effect of the application of the force F1 at various places along the board. When applied to the center ( at the fulcrum) we observe nothing. The fulcrum will push up on the board with a compression force to maintain equilibrium. It is when we move the force application point toward the end of the board that we notice a rotation about the fulcrum; a change in the angular velocity of the board. An angular acceleration. Experience suggests that as either the force value is increased, or the application point is moved further from the fulcrum, the angular acceleration increases. The cause of angular acceleration involves the force magnitude, the forces direction, and the location of application. We will call this Cause variable TORQUE.     The Torque, due to force F, about axis p (out of the page), is found in the following procedure: Extend the line of action of the force. Locate and measure the perpendicular distance from the line of action of the force, back to the axis. d^  must be both perpendicular to the  line of action and the axis. Compute the magnitude of the Torque as: t = d^ F Assess the direction of the angular acceleration; this is also the direction of the Torque. This can be done by physically holding down the paper at axis p and pulling on the paper at the force application point. In the diagram above, the direction would be counter clock wise (or out of the page if you think thumb instead of fingers). Granted, the above procedure seems crude; it is. It does however introduce at a fundamental level, what Torque is. There are some important points to emphasize here. The exact coordinate of the application of the force must be determined. The translational coordinate system must be specified. Choosing the axis is in effect choosing both the axis direction and the origin of the coordinate system. The magnitude and direction of the Torque will depend on the coordinate system axis. Note that if the line of action should go through the axis, d^ will be zero, and thus the Torque about that axis will also be zero. The direction of the torque ( thumb ) is always perpendicular to the plane defined by the force vector and d^ ( in the direction defined by the axis). When considering Torque, the sentence:  Torque due to _____, about _______ axis. should be in ones mind. The first blank indicating the Force, or Force collection, and the second blank indicating the axis for the computation. This procedure has been formalized as a mathematical operator, similar to the DOT Product (scalar product) used to compute work. The operator is called a CROSS Product (vector product). The notation is:  EMBED Equation.3  where R is a location vector identifying the location of the force application point in the chosen coordinate system. Using the location vector, it is easy to show that d^= R sin f, where f is the angle acquired when the location vector and the force vector are positioned tail to tail, R is swept into F.       Torque is the CAUSE variable; the cause of angular acceleration. 5. Mathematical Interlude. At this point, we present the tools of Vector Arithmetic. A right hand coordinate system is assumed. The presentation is done without derivation. Numerous texts on the subject may be consulted by the student unfamiliar with the topic. Here, we will only remind the student of their existence. The Dot Product or scalar product as it is often called was used in the definition of work. It is:  EMBED Equation.3  The Cross Product or vector product as it is often called was used in the definition of Torque. It is possible execute the cross product operator as a pure vector operation using i,j,k notation. This is accomplished by the evaluation of a 3 by 3 determinate, we also expand by minors:  EMBED Equation.3  The magnitude of the cross product may be computed as:  EMBED Equation.3  The Cross Product is not commutative! Remember however, given all the fancy notation above, the fundamental meaning of  EMBED Equation.3 is: extend the line of action of the B vector (at location A), determine the perpendicular distance from that line of action to the axis, multiply B by that distance, and assess the direction as along the axis (+ or -). Relationships using these two operators are listed below.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 , not commutative, the order is important.  EMBED Equation.3  , distributive.  EMBED Equation.3   EMBED Equation.3   To complete our mathematical interlude, we return to the circular to angular transformations. These relationships, previously presented as scalar relationships, can now be interpreted as vector relationships.        Consider a particle moving in a circular path of radius R at a speed vt. Examination of the three vectors, shows that their directions are all perpendicular to each other.     Crossing w into R will result in a vector perpendicular to the plane defined by w and R, vt. Thus, we may describe each of the circular to angular transformations using the Cross Product operator.  EMBED Equation.3  A right hand coordinate system is assumed. Vt and at are in the frame of reference of "R=0". This concludes our mathematical interlude. 6. Angular Relationships; The Strategy. Prior to developing the angular relationships, it is prudent to review the reasons we do so. The reason is convenience. We may always view an object as a large collection of small point masses; this view is what we will call the MICRO view. The MICRO view is sufficient to understand all properties of motion of an object. Using the MICRO view to compute motion properties is however very cumbersome. It requires many equations, and tedious mathematics. If there should exist a single relationship, capable of similar computational power, it would seem wise to search it out. If found, this equation will be considered a MACRO view. This then outlines the strategy we will follow. For the case of Vector relationships, there is a trick which we will commonly use, we will Cross Product the translational equation by a location vector from the left, followed by a series of simplifications and interpretations. For the case of Scalar relationships, we will perform a simple summation, followed by simplification and interpretation. In all cases, we will start with a physical relationship well founded in science. 7. Cause and Effect. Consider a collection of particles forming a rigid solid, constrained to rotate about the z axis.          We start by applying F=ma to this particle at the instant shown, in the direction tangent to its motion in the x y plane.  EMBED Equation.3  (single point) Next, we cross product both sides of the above equation by the particles location vector.  EMBED Equation.3  The left side of the above equation is now interpreted as the Torque, due to Fi, about the z axis.  EMBED Equation.3  Next, the scalar mi is factored out of the right side of the equation and the third circular to angular transformation replaces at.  EMBED Equation.3  The triple Cross Product is next expanded using eq. 6,  EMBED Equation.3 ,  EMBED Equation.3  We are now in a position to interpret,  EMBED Equation.3 , and  EMBED Equation.3  This yields the result,  EMBED Equation.3   EMBED Equation.3  If we were to perform the above derivation on each and every particle, and add the individual results, we would arrive at  EMBED Equation.3  Finally, for a rigid collection of points, all ai are identical and may be factored out of the summation giving  EMBED Equation.3  (fixed axis) We are now in a position to interpret and summarize the above equation.  EMBED Equation.3 represents the net Torque, due to all external forces, about the z axis. It is the Cause Variable.  EMBED Equation.3  is the resulting angular acceleration of the rigid solid about the z axis. It is the Effect Variable.  EMBED Equation.3  , a scalar, must be the objects resistance to being angularly accelerated about the z axis. Each Ri is the location vector, measured perpendicular to the z axis, of each particle in the object. We will provide a new name and variable for this macro quantity, moment of inertia, Iz about the z axis. Our Macro equation becomes,  EMBED Equation.3  (rigid body, fixed axis) 8. Moment of Inertia, a Macro Quantity. We have coined the term moment of inertia and assigned the meaning, resistance to angular acceleration. It is interesting to note that the word moment, which we now associate with a time interval, was in fact defined as a rotation. When we say wait a moment, we are in fact saying wait for a rotation of the shadow on a sun dial clock! The moment of inertia of a point mass about an axis is:  EMBED Equation.3  where R^ is the distance, measured perpendicular to the axis, from the axis to the point. For a collection of points in a rigid solid, their moment of inertia is given by:  EMBED Equation.3  The above summation can be performed over a rigid body as an integral to compute the moment of inertia of that body about a given axis:  EMBED Equation.3  The above computation maybe performed for a wide variety of objects, about various axis, and tabled as MACRO quantities for the convenience of later use. As a parting thought, we remind the student that the moment of inertia is not shape dependent, it depends on how the objects mass is distributed as viewed from the chosen axis. 9. Newtons Second Law for Rotation. In the previous section, Cause and Effect, we developed the angular view of F=ma. We understand that Newtons second law was stated in a much more general way, utilizing the concept of momentum. We will here perform much the same derivation as in the Cause and Effect section, on Newtons second law. Consult the figure there for clarity. Newtons second law for a particle is:  EMBED Equation.3  We will assume that F is an external force acting in the x y plane on a particle at position R. We Cross Product the above equation, on the left, by the position vector R.  EMBED Equation.3  We recognize the left side of the equation as the Torque, due to F, about the z axis on the particle. On the right side of the equation, we move the constant R, into the derivative.  EMBED Equation.3  It us useful at this time, to assign a new variable to the quantity within the brackets,  EMBED Equation.3  , whose name will be angular momentum. The result is Newtons second law for angular motion:  EMBED Equation.3  While the definition of angular momentum,  EMBED Equation.3  , may be useful for a single particle ( or the center of mass particle), we will continue to expand it by substituting the second circular to angular transform.  EMBED Equation.3  We factor out the scalar, m, and expand the triple cross product,  EMBED Equation.3   EMBED Equation.3  We are now in a position to interpret,  EMBED Equation.3 , and  EMBED Equation.3  The resulting outcome is:  EMBED Equation.3   EMBED Equation.3  and it can be seen that the angular momentum of a rigid solid can also be expressed as:  EMBED Equation.3  (Rigid solid, fixed axis) 10. Angular Impulse and Momentum. As one may suspect, one may reinvent Newtons law for angular motion, and move it into a very useful form. Starting with  EMBED Equation.3  we multiply both sides by dt, and integrate.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  (fixed axis) Application of angular impulse and momentum is in many ways similar the translational version. One interesting exception, is in the treatment of the Torque. With clever selection of axis, it is often possible to make the torque due to an external force equal zero and further simplify the application. 11. Energy Relationships; The Strategy. The total kinetic energy of any collection of particles, is the sum of the individual kinetic energies of the individual particles. Here in lies our basic strategy. There are however a few cautions which must be considered once a Macro form of that total energy is developed. Kinetic energy is frame of reference dependent in that the underlying velocity (speed) is itself dependent on the frame of reference from which it is measured. Work also depends on the frame of reference in that the displacement is depends on the frame of reference. For example, a ball rolling across a table fixed to the earth will have kinetic energy in the earth frame of reference, whereas in the balls center of mass frame of reference, the balls velocity and kinetic energy will be zero. This is not an energy violation; it is a mater of view. The energy distribution in a system, and what forces may or may not do work is dependent on the frame of reference. When doing any energy audit, the student must clearly state the frame of reference, and then consider the forms and amounts of energy involved; taking care not to ignore an amount of energy, nor collect a single amount of energy twice! 12. Rotational Energy; A Macro Form. We start with a collection of identical particles which form a rigid solid, rotating about the zcm axis (which is also the collections center of mass) as shown.       We compute the ith particles kinetic energy from the frame of reference of the particles center of mass.  EMBED Equation.3  For all particles, the total kinetic energy is the sum the individual kinetic energies.  EMBED Equation.3  We now examine the second circular to angular transformation to arrive at a substitution for v2.  EMBED Equation.3  We substitute in the kinetic energy equation,  EMBED Equation.3  Note that for a rigid solid, although the vi are not identical, w1= w2= w3 & = wcm. We factor out of the summation, , and wcm.  EMBED Equation.3  The quantity within the parentheses is recognized as the objects moment of inertia about the cm axis. We now define the Macro Rotational Energy as:  EMBED Equation.3  The above rotational energy is in fact the micro kinetic energy of the collection of particles in the rigid solid, rotating about the axis for which the moment of inertia ( and angular velocity) is computed. 13. Work; An Alternative View. Contemplate the task of starting a power lawn mower. In pulling the mowers cord, we apply a force over a displacement and thus do work. We may however wish to view this task in another way; we apply a torque by means of the tension in the cord and angularly displace the shaft of the motor. Both views are of the same work. Only the view of it is different. We derive the equivalent.  EMBED Equation.3  Choosing our axis origin through the shaft of the motor, we both multiply and divide the force by the perpendicular distance from the shaft axis to the line of action of the force, d^.  EMBED Equation.3  The product d^ F is the torque due to the force, about the axis. The quantity dx/ d^ is the angular displacement, dq. Substitution gives,  EMBED Equation.3  (fixed axis) We again remind the reader that these are not two separate works, only two alternative views of the same work. 14. Parallel Axis Theorem. Consider a collection of particles which form a rigid solid, rotating about a fixed z axis (not through the center of mass) as shown below. We may audit the objects energy of motion from the frame of reference of this fixed z axis in a number of ways. First, as in the previous section, we can accept the Micro view; the sum of all the individual kinetic energies of the individual particles. Second, we can accept the Macro view; the rotational energy, I w2, about the z axis. There is however a third  combination view of the objects energy which is often quite useful. In the combination view, we add the kinetic energy of the objects center of mass as if it is a single particle, to the objects rotational energy about the objects center of mass axis which is parallel to the z axis. See Appendix II. All three views of the total energy are identical; they produce the same total energy.          We equate the pure rotational view of the energy about the z axis to that of the combination view:  EMBED Equation.3   EMBED Equation.3  We may use the second circular to angular transform to replace vcm in the above equation.  EMBED Equation.3 .  EMBED Equation.3  EMBED Equation.3  Recalling that an angular displacement about an axis is equivalent to the same angular displacement about any parallel axis,  EMBED Equation.3 , we cancel both and wz from the above equation to give the Parallel Axis Theorem.  EMBED Equation.3  The above derivation not only illustrates various perspectives and interpretations of rotational energy, but provides a most useful formula for computing the moment of inertia of a rigid body. Most tables of moment of inertia list those values for axis through the center of mass of the object. The parallel axis theorem provides the to shift the axis of computation to other locations. 15. Rotational Equilibrium. Beginning students of physics are often introduced to the concept that when an object is at rest,  EMBED Equation.3  and  EMBED Equation.3  The above being described as the first, and second, condition of equilibrium. Nearly all the examples presented the student involve objects at rest, that is zero translational acceleration and zero angular acceleration. We now wish to consider the case of non-zero translational acceleration. Is an object, whose center of mass is accelerated, but is not spinning otherwise, angularly accelerated? As for example, a car on a level road, which is applying its brakes to stop; is the car angularly accelerated? Before we formulate an answer, let us look at F=ma applied to the center of mass of the object.  EMBED Equation.3  We cross product both sides of this equation by the instantaneous location vector of the center of mass,  EMBED Equation.3  The quantity on the left of the equal sign is identified as the sum of the external torques,  EMBED Equation.3  We arrive at the result,  EMBED Equation.3  With the above relationship, we now answer the question; It is a mater of perspective and interpretation it is a mater of coordinate system! If the axis origin is located at the objects center of mass ( or anywhere along the line of action of the acceleration vector acm, the cross product,  EMBED Equation.3  and we may conclude that the angular acceleration about that axis is zero. This also leads to the fact that the sum of the Torques about that axis will also be zero. If however, the axis origin is located any where else, the cross product is not zero; and with substitution of the third circular to angular transformation  EMBED Equation.3 , followed by the expansion of the triple cross product (similar to that used in the section Cause and Effect), we arrive at  EMBED Equation.3  where it is clear that the object possesses a non-zero angular acceleration. Perspective and interpretation are always considered from a previously selected coordinate system s frame of reference. 16. Examples. A solid sphere of mass .5 kg and radius .4 m is released (vi=wI=0) on a 10 degree incline and rolls down without slippage. We wish to determine the acceleration of the center of mass of the sphere, the final velocity of the center of mass of the sphere when its altitude is lowered by 2 meters, and the corresponding angular values as well. We start with the drawing below. A force analysis results in three forces externally acting on the sphere: the weight of the sphere placed at its center of mass, the compression and friction between the incline and sphere acting at the point of contact. Coordinate system directions for translation are chosen; we will hold off the decision as to where the origin is located momentarily. We can see that the lines of action of all forces do not intersect at a single point in space; this lets us know that the problem will involve angular components.       Finding acm, solution 1. We start with a choice of origin; it will be placed at the point of contact of the sphere and the incline. Z axis out of the page. The coordinate system is thus fixed to the earth frame of reference and instantaneously at rest. About this axis, the torque due to both C and Ff will be zero. This is clever for two reasons: we do not know the value of C, and we do not know the value of Ff ( remember that Ff=mC is invalid since there is no slippage). The only torque producing force is mg, and this we know. We perform a torque analysis,  EMBED Equation.3 about this axis. The perpendicular distance from the z axis to the line of action of the force mg is  EMBED Equation.3 , and thus the torque due to mg about the z axis is  EMBED Equation.3 . Looking up the moment of inertia for a sphere we find Icm=2/5 mR2. Since our axis is the z axis, not the center of mass axis, we use the parallel axis theorem to offset the axis a displacement of one radius.  EMBED Equation.3  The equation,  EMBED Equation.3 , is applied.  EMBED Equation.3   EMBED Equation.3  The mass is canceled, and the radius also giving  EMBED Equation.3  Finally, we recognize the quantity on the right as acm.  EMBED Equation.3  Note that the outcome does not depend on radius or mass. Finding acm, solution 2. In this solution approach, we place our z axis origin at the center of mass of the sphere for the torque computations only. This means that the z axis is accelerated in the earth frame of reference, but is at rest in the center of mass frame of reference. The forces C and mg produce zero torque about the center of mass axis; the force of friction produces a torque of the amount, R Ff. The required moment of inertia is about the center of mass axis.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  It is at this point that we observe the first complication; we do not know Ff, nor do we have a convenient substitution for it. We thus resort to Newtons second law, applied to the center of mass particle, in the y dimension along the incline, for the required substitution.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  We equate these two results,  EMBED Equation.3  A few steps of algebra yield,  EMBED Equation.3  To complete the derivation, we recall that the angular acceleration about the center of mass axis is equivalent to the angular acceleration about any other parallel axis. We choose the z axis at the point of contact,  EMBED Equation.3 , and substitute,  EMBED Equation.3  As seen in solution 1, we now identify R az as acm based on the third circular to angular transformation and substitute.  EMBED Equation.3  We solve for acm.  EMBED Equation.3  The result is of course identical. The angular acceleration is most easily found by applying the third circular to angular transform,  EMBED Equation.3 . Specific values for the posed problem give:  EMBED Equation.3   EMBED Equation.3  Finding Vcm, solution 1. In this approach, we simply use the linear motion equation,  EMBED Equation.3 , and the acceleration obtained from the previous solution. Here, vi=0, and  EMBED Equation.3 Ds=Dh/sin q .  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3  This is the quickest way to solve for vcm if acm is known. Finding vcm, solution 2. In this approach, we use the work energy theorem. We select the earth frame of reference; the z axis origin is placed at the point of contact of the sphere. The version of the work energy theorem we will use is:  EMBED Equation.3  Reviewing the work done, C does no work since the angle between C and the displacement is 90 degrees; Ff does no work since the displacement is zero ( no slippage), and the work done by mg is excluded. Thus, no work is done. We will treat the kinetic energy change as purely rotational about the z contact axis. The gravitational potential energy change will be negative (we control the sign explicitly).  EMBED Equation.3  As in the previous examples, Iz is computed using the parallel axis theorem as Iz=7/5 mR2, and upon substitution we have,  EMBED Equation.3  a few steps of algebra gives,  EMBED Equation.3  We identify  EMBED Equation.3  using the second circular to angular transformation and replace in the above equation to give,  EMBED Equation.3 =5.23 m/s The value obtained is identical to solution 1. Finding vcm, solution 3. This approach is quite similar to solution 2; we use the work energy theorem. We select the earth frame of reference; the z axis origin is placed at the point of contact of the sphere. The version of the work energy theorem we will use is:  EMBED Equation.3  Reviewing the work done, C does no work since the angle between C and the displacement is 90 degrees; Ff does no work since the displacement is zero ( no slippage), and the work done by mg is excluded. Thus, no work is done. The work analysis is identical to that in solution 2. What is different is that we will treat the kinetic energy change as a combination of the kinetic energy of the center of mass particle and add the rotational energy about the center of mass axis. The gravitational potential energy change will be negative (we control the sign explicitly).  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  At this point, we recall  EMBED Equation.3 , and  EMBED Equation.3 . Substitution in the above gives,  EMBED Equation.3   EMBED Equation.3  and solving for vcm, we arrive at our result,  EMBED Equation.3  which is identical to solution 1 and 2. The combination approach using the kinetic energy of the center of mass plus the rotational energy about the cm axis often seems to be the easiest for beginning students since it simply adds the change in rotational energy term to the already familiar form of the work energy theorem. Finding vcm, solution 4. This approach is by far the least efficient solution. It is presented here only to illustrate how the frame of reference effects the work energy distribution. We apply the work energy theorem of form,  EMBED Equation.3 . What is different is that we will select as a frame of reference, the center of mass of our sphere. In this frame of reference, we observe neither a change in altitude, nor a change in center of mass velocity; only a change in the micro kinetic energy of rotation. We will use as the form of the work calculation,  EMBED Equation.3  which reduces to  EMBED Equation.3  for a constant torque. The work audit suggests that the forces C and mg produce no torque about the center of mass axis (note mg would be excluded anyway do to the presence of the change in gravitational potential energy) and thus does no work. The Ff is another mater. In this frame of reference, the force of friction produces a constant torque about the center of mass axis which results in an angular displacement. Work is done by Ff. Our work energy equation becomes,  EMBED Equation.3   EMBED Equation.3  We substitute for the moment of inertia and reduce,  EMBED Equation.3   EMBED Equation.3  Note that from the first circular to angular transform, RDq=Ds.  EMBED Equation.3  Here again, we run into the problem of a suitable replacement for Ff. We utilize Newton s second law applied to the center of mass particle in the y dimension along the incline;  EMBED Equation.3 . We solve for Ff to give  EMBED Equation.3 and replace the force of friction in the above equation,  EMBED Equation.3   EMBED Equation.3  Let us recognize what we have above;  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 . Substitution gives,  EMBED Equation.3  This now leaves us with the task of eliminating acm. Recall from linear motion,  EMBED Equation.3 . Setting vi=0 and rearranging we get  EMBED Equation.3 which we will substitute.  EMBED Equation.3  Final rearrangement produces the desired and identical result,  EMBED Equation.3  Clearly a scenic exercise. The journey does however illustrate the fact that the work and energy quantities may distribute them selves in different forms and places when viewed from different frames of reference. Example. A 200 lb bicycle(and rider) suddenly applies both brakes to stop. The bicycle slides along the road way. Coefficient of friction between the tires and road is .6 . We wish to determine the center of mass acceleration and the friction forces on each tire. The geometry of the bicycle and rider, and forces involved, are shown in the figure below.       We choose the translational coordinate system as shown, the location of the z axis origin will be at the contact point of the front tire. With this choice, we recall there is a non-zero angular acceleration (see Rotational Equilibrium). The advantage of this selection is that many of the forces involved produce zero torque about the axis. We perform a linear force analysis to obtain acm. The system equations are:   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The friction model equations are substituted into the y dim equation to give,  EMBED Equation.3  The equation is factored,  EMBED Equation.3  The x dim equation is substituted for the sum of the compression forces,  EMBED Equation.3  The mass is canceled and we have our result.  EMBED Equation.3  We now address the issue of torques. The equation is:  EMBED Equation.3  With the chosen axis, only mg and C2 produce torques. Our equation is:  EMBED Equation.3  Using mg=200 lbs, m=6.25 slugs, and acm=-19.2 f/s2, we solve for C2 to get C2=36 lbs From the vertical force balance, C1=164 lbs. The friction forces may now be obtained from the friction model equations as Ff1=98.4 lbs and Ff2=21.6 lbs or a total friction force of 120 lbs. As a quick check, we evaluate |macm|= 6.25 slugs 19.2 f/s2 = 120 lbs! We will leave it to the reader to verify that if the z axis were placed at the center of mass, the sum of the torques would equal zero about this axis, and that analysis would yield the same result. 17. Appendix I, Moment of Inertia Computations. Here we derive the specific formula for the moment of inertia of selected objects about various axis through implementation of the definition,  EMBED Equation.3  Before the student of physics reviews these derivations however, a few common thoughts must be emphasized: We draw attention to the coordinate system, and its origin. When we select the axis of revolution, for example the z axis, any element of matter (dm) that lies on that axis (x=y=0) must contribute zero moment of inertia to the object. Its Rz must be zero. Where you put your origin is extremely important. An often overlooked fact is that the coordinate axis has both a plus and minus direction. Recall we are using the coordinate system to target (address) elements of matter. Consideration of the objects symmetry. If the object possesses symmetry about the chosen axis, it will be more efficient if we take advantage of that symmetry. The variable dm in the moment of inertial definition has important physical significance but is useless in the evaluation of the integral itself. We will consistently use the fact that the object is homogeneous and uniform, and thus of constant density (r). Therefore,  EMBED Equation.3  and  EMBED Equation.3 may be used to replace dm in the integral. Finally, our tactic will in general be to: Sketch the object. Identify the axis. Identify the differential element taking advantage of symmetry. Target the element using the coordinate system. Identify the limits of the integral. Perform the integral. We proceed. Long, infinitely thin object, cm axis. By definition, infinitely thin implies that the cross section of the object has no distinguishing shape; the object can be considered a long thin round rod, a long thin square rod, etc. Also, infinitely thin means that we need only consider the objects length in computing the moment of inertia.     Above, we have our sketch. The axis is identified as out of the page and the origin is at the center of mass. The differential element is located at x and its length is dx. The limits are identified. We present the integral:  EMBED Equation.3  To replace dm, we observe the one dimensional nature of this integral and will thus use the linear mass density: the mass per unit length.  EMBED Equation.3  and so the substitution becomes  EMBED Equation.3 ,  EMBED Equation.3  The integral is executed to give,  EMBED Equation.3  and evaluated,  EMBED Equation.3   EMBED Equation.3  At this point, we replace the linear mass density with  EMBED Equation.3  , and arrive at our final form,  EMBED Equation.3  Circular Cylinder of length L, axis along length, through cm. Below is our sketch, top view, with steps 1 4 complete. The axis is out of the page and through the center of mass. We will take advantage of the circular symmetry by selecting the differential element as a ring of thickness dr and length L. The limits of integration will be from r=0 to r=R. Since the object has area, we will use the area mass density; the mass per unit area.       We start with our moment of inertia definition,  EMBED Equation.3  To replace dm, we utilize the mass pre unit volume;  EMBED Equation.3  and solving for dm we get,  EMBED Equation.3 . The volume of our differential ring, from geometry, is,  EMBED Equation.3 . Thus  EMBED Equation.3 and we substitute into the integral.  EMBED Equation.3  The integral is executed and evaluated to give,  EMBED Equation.3  We replace the density,  EMBED Equation.3  to get,  EMBED Equation.3  Notice that the length of the cylinder does not appear in the final result. Flat Circular Plate, infinitely thin, axis within the plane of the area. This object is somewhat different than the long thin rod in that it has a finite area and is thus two dimensional. We will take this opportunity to illustrate a trigonometric substitution of variable. Below is our sketch with steps 1 through 4 complete. The axis is within the page as shown and through the center of mass. A vertical strip will be chosen as the differential element. The perpendicular distance to the differential element is x.   The integral to evaluate is:  EMBED Equation.3  . To put this in executable form we require a few substitutions,  EMBED Equation.3  and  EMBED Equation.3  Since q is a function of x, we will replace x with q and take the integration limits from q=p (formerly  R) to q=0 rad (formerly +R). From trigonometry,  EMBED Equation.3  and differentiation of the above gives,  EMBED Equation.3  Replacement of dm with much substitution gives,  EMBED Equation.3  Substitution of x and dm into the integral gives,  EMBED Equation.3  Now from trigonometry  EMBED Equation.3  . We substitute and simplify,  EMBED Equation.3  From tables of integrals, we find  EMBED Equation.3  and apply it to our integral,  EMBED Equation.3  which simplifies to,  EMBED Equation.3  Replacing the density with  EMBED Equation.3  , we arrive at the final result,  EMBED Equation.3  Circular Hoop, Infinitely Thin, Axis normal to Hoop through cm. Since all the mass of the hoop is at a perpendicular distance R from its center, the moment of inertia formula is identical to that of a point mass. No derivation is necessary.   EMBED Equation.3  Rectangular Slab, cm Axis.  We take this opportunity to introduce an interesting and efficient technique called a multiple integral to compute the moment of inertia of a rectangular slab. The multiple integral is often used to address points in space on an object when the object and the coordinate system have similar symmetry. The technique is based on the fact that the vector components are independent of each other (orthogonal) and may therefore be changed (moved) independently of each other to address (target) each point in the object. Consider the following diagram.            Our differential element, dm, has a volume dV= c dx dz . We will use the volumetric mass density,  EMBED Equation.3 . Our integral for consideration is,  EMBED Equation.3  In the above integral,  EMBED Equation.3  and so we replace it.  EMBED Equation.3  As you are observing, we are systematically replacing our variables with functions of x and z. Once this is accomplished, the double integral will effectively move our differential element over the x z surface, collecting each elements contribution to the moment of inertia of the slab. Next, we replace dm and show the result as a double integral. Notice the limits of integration will sweep the x z surface of our slab.  EMBED Equation.3  At this point, we explain the procedure to evaluate the above double integral. In our case, we have two variables, x, and z. We will start with x and integrate considering z to be constant. Then we will integrate again on the remaining variable, z. The order should not be important since x and z are orthogonal. This method in effect, sums the x components first, and then the z compon$%&':BFX *:?@eghjy456:?A(/68^_abcdefBJ LNοοοηΤΔΔΔΔΔΔ΍΍΍ h~86]jh~8CJUmHnHu h~8CJh~86>*] h|\h|h|\h| h~8>*CJhNyh~8 h~85\ jhEmz5U\mHnHujhEmzUmHnHu h~8>*CJ6$'5:CDEFXY` j  p  o  d    !X  [$a$gdNy$a$eO s ?@Aef1345789:@A&'(01234`$a$$gd| 4569]^`aceghi,-tvxz|T$a$`NR)1GIdn|RT TV|~IO󪝽}jSh|h|EHU%j&K h|CJOJQJUVaJjh|h|EHU%j.&K h|CJOJQJUVaJjh~8U h~85\jh~8CJUmHnHu h~8>*h~85OJQJ\ h~86]h~8h~85H*\--.m!!!!!!!!!!!!!!!!p"q"r"#`$a$!!!!!!!!!!!!!!!!!!!!s"v"w""$$ $$$$"$$$&$,$$$%%&&&&&&һһҰⰻ⻰⡰}!jh~85EHOJQJU\%j+A h~8CJOJQJUVaJjh~85OJQJU\h~85OJQJ\,jh~85CJOJQJU\mHnHujh~8CJUmHnHuh~8 h~85\$jh~85CJU\mHnHu.# $ $$$$$$($*$,$%&P&&&)F)) *:*++,,,Z-[-_-`-$a$`&&&&&&m'n'((((((())>)@)B)D)))))**** * *2*4*6*8*H*J**qgj< h~8EHU%j3A h~8CJOJQJUVaJj h~8EHU%j2A h~8CJOJQJUVaJh(xjh~8EHU%j 2A h~8CJOJQJUVaJ h~85\h~8OJQJ h~8H*jh~8EHU%je/A h~8CJOJQJUVaJh~8jh~8U%*******++++++,, ,&,',W,Z,,,Z-[-^-`-f-h---^0g0l0p0q0r0v0w0x0y0z00011222222ꈂ h~8CJh~8CJOJQJ h~86] h~8H*,jh~85CJOJQJU\mHnHu hR%h~8 hR%H*hR%jLh~8EHU%j4A h~8CJOJQJUVaJjh~8Uh~8 h~85\h~85OJQJ\2`-g-h-h0j0l0q0s0t0u0v0x0z0{0012244z556|89::; & F & F & F`222222224477T9\9^9`99):;;;;;;<<<<<<<==================䫡zzzzzjh~8CJUmHnHu,jh~85CJOJQJU\mHnHujh~8EHU%jbA h~8CJOJQJUVaJjh~8U h~86] h~85\ h~8CJh~8CJOJQJh~8h~85OJQJ\h~85CJ OJQJ\0;;6<=================;>?????A$`a$ !`$a$=;>U>>>??????@@ A!A4A5A6A7ApAqAAAAAAABBBBBB1C2CECukj&h~8EHU%j~A h~8CJOJQJUVaJjzh~8EHU%jYA h~8CJOJQJUVaJjhR%hR%EHU%jO'K hR%CJOJQJUVaJ h~86]jh~8EHU%jfA h~8CJOJQJUVaJjh~8UhR% h~85\h~8#A A8A9ApAAAABB1CICaCCCCCCCCDDDDDDh^h & Fdh$a$$`a$`ECFCGCHCICJC]C^C_C`CaCbCuCvCwCxCCCCCCCCCCCCCCCCù֦։lbj#(h~8EHU%jGA h~8CJOJQJUVaJj%h~8EHU%jZA h~8CJOJQJUVaJji#h~8EHU%jA h~8CJOJQJUVaJjT!h~8EHU%joA h~8CJOJQJUVaJh~8jh~8Uj0h~8EHU%jA h~8CJOJQJUVaJCCCCDDDDDDDDDDDDDDDDDD*j[h~8EHU%j/A h~8CJOJQJUVaJjYh~8EHH*U%jiA h~8CJOJQJUVaJ h~8H*jh~8H*Uh~8CJOJQJj_Wh~8EHU%j,A h~8CJOJQJUVaJ h~86] h~85\h~8h/8jh~8Uj Uh~8EHU%YlZZZ#[[[S]T]{]]B^Z^_)____``aQaRaja !`gdi$a$gdu$a$gdi$`a$`$a$]]]]B^C^V^W^X^Y^__%_&_'_(______________``Ӡӏ|oӏhU%j"A h~8CJOJQJUVaJ h~86]jfh~sh~sEHU%jCA h~8CJOJQJUVaJhujch,h,EHU%j+K h,CJOJQJUVaJj`h,h,EHU%j+K h,CJOJQJUVaJh~8jh~8Uj'^h,h,EHU%j+K h,CJOJQJUVaJ```=`>`Q`R`S`T``` a a aaRaSafagahaiakalaaaaaaaaa涩斉viV%jA h~8CJOJQJUVaJjsh,h,EHU%j,K h,CJOJQJUVaJjph,h,EHU%ja,K h,CJOJQJUVaJjvmh,h,EHU%jG,K h,CJOJQJUVaJj#kh~8EHU%jCA h~8CJOJQJUVaJh~8jh~8Ujhh~sh~sEHUjakaaabb+bbbbbbbbcccccKdLd}e~eeeg`gdu $ !a$ !$a$aaaaaaaaaaabbbbbbb'b(b)b*b+bbbbbbbbbb cc鹬騠|i\U h~85\jh~sh~sEHU%j/K h~sCJOJQJUVaJhuj#h,h~sEHU%jF/K h~sCJOJQJUVaJjh~sUh~sj8|h,h~sEHU%j".K h~sCJOJQJUVaJjlyh~8EHU%jA h~8CJOJQJUVaJh~8jh~8Ujwh~8EHU!ccccccccccccccccccc d!d4d5d6d7dJdeeSjTjUjyjù{tdQt$jh~85CJU\mHnHujh~8CJUmHnHu h~85\jh~8EHU%j%A h~8CJOJQJUVaJhujh~8EHU%jA h~8CJOJQJUVaJj`h~8EHU%jA h~8CJOJQJUVaJjh~8EHU%j"A h~8CJOJQJUVaJh~8jh~8UgciOjPjQjRjSjkk"k#k$k(k+k-k/k2k3k4k5k6kkkl'llll$a$ !`yjjjk!k$k'k(k*k+k,k-k.k/k1kFkHkkkkkkkll#l$l%l&lllllllllllllǽp%jA h~8CJOJQJUVaJjh~8EHU%jMA h~8CJOJQJUVaJj[h~8EHU%jXA h~8CJOJQJUVaJjh~8EHU%jgA h~8CJOJQJUVaJjh~8U h~8H*jh~8CJUmHnHu h~8H*h~8(lllmmnnn nnnn"n&n|n~nnnnnnnnooppp ppprrrrrrtt t t2tɿ鬢~th~8CJOJQJjh~8EHU%j2 A h~8CJOJQJUVaJ h~85\jh~8EHU%jeA h~8CJOJQJUVaJjHh~8EHU%jCA h~8CJOJQJUVaJh~8OJQJ h~8H*h~8jh~8Ujh~8EHU)llnnnoo p pprr t:tRuTuuuAvBvxzzzzzz` $ !a$ !$a$2t4t6t8tVtXttt"u$uTuVu|u~uuuuuuAv]vZx\x^xxxHzpzqz|z~zzzzֲֹ~wrjrfVjh~8CJUmHnHuhuh~85>*\ h~8>* h~86] h~8H*huhu\ hu5\jh~85EHU\%j]6 A h~8CJOJQJUVaJ h~85\jh~85U\h~8OJQJh~8CJOJQJh~8jh~8Ujh~8EHU%j5 A h~8CJOJQJUVaJ!zzzzzzzzzzzzzzz{{g{h{{{|{}{~{{{{{{{{{{{ ||||||@|B|D|H|ukjh~8EHU%jF?A h~8CJOJQJUVaJjh~8EHU%j>A h~8CJOJQJUVaJ h~8H*jh~8EHU%jk>A h~8CJOJQJUVaJjdh~8EHU%j=A h~8CJOJQJUVaJjh~8Ujh~8CJUmHnHuh~8*zzzz{{{{g{{{||v|$~%~=~>~H|ʀ߂`$`a$` !$a$H|n|p|r|t|p}r}}}}}}}} ~"~%~&~9~:~;~<~HI\]^_dexĺndjh~8EHU%j5A h~8CJOJQJUVaJ h~85\jh~8EHU%jBA h~8CJOJQJUVaJh~86>*] h~8H*h~8OJQJjph~8EHU%jNAA h~8CJOJQJUVaJjh~8Ujnh~8EHU%j@A h~8CJOJQJUVaJh~8 xyz{ ߂`atuvwՃփ02ּ֟ւxe[V h~8H*jgh~8EHU%j"A h~8CJOJQJUVaJjh~8EHU%jA h~8CJOJQJUVaJjh~8EHU%jزA h~8CJOJQJUVaJjTh~8EHU%j-A h~8CJOJQJUVaJ h~86]h~8jh~8Uj&h~8EHU%jjA h~8CJOJQJUVaJ!`xՃ89QR468V\ŒÌČ$ !`$a$2GH[\]^789:MNOP8RԈֈ؈ڈ܈0PnjȌɌˌ̌ĺr_rr$jh~85CJU\mHnHujh~8CJUmHnHu h~86]h~8OJQJ h~85\jch~8EHU%joA h~8CJOJQJUVaJ h~8H*jh~8EHU%jOA h~8CJOJQJUVaJjh~8EHU%j:A h~8CJOJQJUVaJjh~8Uh~8%ČŌƌnjɌ̌όЌьӌ֌׌،ٌڌی܌݌ʑ-Ev !$ a$  ̌͌ΌьҌӌԌՌތ"$&(*+,-ސߐٺٲٟٲقxٲe%jH9 A h~8CJOJQJUVaJjh~8EHU%jW8 A h~8CJOJQJUVaJjh~8EHU%j07 A h~8CJOJQJUVaJjh~8Uh~8OJQJ h~8H*h~86>*H*]h~86>*]h~8jh~8CJUmHnHu$jh~85CJU\mHnHu#029:ʑˑޑߑ)*+,-.ABCDvwx鯥针uk[jh~8CJUmHnHujh~8EHU%j6K A h~8CJOJQJUVaJjh~8EHU%j)K A h~8CJOJQJUVaJjh~8EHU%j07 A h~8CJOJQJUVaJjDh~8EHU%jL: A h~8CJOJQJUVaJ h~8H* h~8H*h~8jh~8Ujh~8EHU"Ȓʒ̒͒Β )+8/01234֮֜ubXjh~8EHU%jDP A h~8CJOJQJUVaJjh~8EHU%jK A h~8CJOJQJUVaJh~86>*H*]h~86>*]jh~8EHU%jB A h~8CJOJQJUVaJ h~8H*jh~8CJUmHnHuh~8jh~8Uj;h~8EHU%jJK A h~8CJOJQJUVaJ"̒3K`xŖƖޖ.^R֙  ! $ !a$ !4GHIJ`atuvwxyƖǖږۖܖݖ{h^jh~8EHU%jS A h~8CJOJQJUVaJjh~8EHU%jR A h~8CJOJQJUVaJjh~8EHU%jQ A h~8CJOJQJUVaJjxh~8EHU%jwQ A h~8CJOJQJUVaJ h~8H*jh~8Uj6h~8EHU%j4R A h~8CJOJQJUVaJh~8! .0VXZ\ĘRTz|~ΙЙҙù֦֔֏|r֏_Ujh~8EHU%jB A h~8CJOJQJUVaJj8h~8EHU%j[V A h~8CJOJQJUVaJ h~8H*h~8OJQJjh~8EHU%jwU A h~8CJOJQJUVaJjh~8EHU%jT A h~8CJOJQJUVaJh~8jh~8Ujh~8EHU%j T A h~8CJOJQJUVaJ!ҙԙrs͚̚ΚϚКњABUVWXù󏅏rhc h~8H*j*h~8EHU%jr A h~8CJOJQJUVaJh~86>*H*]h~86>*] h~8>*jh~8EHU%jX A h~8CJOJQJUVaJjh~8EHU%jW A h~8CJOJQJUVaJjh~8EHU%jiW A h~8CJOJQJUVaJh~8jh~8U$К PޜVXNf4١ .G $ !a$  ! ! $ !a$ "HJLNPRxz|~֜؜ڜܜ.2<λΞ΁wdZU h~8H*j h~8EHU%j A h~8CJOJQJUVaJjh~8EHU%j A h~8CJOJQJUVaJj;h~8EHU%jW A h~8CJOJQJUVaJjh~8EHU%j A h~8CJOJQJUVaJh~8h~8OJQJjh~8Ujh~8EHU%j A h~8CJOJQJUVaJ!<@ZlpNObcde0ɿukjh~8EHU%j A h~8CJOJQJUVaJjh~8CJUmHnHu h~8H*j\h~8EHU%j A h~8CJOJQJUVaJ h~8>*j h~8EHU%j A h~8CJOJQJUVaJjh~8Uh~86>*H*]h~86>*]h~8 h~8H*%012345ABUVWX̡͡Ρϡ "*+,-FGlֳ֖քzg]XS h~8>* h~8H*jh~8EHU%j A h~8CJOJQJUVaJh~86>*H*]h~86>*]j\h~8EHU%j> A h~8CJOJQJUVaJjh~8EHU%jг A h~8CJOJQJUVaJjh~8CJUmHnHuh~8jh~8Ujh~8EHU%j A h~8CJOJQJUVaJ Glͥ>Vnަ Yqլ@دT $ !a$  !lmɥʥ˥̥ùlbj(h~8EHU%j r A h~8CJOJQJUVaJj&h~8EHU%jq A h~8CJOJQJUVaJj8#h~8EHU%j4q A h~8CJOJQJUVaJj h~8EHU%jp A h~8CJOJQJUVaJj"h~8EHU%j; A h~8CJOJQJUVaJh~8jh~8U >?RSTUVWjklm ù֦֗քzrhrU%jx A h~8CJOJQJUVaJh~86>*H*]h~86>*]j1h~8EHU%j> A h~8CJOJQJUVaJ h~8H*j|/h~8EHU%jr A h~8CJOJQJUVaJj,h~8EHU%jr A h~8CJOJQJUVaJh~8jh~8Uj*h~8EHU%j8r A h~8CJOJQJUVaJ;<OPQRdexyz{xy34YZmnopqrѬҬ鹯闍zp]%jz A h~8CJOJQJUVaJj=h~8EHU%jy A h~8CJOJQJUVaJj:h~8EHU%jy A h~8CJOJQJUVaJ h~8H*j8h~8EHU%jzw A h~8CJOJQJUVaJj6h~8EHU%j'w A h~8CJOJQJUVaJh~8jh~8Uj\4h~8EHU"ҬӬԬլ֬8:<>ƮήίЯүԯTUh鱧鏅rhj7Ih~8EHU%j| A h~8CJOJQJUVaJjGh~8EHU%jM| A h~8CJOJQJUVaJ h~8H*jDh~8EHU%jx~ A h~8CJOJQJUVaJh~8OJQJj#Bh~8EHU%j'{ A h~8CJOJQJUVaJh~8jh~8Uju?h~8EHU%hijklm°ðְװذٰ߰ !ù֦։lbjTh~8EHU%jb A h~8CJOJQJUVaJjRh~8EHU%j A h~8CJOJQJUVaJjPh~8EHU%j؂ A h~8CJOJQJUVaJjNh~8EHU%j A h~8CJOJQJUVaJh~8jh~8UjxKh~8EHU%j~ A h~8CJOJQJUVaJTl %=UV./0ôĴ $ !`  ! $ !a$!"#$VXvwȱɱʱ˱=>QRST0־֡քzg]j`h~8EHU%j> A h~8CJOJQJUVaJj6^h~8EHU%j@ A h~8CJOJQJUVaJj[h~8EHU%jʅ A h~8CJOJQJUVaJj|Yh~8EHU%j A h~8CJOJQJUVaJ h~8H*h~8jh~8UjWh~8EHU%jЄ A h~8CJOJQJUVaJ 0IQ´ƴʴRTqtuv¶&߯ߒukjQhh~8EHU%jŘ A h~8CJOJQJUVaJjeh~8EHU%jї A h~8CJOJQJUVaJj5ch~8EHU%j A h~8CJOJQJUVaJjh~8U h~8H*jh~8CJUmHnHuh~8h~86>*]'jh~86>*CJU]mHnHu'ĴŴƴ˴̴ʹ#quöĶ*D\:;Sǹ$a$ !&'()DEXYZ[;<OPQRvwù֦։lb] h~8H*jsh~8EHU%j A h~8CJOJQJUVaJjUqh~8EHU%j A h~8CJOJQJUVaJj;oh~8EHU%js A h~8CJOJQJUVaJjlh~8EHU%j A h~8CJOJQJUVaJh~8jh~8Ujjh~8EHU%j A h~8CJOJQJUVaJ ظڸ*+&(˻̻ͻλ01 "HJļĵ֢֑։v%j$YA h~8CJOJQJUVaJh~8OJQJ h~86]jxh~8EHU%j/A h~8CJOJQJUVaJ h~85\h|5>*\h~85>*\ h~8H* h~8H*h~8jh~8Ujqvh~8EHU%j\ A h~8CJOJQJUVaJ*ǹȹ׺غٺںۺܺݺ޺ߺ`ϻл>s$ž2X~/Tjvw & F  & F $`a$` !JLNXZ~rs顗zg%jA h~8CJOJQJUVaJjh~8EHU%jA h~8CJOJQJUVaJjh~8EHU%jA h~8CJOJQJUVaJjh~8CJUmHnHuh~86>*]j}h~8EHU%jYA h~8CJOJQJUVaJh~8jh~8Uj-{h~8EHU$wxyz{|}~r% !$a$`%&9:;<>?RSTU鹯霒ubXj4h~8EHU%jA h~8CJOJQJUVaJjh~8EHU%jA h~8CJOJQJUVaJj5h~8EHU%joA h~8CJOJQJUVaJjh~8EHU%jsA h~8CJOJQJUVaJj2h~8EHU%j4A h~8CJOJQJUVaJh~8jh~8Ujh~8EHU"%=>V   !` !$a$ @ATUVWst±wd%jvVA h~8CJOJQJUVaJj֖h~8EHU%jRVA h~8CJOJQJUVaJjh~8EHU%jWA h~8CJOJQJUVaJ!jh~8>*CJUmHnHujh~8CJUmHnHu h~8>*jh~8UjZh~8EHU%jA h~8CJOJQJUVaJh~8% !4567hi|}~鹯霒ubXjMh~8EHU%jSWA h~8CJOJQJUVaJjh~8EHU%j$WA h~8CJOJQJUVaJjIh~8EHU%jWA h~8CJOJQJUVaJj(h~8EHU%jVA h~8CJOJQJUVaJjh~8EHU%jVA h~8CJOJQJUVaJh~8jh~8Ujh~8EHU" 8hij*+,:=>?@AFIJKLM !$a$`i+,9;<=AEqrhjtjbbbh~8OJQJjvh~8EHU%jdA h~8CJOJQJUVaJjh~8EHU%jKdA h~8CJOJQJUVaJj¨h~8EHU%joyA h~8CJOJQJUVaJ!jh~8>*CJUmHnHu h~8>*jh~8Uj~h~8EHU%j \A h~8CJOJQJUVaJh~8&MNOPQRS8h$<nC[ 4 !4$ 4a$ $ 4`a$ 4``8:`bdf$%89:;noxe[jh~8EHU%jlA h~8CJOJQJUVaJjh~8EHU%j۹A h~8CJOJQJUVaJjh~8EHU%jThA h~8CJOJQJUVaJjӱh~8EHU%jfA h~8CJOJQJUVaJjɯh~8EHU%jfA h~8CJOJQJUVaJjh~8Uh~8OJQJh~8# !"#$CDWXYZpqùlbjh~8EHU%jjuA h~8CJOJQJUVaJjDh~8EHU%jrA h~8CJOJQJUVaJjh~8EHU%juA h~8CJOJQJUVaJjih~8EHU%jpA h~8CJOJQJUVaJjȻh~8EHU%jtA h~8CJOJQJUVaJh~8jh~8U [p?$ a$  V^ 4$ 4a$ !4>?D8Coqsuvz{|};<O־־֫ξ־־־־־־־־־־־֎j#h~8EHU%j5A h~8CJOJQJUVaJj h~8EHU%jA h~8CJOJQJUVaJjh~8CJUmHnHuh~86>*]h~8jh~8Ujh~8EHU%juA h~8CJOJQJUVaJ38lmnorsv{};S$a$ OPQRjk~abuvwx*+,-GH[ù֦֑wjh֑w[jqh~86EHU]Ujh~86EHU]%jAA h~8CJOJQJUVaJ h~86]jh~86U]j6h~8EHU%jɨA h~8CJOJQJUVaJjh~8EHU%jIA h~8CJOJQJUVaJh~8jh~8Ujh~8EHU%jA h~8CJOJQJUVaJ ay.G_mnɄׄ؄IJbc$a$`ents. Starting with,  EMBED Equation.3  First we integrate on x,  EMBED Equation.3  and evaluate.  EMBED Equation.3  Next, we integrate on z (a is constant),  EMBED Equation.3  and evaluate.  EMBED Equation.3  Finally, we replace the density with  EMBED Equation.3  , and arrive at our result:  EMBED Equation.3  Note that the dimension, c, does not appear in the final result. This is as expected since that dimension is parallel the axis of the calculation. Solid Sphere, cm axis. Rather than using the definition,  EMBED Equation.3 , to compute the moment of inertia, our tactic will be to select a differential element in the shape of a thin circular disk; a slice of the sphere. We then sum (through integration) the contribution of each slice to the moment of inertia of the entire sphere.        We start with our summation equation,  EMBED Equation.3  Borrowing the result for the moment of inertia of a cylinder, our disk is infinitely thin and of infinitely small mass, and as such has an infinitely small moment of inertia,  EMBED Equation.3 . We will require a replacement for both dm and x2. For the dm substitution, we replace as dm = r dV= r px2 dy. For the x substitution, we rely on the Pythagorean theorem, R2 = x2 + y2 , thus x2 = R2 y2. We now replace.  EMBED Equation.3   EMBED Equation.3  We expand,  EMBED Equation.3  and change the limits of integration,  EMBED Equation.3  and perform the integration and evaluation,  EMBED Equation.3   EMBED Equation.3  and finally, we replace the density with  EMBED Equation.3 , to arrive at our result,  EMBED Equation.3   Long circular rod, cm axis, normal to the length. In this final calculation, we will use yet another method. We will incorporate the result of the Flat Circular Plate, infinitely thin, axis within the plane of the area derivation outcome, and the parallel axis theorem. The tactic is to first select the flat circular plate as our differential element, to offset the axis of our flat circular plate using the parallel axis theorem, and to sum, using integration, the contribution of each plate in the circular rod.      We start with the summation,  EMBED Equation.3  . The moment of inertia of the infinitely thin disk about its own axis, as derived, is  EMBED Equation.3 . Using the parallel axis theorem, we adjust the moment of inertia to the Rods cm axis as,  EMBED Equation.3  and replace it in the integral.  EMBED Equation.3  We now seek a substitution for dm using the density.  EMBED Equation.3  and so  EMBED Equation.3 . We substitute to get,  EMBED Equation.3  Removal of the constants from the integral gives,  EMBED Equation.3  and a limit change for ease of evaluation gives,  EMBED Equation.3  We integrate to give,  EMBED Equation.3  Evaluation of limits gives,  EMBED Equation.3  We now replace the density as  EMBED Equation.3 , and arrive at our final form,  EMBED Equation.3  In comparing this result to the infinitely thin object, we see that the above formula carries an extra term for the finite round rod. It may also be interesting to note that for a rod with a length to diameter ratio of 10:1, the additional term affects the result at its third significant figure. 18. Appendix II. The Combination Method for determining Total Kinetic Energy. The total energy of motion of a collection of particles is simply the scalar sum of the individual kinetic energies of all of the particles in the collection.  EMBED Equation.3  Since this total energy is based on each particles velocity, which is in turn based of some given frame of reference, the total energy must also be based on the frame of reference from which the velocities are measured. Here, we wish to consider the kinetic energy of a group of particles as determined both from some fixed frame of reference (such as the earth frame of reference), and from the frame of reference of the collections center of mass, which is moving with respect to the fixed earth frame of reference. Let  EMBED Equation.3  be a particles velocity measured in the frame of reference of the collections center of mass, and let  EMBED Equation.3  be the particles velocity measured in the fixed earth frame of reference, and let  EMBED Equation.3  be the velocity of the center of mass as measured in the fixed earth frame of reference. The Galilean transformation relating these three velocities is:  EMBED Equation.3  The Total kinetic energy of the collection, as determined from the earth frame of reference, is thus  EMBED Equation.3  on application of the Galilean transformation,  EMBED Equation.3  next, we expand the squared quantity,  EMBED Equation.3  we distribute the summation,  EMBED Equation.3  we identify the first and third summation,  EMBED Equation.3  Since  EMBED Equation.3  from the definition of the center of mass velocity, the center summation becomes,  EMBED Equation.3  and the velocity of the center of mass of the collection, in its own frame of reference (the frame of reference of the center of mass) is equal to ZERO.  EMBED Equation.3  Thus, the total kinetic energy of the system of particles in the earth frame of reference is equal to the total kinetic energy of the system of particles as determined from the frame of reference of the center of mass, PLUS the kinetic energy of the center of mass particle in the earth frame of reference. For the case of a rigid solid which is rolling on a surface in the earth frame of reference, we may compute the total kinetic energy in the earth frame of reference by adding the rotational energy , computed from  EMBED Equation.3  (center of mass frame of reference), to the translation kinetic energy of the center of mass particle, computed from EMBED Equation.3 in the earth frame of reference. This is the Combination Method.     PAGE 1 PAGE 1 G. Kapp,  DATE \@ "M/d/yy" 4/2/08 front cm axis Y axis Y axis Y axis x back cm axis back y z Thumb fingers qx qy qz Dq Ds r F1 F1 F1 F  p axis d ^ d ^ F R f f f R F x y z vt R w w R vt x y z mi Ri Fi at az zcm Ri vi mi wcm Zaxis cm axis wz wcm d^ vcm C Ff mg +x +y q q acm R az acm vcm R R +z Dh F[\]^nońƄDŽȄ؄ل)*+,JK^ù֦։lbjh~8EHU%j5KA h~8CJOJQJUVaJjh~8EHU%jJA h~8CJOJQJUVaJj6h~8EHU%jHA h~8CJOJQJUVaJj\h~8EHU%jGA h~8CJOJQJUVaJh~8jh~8Uj^h~8EHU%jiEA h~8CJOJQJUVaJ^_`a "GH[\]^efhjklmnoprstuyzֈ؈º֧֍֍֍zpj$h~8EHU%jA h~8CJOJQJUVaJjh~8CJUmHnHujh~8EHU%jܴA h~8CJOJQJUVaJh~86>*]'jh~86>*CJU]mHnHuh~8jh~8Ujh~8EHU%jKA h~8CJOJQJUVaJ%deghknptvwxy{|}njk !$a$`hjȉʉԉډ܉މ2378<=LMQRVWklֶ֙|rjh~8EHU%jKA h~8CJOJQJUVaJjwh~8EHU%j~A h~8CJOJQJUVaJjh~8EHU%jA h~8CJOJQJUVaJh~8OJQJ h~8H*h~8jh~8Uj>h~8EHU%jA h~8CJOJQJUVaJ,()<=>?@ATUVWɋʋˋ̋Ћًڋù֦։lbRjh~8CJUmHnHujh~8EHU%jA h~8CJOJQJUVaJj`h~8EHU%j]A h~8CJOJQJUVaJjh~8EHU%jA h~8CJOJQJUVaJjzh~8EHU%jA h~8CJOJQJUVaJh~8jh~8Ujh~8EHU%jA h~8CJOJQJUVaJ(@X͋΋ϋЋ$a$ !ڋ q*+,-45HIxe%j7A h~8CJOJQJUVaJjF h~8EHU%jA h~8CJOJQJUVaJj h~8EHU%j A h~8CJOJQJUVaJjh~8EHU%jA h~8CJOJQJUVaJjh~8Ujh~8CJUmHnHu h~8>* h~86]h~8h~86>*]$4Lҏ4e}ǐߐ45MN{|Вђrsij$a$gd|gd|$gd| !`$a$IJKҏӏ0123efyz{|鹯霒ubXjh~8EHU%j9A h~8CJOJQJUVaJj1h~8EHU%jUA h~8CJOJQJUVaJjh~8EHU%j A h~8CJOJQJUVaJjhh~8EHU%jA h~8CJOJQJUVaJjh~8EHU%jgA h~8CJOJQJUVaJh~8jh~8Uj h~8EHU"ǐȐېܐݐސ56IJKL{|}˒Вstĺ|umu|e|VjʜK h|CJUVaJjh|Uh|5>*\ h|5\h|j)$h~8EHU%jWA h~8CJOJQJUVaJj!h~8EHU%jؤA h~8CJOJQJUVaJjUh~8EHU%j>A h~8CJOJQJUVaJjh~8Ujh~8EHU%jA h~8CJOJQJUVaJh~8#$789:BCVWXYŗƗٗڗۗܗ !"騞鏅vl]jٜK h|CJUVaJjN1h|EHUjڜK h|CJUVaJj/h|EHUjTK h|CJUVaJj-h|EHUjK h|CJUVaJj(+h|EHUjќK h|CJUVaJj8)h|EHUjԜK h|CJUVaJh|jh|Uj&h|EHU$jBZ[ŗݗ  %KLdƘޘNfST$a$gd|gd|"#$LM`abcƘǘژۘܘݘNObcde騞鏅vl]jK h|CJUVaJjDh|EHUj"K h|CJUVaJjyAh|EHUjK h|CJUVaJj>h|EHUj4K h|CJUVaJj:h|EHUjK h|CJUVaJjZ7h|EHUjAK h|CJUVaJh|jh|UjI4h|EHU$01DEFGҜӜԜ՜ !#$&'-./0134:;<=>G鯫wjh~86B*CJ]ph3fhNy0JmHnHuh~8h7k#0JmHnHujh7k#0JU h~80Jjh~80JUh6mjh6mUjLh|EHUj\K h|CJUVaJj#Jh|EHUjK h|CJUVaJh|jh|Uj3Gh|EHU%T "#%&123>cdekltu|}&`#$gd|GHZ[abcde   "028:@BL^bdfhlnprʵʵʵʞʖʨʨʨʵʵʖʵʵh~8OJQJh~8CJOJQJh~8OJQJh~8CJOJQJ h~8H*h~85H*\h~85OJQJ\ h~8CJh6mh~8$h4M6B*CJ]mHnHph3fuh~86B*CJ]ph3f"jh~86B*CJU]ph3f2 &(,.446<>DFJL^`hjrtxz~ !žĞȞʞΞО֞؞ܞޞ !ĞƞОҞ֞  "&*.246<@JR\^`dfhnrtzƟȟ̟П֟ڟ Uh~8B*ph3fh~8CJOJQJh~8OJQJ h~8H*h~85OJQJ\h~8h~85H*\ h~85\L "$*,24<>JL\^dfnpvxğƟ̟Ο֟؟ ! "(*028:BDJLRTZ\jlvx !f1 wz Ff2 C1 C2 mg cm acm +z +y +x 3.5 ft 2 ft 3 ft x=0 x dx x=L/2 x=-L/2 cm axis dm r dr dm r=R r=0 x=-R x=+R x=0 x dx q R dm Axis Axis R +x +y +z Axis x z a b c dx dz dm L y disk dm R x +x +y axis $(<BDJLRTZ\jlvx04bhjprxGJKNOPh|hEmzUh~8OJQJ h~8CJh~8B*ph3f h~8H*h~8&x  !  $&.046:<BDNPZ\`bhjprxz ! !   "#(),-12:; ! dy x dm disk dm x dx +L/2 -L/2 dm x=0 cm axis R cm axis +z C ;=>FGJKMNOPgd| 0&P1h/ =!"#$% SDd lw~b Q c $AP? ?3"`?2iʔ{eKyD`!qiʔ{eK68?xuQK@޻TY0TQء"vQh@ k'W;;\tvR%-ߏw"I>X9"8MSi1JP( Na !Õ>ԗJDsJjf39Be ~`9fO>ʘWvB6n#k)o=zII6kS_2Yߢ&ƿu<=svGr̺#^90F]~ k dݳ,Ӷw#J=~BC OjXDd 4lllb P c $AO? ?3"`?2peƽqIY~`!vpeƽqIY6 pDxuQJAf6Q h3"@@ %rp\g)"M k;Bl| "ٽK5 s||?sI>(ÜT2D$E4MyE)7ZCFI{A^R8Eyn=> TaD ;lд󓏾R湝0(  )w=z;ԓl֪>SeӾAOLǟ-7g?_9 T7r6 {r̺#E90vw6+uOJ]׾1Wjq+T/D+0nRV zh Dd 8J R C AQ? "2m]s%I`!A]s%v Hxcdd``bd``beV dX,XĐ 6! KA?H: l @P5< %! `fjvF ,L ! ~ AyP9WMV"8_ 7D&2AMdLۘP 27)?"~F>B XcڿO-o炆(8)c 'F&&\@@"|xcdd``gd``beV dX,XĐ ɁɁURcgb x@eOjx|K2B* R. XB2sSRsl.Jsi#h"8ߜ/🐉P_M*pAALbFpx +ssp)813`ՏiQmaGfR(#RpeqIj.C]TjLDd lJ T C AS? "2ګgN)YO4 `!ګgN)YO`hn PxڅRJA}3w^0hCv hL$ $p?`/X؈ {\ .vPR֠ǂ*kUD!βL2HfV/qӭ(o[:2 B]O{PV[GalM 䬓#U BE{*ys\i _|m79z<&Q4 [7ĝLXT剧k/l|$OL< W{б/ 0AlWzLWh1Pt+Poh9L&3x=Y>|ھb{6́GCl$1,sr^Dd HhJ V C AU? "2rT_>XsyN `!FT_>Xsy>@@"|xcdd``gd``beV dX,XĐ ɁɁURcgb x@eOjx|K2B* R. XB2sSRsr.Jsi#h"8ߜO🐉5PzJ0@mnbQhdBܤ.i`KǴ.' \а#P3E@cI)$5!M`uOT^Dd [$~pV W c $AV?"2,U„3  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~l&Root Entry FP^@Data OWordDocument2ObjectPoolp]P^_1268655662Fp]p]Ole CompObjfObjInfo  #&),/0369<=>?BEFGHIJKLORSV[^adgjmpqtwxy| FMicrosoft Equation 3.0 DS Equation Equation.39qC* tE a"ddt FMicrosoft Equation 3.0 DS EqEquation Native F_1268655637# Fp]p]Ole CompObj f      "!#$%'(O)*,+-/.0124356789:;=<>?@BACDEFGIHJLKMPNQRSUTVWXZY[\]_^`abcdefhgijmnopqrstuvwxyz{|}~uation Equation.39qC*A  a"ddt FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native  F_1090661358Fp]p]Ole  CompObj fObjInfoEquation Native D_1090662245Fp]p]h(`mI\yI S= r FMicrosoft Equation 3.0 DS Equation Equation.39qhD`mI\yI dSdt=ddtrOle CompObjfObjInfoEquation Native `_1090662925Fp]p]Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qh,`mI\yI v t = r FMicrosoft Equation 3.0 DS EqEquation Native H_1090663106'Fp]p]Ole CompObj fuation Equation.39qhX`mI\yI dv t dt=ddt r FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo!!Equation Native "t_1090663187$Fp]p]Ole $CompObj#%%fObjInfo&'Equation Native (H_1090663557",)Fp]p]h,XInI a t = r FMicrosoft Equation 3.0 DS Equation Equation.39qht`mI\yI S =  rv t = ra t =Ole *CompObj(*+fObjInfo+-Equation Native . r FMicrosoft Equation 3.0 DS Equation Equation.39q,`mI\yI " =2R 2F _1090740751.Fp]p]Ole 1CompObj-/2fObjInfo04Equation Native 5H_1090751590@E3Fp]p]Ole 7CompObj248f FMicrosoft Equation 3.0 DS Equation Equation.39qP`mI\yI 2A " 2B a"(A x B x )+(A y B y )+(A z B z )=2A 2B cos A!BObjInfo5:Equation Native ; _1268655951h8Fp]p]Ole @ FMicrosoft Equation 3.0 DS Equation Equation.39qC0=  2A 2B a"2i 2j 2k A x A y A z B x B y B z =(A yCompObj79AfObjInfo:CEquation Native D _1090750553J=Fp]p] B z "A z B y )2i "(A x B z "A z B x )2j +(A x B y "A y B x )2k  FMicrosoft Equation 3.0 DS Equation Equation.39qOle MCompObj<>NfObjInfo?PEquation Native QPل`mI\yI 2A 2B   =2A 2B sin A!B FMicrosoft Equation 3.0 DS Equation Equation.39q_1090750846BFp]p]Ole TCompObjACUfObjInfoDWEquation Native X<_1090752230GFp]p]Ole YCompObjFHZfP `mI\yI 2A 2B  FMicrosoft Equation 3.0 DS Equation Equation.39qP4`mI\yI 2A " 2A =A 2 ObjInfoI\Equation Native ]P_10907523671TLFp]p]Ole _CompObjKM`fObjInfoNbEquation Native cD_1090752418QFp]p] FMicrosoft Equation 3.0 DS Equation Equation.39qP(InI 2A 2A =0 FMicrosoft Equation 3.0 DS Equation Equation.39qOle eCompObjPRffObjInfoShEquation Native i`PDyItI 2A 2B ="(2B 2A ) FMicrosoft Equation 3.0 DS Equation Equation.39qPt`mI\yI 2A (2B +2_1090752602OYVFp]p]Ole kCompObjUWlfObjInfoXnEquation Native o_1090752839[Fp]p]Ole rCompObjZ\sfC )=(2A 2B )+(2A 2C ) FMicrosoft Equation 3.0 DS Equation Equation.39qP٨`mI\yI 2A " (2B 2C )=(2A 2B )" 2C =2BObjInfo]uEquation Native v_1090753044;`Fp]p]Ole z " (2C 2A )=scalar FMicrosoft Equation 3.0 DS Equation Equation.39qP٠InI 2A(2B 2C )=2B (2A " 2C )"2C CompObj_a{fObjInfob}Equation Native ~_1090822071eFp]p](2A " 2B )=vector FMicrosoft Equation 3.0 DS Equation Equation.39q٨IԂI "S   =  "  "R "v  t =" "R "a  t ="  "R Ole CompObjdffObjInfogEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qCRة  2F  i =m i 2a  i,t_1268656431jFp]p]Ole CompObjikfObjInfolEquation Native n_12686564526woFp]p]Ole CompObjnpf FMicrosoft Equation 3.0 DS Equation Equation.39qC†E 2R  i 2F  i =2R  i (m2a  i ) t FMicrosoft Equation 3.0 DS EqObjInfoqEquation Native _1268656572tFp]p]Ole CompObjsufObjInfovEquation Native _1268656640ryFp]p]uation Equation.39qC‚B  ("  z ) i =2R  i (m2a  t ) i FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjxzfObjInfo{Equation Native C¡ع|F ("  z ) i =m i  2R  i ("  z 2R  i ) FMicrosoft Equation 3.0 DS Equation Equation.39q_1090831944c~Fp]p]Ole CompObj}fObjInfoEquation Native _1268656694Fp]p]Ole CompObjfڄIԂI 2A(2B 2C )=2B (2A " 2C )"2C (2A " 2B ) FMicrosoft Equation 3.0 DS Equation Equation.39qCl@ ("  z )ObjInfoEquation Native _1090832386Fp]p]Ole  i =m i "  z (2R  i " 2R  i )"2R  i (2R  i " "  z )[] FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoEquation Native _1090832565|Fp]p]ڐ`mI\yI 2R  i " 2R   =R i2  which is scalar FMicrosoft Equation 3.0 DS Equation Equation.39q`mI\yI 2R  i " "Ole CompObjfObjInfoEquation Native    z =0  since R and  are at 90 o  to each other. FMicrosoft Equation 3.0 DS Equation Equation.39q_1090832798Fp]p]Ole CompObjfObjInfoEquation Native $_1268656718mFp]p]Ole CompObjf`mI\yI  FMicrosoft Equation 3.0 DS Equation Equation.39qCw  ("  z ) i =m i R i2 "  zObjInfoEquation Native _1268656827Fp]p]Ole  FMicrosoft Equation 3.0 DS Equation Equation.39qCع  i=1n " ("  z ) i = i=1n " m i R i2 ("  z ) iCompObjfObjInfoEquation Native _1090835510Fp]p]Ole CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q٬IԂI   " "  z =( i=1n " m i R i2 ) z "  z_1090835625Fp]p]Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q,`mI\yI "  z " FMicrosoft Equation 3.0 DS EqEquation Native H_1090835763Fp]p]Ole CompObjfuation Equation.39q `mI\yI "  z FMicrosoft Equation 3.0 DS Equation Equation.39q@`mI\yI m i ObjInfoEquation Native <_1090835884Fp]p]Ole CompObjfObjInfoEquation Native \_1090836333Fp]p]" R i2 FMicrosoft Equation 3.0 DS Equation Equation.39q\`mI\yI "  z =I z "  "  zOle CompObjfObjInfoEquation Native x_1090837036Fp]p]Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qPpmIlyI I point =m R "2 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native l_1090837353Fp]p]Ole CompObjfObjInfoEquation Native _1090837551F`^`^Ole    #&),/014789<?@ABEHIJKNQRUXYZ[^adgjmpsvy|ـpmIlyI I collection =m i " R "i2 FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoEquation Native l_1268657061Fp]p]P`mI\yI I z =R "z2 +" dm FMicrosoft Equation 3.0 DS Equation Equation.39qCZ(  2F  i =d(m2v ) i dtOle  CompObj fObjInfo Equation Native v_1268657078Fp]p]Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qCŽ DLF 2R  i 2F  i =2R  i d(m2v ) i dtEquation Native _1268657129Fp]p]Ole CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39qC™S\ ("  z ) i =d2R  i (m2v ) i {}dt FMicrosoft Equation 3.0 DS EqObjInfoEquation Native _1090843971Fp]p]Ole CompObjfObjInfo!Equation Native "l_1090844194Fp]p]uation Equation.39q_P`mI\yI 2l a"2R   (m2v ){} FMicrosoft Equation 3.0 DS Equation Equation.39qOle $CompObj%fObjInfo'Equation Native (P_4`mI\yI " a"d2l dt FMicrosoft Equation 3.0 DS Equation Equation.39qC³TF ("  z ) i =d2R  i m_1268657223Fp]p]Ole *CompObj+fObjInfo-Equation Native ._1268657249Fp]p]Ole 2CompObj3f["  z 2R  i ]{}dt FMicrosoft Equation 3.0 DS Equation Equation.39qC»Sd ("  z ) i =dm(2R  i ObjInfo5Equation Native 6_1268657322Fp]p]Ole :["  z 2R  i ]){}dt FMicrosoft Equation 3.0 DS Equation Equation.39qCTtN ("  z ) i =dm(" [2R CompObj;fObjInfo=Equation Native >"_1090845920^]Fp]p] i " 2R  i ]"2R   i  [2R  i " "  z ]){}dt FMicrosoft Equation 3.0 DS Equation Equation.39q_IԂI 2R  i " "Ole CCompObjDfObjInfoFEquation Native G   z =0  since R and  are at 90 o  to each other. FMicrosoft Equation 3.0 DS Equation Equation.39q_1268657698Fp]p]Ole LCompObjMfObjInfoOEquation Native P_1268657990Fp]p]Ole SCompObjTfCŽ?  ("  z ) i =dmR i2 "  z {}dt FMicrosoft Equation 3.0 DS Equation Equation.39qC A   " (" ObjInfoVEquation Native W&_1268658100Fp]p]Ole \ z ) i =d" z [  " mR i2 ]{}dt=dI "z "  z {}dt FMicrosoft Equation 3.0 DS Equation Equation.39q#y3 `!,U„3#y3 `SHD VxڅR1KP^m-XEAtQhW';B*TLX88uqpw8ֱkhww#uCcJl)`ٷE[tD1&u}Lw Dd J Y C AX? " 2mHAP4p%I`!AHAP4p%``/xڍKA߼56٤ $KAK( ]SLZJK/ċXD𠇜3+yWg^f~:2b鸩L}:j=OL dԧ@νώIߛugDԟ_ۏjo&~y5bE|yI٣/g4dၓ}]SIDaqk6_kEm/4޻h2}ԏ¦nku4[..fa~ДE받M $ꦰm":!'B1>Y"iDd 4b b c $Aa? ?3"`? 2X1&⵼R`!X1&⵼R -E xڥAhANv7Mlc5mMAP`EFIikcKLSxЛŃEH"sSj!x$AfW*n}ofD댋 RhfD$Ɏ$h4xvW|6R%V@OF%}7ͱ=>c9s(i2%Ӗ$ۊX߲ٷcTY7Sba8USe{YEq[%Ƭy$`놼d&e|LG_ُH+q{5_ۍ! /ũiSQ5 ?h*NK^g=+YtGA|_oKEsU~{0k6y)x"y ̏:_|_hg؟݉}_S>^SU<WeMٜq|\ʃ5Ȫ3$%{Od8Epa8y&N~Cm!sW}{/X_ cP[,}zӟ,^_oC>͆Hy;'ߎb.;I%.}km~psZ!CLP[Ќ8fX =p.Dd J Z C AY? " 2F#S|B3熗`!F#S|B3熗xڍSO@~(D'5Ȃ08`R#$BY\K''Gb#t`sAɯD{߽^{OB_l"4:}#d#i$2 /ffZ,,ϒ A<:{q[2&RؼpL |`#O\'=GzRXNa9e,Y &,mELXI<Wb*33X6aŲ> Ȋt4wq-bxcZ2"_sEd_́~~b'Qh'sPP)XS>9\oWc5?ɳO: Sw2*> MqUVj:ʿFMP섆@)&M\HGSfP#~" Dd l@J [ C AZ? " 2lgh"+h Hj`!@gh"+h  xcdd`` @bD"L1JE `x0 Yjl R A@ ^7$# AL aA $37X/\!(?71]46:8'8?wdD3%! %E &~Xplb_ `1+ss.wQ97[>#"=ddbR ,.IeCw#3XQA$Dd L@J \ C A[? " 2L\tTsqˌbt`!ZL\tTsqˌ& (xuQJA}3gs xjFLoaa s  ^"IN2&E/ٹ͂!173:m؂95:lEB\Vh[+Ly_dvP&g%}`, iqq:T#x!u,F(gIoPWn- m BbH/49uKT8\|aȮp+}r8F0I{fR[8߂"F.7i8_KA&(V%2t`wk)MEp~3 S.p"X=bdbR ,.IeCw53XȏN.CDd |J ^ C A]? "2[fF4]LC{#`!y[fF4]LC{ `(0Gxcdd``vfd``beV dX,XĐ ) A?d.znĒʂT ~35;a#L ! ~ Ay  9Ư Śp>'a"# &2H& MvMvg3# 17D(PJl6AU OP P 27)?on XA|]8 3o 3BsA ȂA`8e021)W2g[wDd  |J _ C A^? "2@pӌas%`!@pӌasF``` 0{xu=KAg撘\:DB2"&eD%h#XRhk6V kф1gsʧg3A,e! %E ö6W*yATj>T ܨeݘQ |.h>pBӋ(gpbdbR ,.IeCef:Q~Dd |J a C A`? "2eyG -ji|V+`!eyG -ji|V`)0xڅKAߌ?u4“Dҥt,%.ldnxtt:w=h7cB >}o͛Al>փ-/>C8|fLEJ\ũ6QfÐ@cak >*Yvdxߴ[RrEC/'jr4%XᙤqIsϬ^M+cx!.<)8ʜUdǀީOt㵁]aY٧ekƉy^ڋ_(rJE~\%]w3%b ѐg7Fe>(-iW f%m+ᵫ?{+8Us>g_/{bX{ᯇP:m݀`=}hDd L~pV c c $Ab?"2-v_.̏*,_-`!-v_.̏*,_ xڅK@߻Oq "]. 6\#Lk`[ts@7.N"ƻKj ^{w xO21yg,ZYZYc(i˰n٘cf! nyqGj2PJ衈u)BAr:MsikWM,s{T(DŽ.،8m z'}W>>L$m!ʋςaFf_Fk2#-l>@ɓӗ$OK6\5zD,vA5$?,Ma-B`~S;W[ڎ?kE[P :^HXx@وβ bqWx#xF7^zro+ψ됒w>Ex=KBNs\@#|sDd b d c $Ac? ?3"`?2&7&l_(Z0`!&7&l_(@8 _xڝK@ǿwT T .:!҂ '-N:u/tпC)N% XԄ'w<;g"UPO#X11El.fdyeEKE+Y, 0G*TvUQu#["7nnH5a: {^h7}Qx{5Gü[Sj6(@yp?}>g}Dt\9fg\抜?NtYN~WѿR[u읳K?"?Sw:6k?lI2d{ߒ"ADǒΈ)|&;cX5`daS?ADd b e c $Ad? ?3"`?2s /Mi/12`!s /Mi/1z hxڝ1K@wim+:IR;ڭm?b[1Vh t,N t  )Jqw p˽w hx|ТKkLKE:íId gܺπgOE˭U=ؐ?K 8YcF-snc{5he.)۪Ƹ_;'tHrG\W|4s-ʳ$@Ozt]J`yCN v_?P磺>,䤮ܸ%\N 8`!؀*,O%b>%\N PhxڕK@]'"tvuh`  Ɩ6# Iw7P?U\tTJQwhj6侗}/< ] G!/Bq-q,t n`2tt&abP)htEs`eO@N3vU乿p[SbdMHbhp|ݶ"K+jn,P)Ɏ> ~0&?QPC|( eUXp0r{ϼG&0c0˛+Y@^/?Rǂ yD7[pGxZReI>˭&lq_y~^Eys3mɛO oVTq30*uS7=1 'j-Dd |J f C Ae? "2 i@|tE:`! i@|tE`+ 0xuOKAyWs]W]@:Iуџ[P`QPDJt`R!R>DK[cmup wf~<3<,(@0z M#φʟ llokEIQ"Wn{fEP{ P|| nTd6diG >6w~R«Hdvx]8'+3o[-%8;C"v0ocdbR ,.IerP"CDHg!0e< k^Dd x J h C Ag? "2 89 2Ͼm@`! 89 2ϾXbxڕRMK@~(ݣ"m@,4[ԣ?G "x޼ 6`[)&,;o͛ 2@ *Q2HqK$sHhetG^~!E085fpPuƨܠ p~Z7p;p.NbEX\s6ݶ/ܣLòBA[~Lo/ 7ɼD?2K"QBY`gc9]SK'`g&_> .myYM=Jx|zy}&_~'`L%S&e\;P$sDQ8̺yBrDd J i C Ah? "2(Zʶ#6q8IB`!Zʶ#6q8I@-hxڕSNAY0 jHԍZ`|^p.*VVFc ck3w/ܾ"BQ  %9u]p#+}v\1׳jI@\ARݩUXRT# >D瓍9 rnu"vl@,z=FvV34yQ?v?EI%U8 vl͘r)E QMM0iiF11WY8Ќx \xEVnKJf"˔ƛ4l ԮهQ;M/﷟^39ǁJTi4ahjH͔P[Nӄpo/rך>>.xw6 QFO|CQRXuhvbpgpDd TJ j C Ai? "28P N/7E`!8P N/7  ȽXJtxcdd``a!0 ĜL  312Ec21BUs30)0)0Qcgb  P#7T obIFHeA*CT f0 PeDd |b m c $Al? ?3"`?2;SkdfEG`!;SkdfE`@ 0xڥS=KA} i8-DE VHrI^"A\+KR}@6)lb#ADswrp1 уw{o!1DJA"ǡhMy~^c<#d%э\80)^N 9.ž I)9Ŋe6mkQUy&O xց]73<%ZSx1} =_~EH}BЏqXBNF7{g>%\A4lBW WL!$Dd b o c $An? ?3"`?2I,.[(0-/ڑ%I`!,.[(0-/ڑ*`@xcdd`` @c112BYL%bpu2@0&dT20<I? 01d++&1t\> /fJ e*F\tfVyA*8Uއ σ1łΌl/ K !kG@?ցT ;s0{Snʯbȣ'E¸oFp}C8߀L {+ssn\Qho7?.p|` g=! %E l\A_ ,8Abf!l l #V >Bǜxc.Ǐ-<~3@ ?Ѥ>V\R \L``*F&&\P @ ]` bg!t?1BGΔDd J n C Am? "2}$FaUpǩ=L`!}$FaUpǩ= x(+xڕ1K@߽I[ "Zh\vhhi hEC ;xrmZ&w{yB@@hag˴S@&^xr$8ꞏA$eL J_\qPv@Btܮ]vJsZBJv&hi{)9MSpC19V nId ^}SJwt}:x;!أ.E*#j;i7AqX|J;O#Bif]g+cTCNWNTQ]`}{! xwIٛ=ٚV:^T_F?&=S.]^|S5Jִ`n#C\A+I*oDd DJ p C Ao? " 2sEQ`!8]xR I(п3 XJxcdd``> @bD"L1JE `x0 Yjl R A@2 ^t jx|K2B* R.ͤ XB2sSRsL.Jsi#`U`~35C2 +ssF1DA4 7D(f3F ] `>lU `px121)W2䡻=.Dd [J r C Aq? ""2s.F+yhM韵rO@S`!G.F+yhM韵r`ShxڅJ@4֦ P`zhm -7з/݃)vl(,P*3/*Lbkvtk&^W|Jvut Xw ^$uE_n eLl9j.`dyr-Ԥ%DzwffI%i뿓 MݛӿǴ\z/Ki{U4q[7qP ΰ%oځ)Qn1uaNm sY`>Dd DZV t c $As?"$2BQN>pW`!hBQN>e h6xڕR=K`~6-A8(8U`>֟Пpp6p\msf`RY`9]9p=qt]y-aGODd DV u c $At?"%2JP 8eY`!yJP 8e hGxڍR=KA}3wbuXX`+[B$ p}. 636V;Y/pa;y; XY2&qQ٥u1y-8wjbNTyw`_ix0OF>0"7$*n?;n bddG/c?Ώ B`*Oe%sn[y$ 8HYsͳd:t-%~kj+aItXxTfsfWvCSLK]Sdnmyr{yIļٛc?B\TXe;Dd (11V v c $AN?"&2P\(#39w m0\`!eP\(#39w @ @C3xڍRMKPݤM "=AAgb@B O ='iOh|}Ij’};;þjsMcTLX2i8sӦVlw ^3>6Mb LU`d붧>ŦxFJ2C«9lndq"OL%p* *2S@}/B~ğ@'gdYyvQ:#˿GZ'?s7я~n,g;8^ ~[__YwV!v: z\^Dd lb w c $Au? ?3"`?'2F؇[ +Sk^`!F؇[ +S6x xڝR=OP=P(4`twGp~+bhD#q⠳#+!.XGi D}my9{nCnaDxicA}./Ȱ8G4^p|; qt+L&PL 15?RN|ujS+nA^{w̽ɤֺt1p=DD_!Nk@D\ohZgMᰗͪ$9oo3-Ar{C'墕1!ctD_fDd b | c $Ay? ?3"`?)2Mx䘤7,/)\ )c`!!x䘤7,/)\  xdxڥTO@~Zv١ ! !pA= װ a'&z3 1{?O{b"ѳx& P6uݍDL}{_g*`M~uK6-X8f3zR?=;r\m]~֮O1>L)h}FEQũП{IyȅP>۳ݭ!!. ]o]ʭ|R `]A/>z$^@l͡T+5oxs~Eh89Or?U $e|bQ 째ZL-8>ƫ ^.β6UrY s%ժ>bt08c]p62gaE+%I|s\<:ͩ*>Q^7`ȡp'/YDd P y S Aw?"*2rbxf-f`!rbxf-@ PVWxcdd``fd``beV dX,XĐ i A?dmօsC0&dT20Uf* P 27)?"](6X vC5$8@Jk>?$37X/\!(?71&` g50Lfar$7`e)=@\8;8 |=?AOL }C\+J:C "( `p`R..hp$@``S#RpeqIj.ChiHDd TSV z c $Ax?"+2$puP_zi`!r$puP_:@(+@xmQ=KA;?r‘NAMXr\"ɁNMg%?!XYX+xn1{yo"M VA G$s9Q+HEQhe׵Khh+Ӿ{.q5P |9gMxwc{P0GNFBMz ɖ9;r7][?.%q?8/nubU}8o$[x #gk1P EpcI'|' `=_ (7ۆSGӹ'䴦S{6*FӇ9EOHTmd4N3Ѿ_)}PSDd J { C Aw? ",2rbxf-gk`!rbxf-@ PVWxcdd``fd``beV dX,XĐ i A?dmօsC0&dT20Uf* P 27)?"](6X vC5$8@Jk>?$37X/\!(?71&` g50Lfar$7`e)=@\8;8 |=?AOL }C\+J:C "( `p`R..hp$@``S#RpeqIj.ChiDd b  c $A{? ?3"`?-2e{R[aS3Am`!9{R[aS3` dxcdd``g 2 ĜL0##0KQ* Wä/d3H1)fYۀX3PT obIFHeA*P L0d) W&00cAc@`| WZ4 P.P56X Z]Iȼ\"cdQ[ b `@Ɂ8!nhuB2sSRsp FN0%db_YQ姱0Cܥ;4ŮGp`!?B[:>;4Ů d xcdd``.g 2 ĜL0##0KQ* Wä d3H1)fYȹ UXRY T7S`7LYI9 ot >1t\> J{T TV06VW2$Wn@$s󃌨1A|0Aa b Y74cd ! ~ Ay8u# g%_X 21_Ž\8 #X?] s,ldY/dyiLrFzb(V2W&/22.M8߄"]L w(V%2pqE,(Ddx3[Ԟ̏gxƎ_ 1\Dd 8b  c $A}? ?3"`?/2}|R#NAs`!z}|R#NA` D#PHxڭkA߼l$mE,!bx/؝zIz 蹙K}m WFWj`a[E"Z*;<&<Z>W칺S1*LyӺ9*U76R Ź_Y64oZ91?՘}jʷun ^n.;FOgc=>+Y|,h." RrtN|DquYR' 𽡖iOY6;?I^죩'e^4^2l>.Wsoxzu`M]VyDiMsHr>ICȃ|'ψGz$t̫<ŋS%ND.OMuy2C~í9`3=9n=k^Dd x J } C Ag? "02 89 2ϾRw`! 89 2ϾXbxڕRMK@~(ݣ"m@,4[ԣ?G "x޼ 6`[)&,;o͛ 2@ *Q2HqK$sHhetG^~!E085fpPuƨܠ p~Z7p;p.NbEX\s6ݶ/ܣLòBA[~Lo/ 7ɼD?2K"QBY`gc9]SK'`g&_> .myYM=Jx|zy}&_~'`L%S&e\;P$sDQ8̺yBrDd J ~ C Az? "12.X?տ#^d= y`!X?տ#^d= .hxڅS=KAċ^^%bFBX$)-x޻P$a!d1%C\3qsMԨJM;)w"#r n^K$']{`|@K5̎NtNm";* ~Vʨto| [ˉb%Agt4ujm(Ġ ;3ier5nJx@)S=ĶC@J1L[!ΗQE,/?_»7~1[oNxo:tzSfϔM{u =[U:\8W~FXIUοcNK9QtY\h.Os%: ؔt' w e=N (I{fDd `b  c $A? ?3"`?323 6cF[g`!CompObj]fObjInfo_Equation Native `K_1090847160Fp]p]C/  2l a"I " " FMicrosoft Equation 3.0 DS Equation Equation.39q_0`mI\yI "  dt=d2l Ole bCompObjcfObjInfoeEquation Native fL_1090847211  Fp]p]Ole hCompObj  ifObjInfo k FMicrosoft Equation 3.0 DS Equation Equation.39q_HInI " dt=d2l  +"  +" FMicrosoft Equation 3.0 DS EqEquation Native ld_1090847269Fp]p]Ole nCompObjofuation Equation.39q_XIxI "  dt=2l =(I" )  +" FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoqEquation Native rt_1090916967Fp]p]Ole tCompObjufObjInfowEquation Native xx_1090917208Fp]p]\`mI\yI KE i =)12m i v i2 FMicrosoft Equation 3.0 DS Equation Equation.39qڸIԂI KE total =  " KE i = Ole zCompObj{fObjInfo}Equation Native ~i=1n " )12m i v i2 FMicrosoft Equation 3.0 DS Equation Equation.39q`mI\yI v i2 =2v  i " 2v  i =(" _1090917965!Fp]p]Ole CompObjfObjInfo Equation Native ,_1090918139#Fp]p]Ole CompObj"$f i 2R  i )" ("  i 2R  i )= i  2 R i  2 FMicrosoft Equation 3.0 DS Equation Equation.39qڨIԂI KE totaObjInfo%Equation Native _1090918723I(Fp]p]Ole l = i=1n " )12m i  i2 R i2 FMicrosoft Equation 3.0 DS Equation Equation.39qڼ`mI\yI KE totaCompObj')fObjInfo*Equation Native _1090919013-Fp]p]l =)12m i 2R  i2i=1n " () cm2 FMicrosoft Equation 3.0 DS Equation Equation.39qh`mI\yI REa"KE tOle CompObj,.fObjInfo/Equation Native otal =)12I 2 FMicrosoft Equation 3.0 DS Equation Equation.39qxD`mI\yI Worka"2F " d2x  +"_1091187403g52Fp]p]Ole CompObj13fObjInfo4Equation Native `_10911879947Fp]p]Ole CompObj68f FMicrosoft Equation 3.0 DS Equation Equation.39qx``mI\yI Worka"d"d"  +" 2F " d2x  FMicrosoft Equation 3.0 DS EqObjInfo9Equation Native |_1091188317b<Fp]p]Ole CompObj;=fObjInfo>Equation Native _1090993429+DAFp]`d^uation Equation.39qxp`mI\yI Worka"2F " d2x a"" " d"  +"  +" FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj@BfObjInfoCEquation Native 0ڨ`mI\yI )12I z  z axis2 a"KE cm +RE cm axisT FMicrosoft Equation 3.0 DS Equation Equation.39q_1090993771FF`d^`d^Ole CompObjEGfObjInfoH0,ĴI܎I )12I z axis  z axis2 =)12m total v cm2 +)12I cm axis  cm axis2IEquation Native H_1090993883?SKF`d^`d^Ole CompObjJLf FMicrosoft Equation 3.0 DS Equation Equation.39q0\`mI\yI v cm2 = z2  d "2 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoMEquation Native x_1090993990PF`d^`d^Ole CompObjOQfObjInfoREquation Native $_1090994328NXUF`d^`d^0XInI  FMicrosoft Equation 3.0 DS Equation Equation.39q0TIȜI )12I z axis  z axisOle CompObjTVfObjInfoWEquation Native p2 =)12m total  z axis2 d "2 +)12I cm axis  cm axis2 FMicrosoft Equation 3.0 DS Equation Equation.39q_1090994510ZF`d^`d^Ole CompObjY[fObjInfo\Equation Native x_1090994714&:_F`d^`d^Ole CompObj^`f0\`mI\yI  z axis a" cm axis FMicrosoft Equation 3.0 DS Equation Equation.39q0ڜIԂI I z axis =m total d "ObjInfoaEquation Native _1091092533q0dF`d^`d^Ole 2 +I cm axis FMicrosoft Equation 3.0 DS Equation Equation.39qMT`mI\yI 2F  external " =0CompObjcefObjInfofEquation Native p_1091092586iF`d^`d^Ole CompObjhjfObjInfokEquation Native p FMicrosoft Equation 3.0 DS Equation Equation.39qMTXInI "  external " =0 FMicrosoft Equation 3.0 DS Equation Equation.39q_1091088941nF`d^`d^Ole CompObjmofObjInfopMp`mI\yI 2F  ex " =m total 2a  cm FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1091089112l{sF`d^`d^Ole CompObjrtfObjInfouEquation Native _1091089848xF`d^`d^Ole     #$%(-0369<=>ADGJKNQTWZ]`cfilorux{~MڬIԂI 2R  cm 2F  ex " =2R  cm m total 2a  cm FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjwyfObjInfozEquation Native _1091089954v}F`d^`d^MڐIԂI "  ex " =2R  cm m total 2a  cm FMicrosoft Equation 3.0 DS Equation Equation.39qMژ؏IlI "  exOle  CompObj|~ fObjInfoEquation Native  " =m total (2R  cm 2a  cm ) FMicrosoft Equation 3.0 DS Equation Equation.39qML`mI\yI 2R  cm 2a  cm =0_1091091002F`d^`d^Ole CompObjfObjInfoEquation Native h_1091091535F`d^`d^Ole CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39qML`mI\yI 2a  t ="  "R  cm FMicrosoft Equation 3.0 DS EqObjInfoEquation Native h_1091091823F`d^`d^Ole CompObjfObjInfo!Equation Native "_1091254064F`d^`d^uation Equation.39qMIԂI "  ex " =m total (2R  cm 2a  cm )=m total R cm2 "  FMicrosoft Equation 3.0 DS EqOle &CompObj'fObjInfo)Equation Native *8uation Equation.39q`mI\yI =I FMicrosoft Equation 3.0 DS Equation Equation.39q,`mI\yI d"=Rsin_1091254359F`d^`d^Ole +CompObj,fObjInfo.Equation Native /H_1091254600F`d^`d^Ole 1CompObj2f FMicrosoft Equation 3.0 DS Equation Equation.39qL`mI\yI =d"mg=Rsin  mgObjInfo4Equation Native 5h_1091254860F`d^`d^Ole 7CompObj8fObjInfo:Equation Native ;_1091259177F`d^`d^ FMicrosoft Equation 3.0 DS Equation Equation.39q`mI\yI I z =I cm +mR 2 =)25mR 2 +mR 2 =)75mR 2Ole ?CompObj@fObjInfoBEquation Native C` FMicrosoft Equation 3.0 DS Equation Equation.39qDIԂI  z =I z  z FMicrosoft Equation 3.0 DS Equation Equation.39q_1091259190F`d^`d^Ole ECompObjFfObjInfoHtII Rsin  mg=(75mR 2 ) z FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native I_1091259210F`d^`d^Ole LCompObjMfObjInfoOEquation Native Pd_1091257008F`d^`d^Ole RHtI@I 57gsin=R ze FMicrosoft Equation 3.0 DS Equation Equation.39qL`mI\yI a cm =57g sinCompObjSfObjInfoUEquation Native Vh_1091259305F`d^`d^Ole XCompObjYfObjInfo[Equation Native \l FMicrosoft Equation 3.0 DS Equation Equation.39qPIԂI  cm =I cm  cm FMicrosoft Equation 3.0 DS Eq_1091260484F`d^`d^Ole ^CompObj_fObjInfoauation Equation.39qd`mI\yI R F f =25mR 2  cm FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native b_1091260980F`d^`d^Ole dCompObjefObjInfogEquation Native hp_1091260791F`d^`d^Ole jT~IPI F f =25mR  cm FMicrosoft Equation 3.0 DS Equation Equation.39q@`mI\yI 2F =m2a  cm "CompObjkfObjInfomEquation Native n\_1091260862F`d^`d^Ole pCompObjqfObjInfosEquation Native tp FMicrosoft Equation 3.0 DS Equation Equation.39qTInI mgsin"F f =ma cm FMicrosoft Equation 3.0 DS Eq_1091261115F`d^`d^Ole vCompObjwfObjInfoyuation Equation.39q\`mI\yI F f =m(gsin"a cm ) FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native zx_1091261324F`d^`d^Ole |CompObj}fk     ST !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRUWVXZY[\]_^`abdcefghiljmonpqrstuvxwyz{}|~ObjInfoEquation Native _1091261450F`d^`d^Ole xIԌI m(gsin"a cm )=25mR cm FMicrosoft Equation 3.0 DS Equation Equation.39qh`mI\yI gsin=CompObjfObjInfoEquation Native _1091261684F`d^`d^25R cm +a cm FMicrosoft Equation 3.0 DS Equation Equation.39q4`mI\yI  cm a" zOle CompObjfObjInfoEquation Native P_1091261815F`d^`d^Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39qdIԂI gsin=25R z +a cmd FMicrosoft Equation 3.0 DS EqEquation Native _1091262043F`d^`d^Ole CompObjfuation Equation.39qڌIԂI gsin=25a cm +a cm =75a cm FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native _1091262313F`d^`d^Ole CompObjfObjInfoEquation Native P_1091262422F`d^`d^4`mI\yI =a cm R FMicrosoft Equation 3.0 DS Equation Equation.39qڼ`mI\yI a cm =Ole CompObjfObjInfoEquation Native 57 (9.8m/s 2 )sin10 o =1.21 m/s 2 FMicrosoft Equation 3.0 DS Equation Equation.39qڨInI  cm =_1091262681F`d^`d^Ole CompObjfObjInfoEquation Native _1091282802*F`d^`d^Ole CompObjf1.21 m/s 2 .4 m=3.04  rad/s 2 FMicrosoft Equation 3.0 DS Equation Equation.39q``mI\yI v f2 "3 6cF[@#oxڥUkA~f&hҤ) *%AC=9F,4VhI,ު ⡢)^,:ofvYפ v/7vA (_j^h|3 sjpvǯqK&Qẙ͉ҙv|߄jǀ$Q FOGPed5D6t} hj`ev1֞Wz=Yp"mѢLKj+iB&+zXj+tdz],0ߊ`M<|V]_=^b[ 1gbyVp_ohv+쾪h4PtswS%J{|Jw輂wJ'Qs9~BH~ؗ"3}hu[c(<]`˪?X󨏟##uȫ:raB}yVhI}Kt\N WNKki{j6&L7xDd |>Hn  0A??3"`?420R,Oc`!0R,Oc`80XxڍRJA}3IL  ⴰ3?1i#($s;J@0P๻لdavg7｛]B^ igLd3(LV಼H8*F#PeO*8(|b(U[GyF/jU3rek w3XN)8%R;rTq/Z*X RqMq.{ ֠qawOyoaW!V'҆g9fs^wWO'kMٯf.5)- ȨsHc1ϕe(hYܘuF)fxBDd P  S Ax? "52$puP_zb`!r$puP_:@(+@xmQ=KA;?r‘NAMXr\"ɁNMg%?!XYX+xn1{yo"M VA G$s9Q+HEQhe׵Khh+Ӿ{.q5P |9gMxwc{P0GNFBMz ɖ9;r7][?.%q?8/nubU}8o$[x #gk1P EpcI'|' `=_ (7ۆSGӹ'䴦S{6*FӇ9EOHTmd4N3Ѿ_)}P$Dd TJ  C A? "62S 5"1uO b`!ZS 5"1uO V L XJ(xڍ=K`wMM FA"N]`:DPiZil~G~D=!4﹗!t6U1&qQ٥u0U]qh^ [_t.ޣX.i/Rحrz"ʮ`_ >KA@v4'Mp:I1J#Nj=rzul{Brc]NDƸ/;Nx'\um8C_:Yz>&-QMhngs,% H$kZoUw)Hǂ'ib IRL-|WV+Dd J  C A? "72|d,޻y_ !iȉ`!a|d,޻y_ !\@0= @C/x}Q1KPM FAfA. u" 44__#.^|J w|B!ikD.?B RYfv+vTPVs6"ɃӅA]uyeSF^_'ӱq%wcP^-~p 6:uPW#}zɺ\^&q74&(UkJ!xd#FJS4l]+8 ?^;Eo: j3nUh<>Oo$i'0Y( J2VwhDd V  c $A?"82t`sY?X[re`!t`sY?X[re@ h@C`xcdd``nfd``beV dX,XĐ  A?d0CP=7T obIFHeA*P7Ӂ`0LYI9E9H'1p8@J]|9v`E Ŵh_Hfnj_jBP~nb&p4i> W#>K/CAKH\ZQRЁŧNM,ؐ!|QL L?'  9p (pA8#^ `pddbR ,.IeC3XmDDd J  C A? "929""Y [`!z9""Y @ PVHxڕRMK@}3I ?@z =xRkE{h{ҋ"Vh<'ߐ_?qg6Hq&&o޾]Bpɼ3 ,k&i8cӾfjL ue,l b)g"AfK ;{; U]._J߸r^cL( 1HTyOhrVaI**J3V9=kXFYeN %ݤĹ?wwd)^"P ;wG喷ۨq+tϣɚFoOzE?RW]8"Xu Ja4KzbjqDd J  C A? ";2$6Mv9D4`!6Mv9D۪8'hxڕ;KAg1j >0ZZX( P0Qp1]JKJ`J'ZX"x\$"ޱ~;3;}U @! S>ja"fv4֤*zT]$C{#=4~_ifx|] 5i'] 1Ul5<DžxN}`^FDd P J  C A? "=266#L:Z25=]`! 6#L:Z25=]lxڕ/A{~W"W]:WF \t";ar QFA'~Wa͛.~fv3of! ` x@'-JܳE#D9.9Oq]vDɁ$< \w,ڰךLdLi5Pe/Bj_2c/|{d]+Azj2*zΪ(NqaqZcE˄20*IUi:iiB*SwջL:X/8OQ05U$~/cW/ٕuWTF21{GݏHtɛ/4BN ~̯O,}a+~| |k^e3io?]?Ո`}x''2M- ۍ5I,vBXҙ2Pq1 -[J``!Pq1 -[J exڅRJA3 H*$ 6'h2B.r,6+ba-ZsgrE{;07;`7~ j֜ڄ($Iijk6M㮴E(CI*X@4ƁVt\#3xU+t[ӖzKtE3ۼ wz7lu`;eAXPg7k߳)1Ŵni&蛌R_f%AxmCHQ?{SpFmyec ܦcqϿ"x_͖0C{uV{niELףsb,I;\( zCJcЋ:y߹l(Dd @J  C A? "?2 $3c^,b>8pf͝`!^ $3c^,b>8p. p@C,x}QJP`* P\G (FҠͮKA\dhb996k _M2HEQ(uy Uv+$8Hm4鵟̫݇rN63)@,ʎvm}S.O7?s% Un/==%ypYHUDd < J  C A? "A2>t2CBfkS`!>t2CBfk`Hb@CYx}N@gϖ*ՀO oxěc"47}g0^ٺlvjH@Pd2Hb"[8f6ne0QϢ8me t[p$Nr9FB_H4Ԛavc>d<5C}buA}UnM{iS`(2mp5 ~;i 9ӓܩZ񾏷Ӟtzr:81 1ɥk>^2_FVC@½ rQArۑ(Wڦ͠ǐ} x=Dd J  C A? "C2:G!d5{n7qy )lv _L,xNR~|1lHIt!_?MuZ^ÅdΆԖHADd |J  C A? "D2%|.|O9 `!w%|.|O9 ` 0ExڕRJ@f66-)x`#ԃYR5`ZQGO!|y&w'k PMv|ofPK0OU[WD!NT",-&^N汊Ԑ4~޽t~4YTDCuOpcoh }N-/5kpVkV[|+%tU5$_U8:"WQo%od;Nh權JzNr K@Y79TYfНc7gƓ͒˳sTp=.Z'i}BnmpDd TJ  C Ai? "E28P N/7B`!8P N/7  ȽXJtxcdd``a!0 ĜL  312Ec21BUs30)0)0Qcgb  P#7T obIFHeA*CT f0 PeDd J  C A? "F2d\ӵM"^@`!8\ӵM"^@(.xcdd``.c ĜL  312Ec21BUs3010; Yjl R A@2OaP=7T obIFHeA*[~1AtF]F"LQ oA,aeLpI0w0b01VQ~3)97&br8: .N.N .9w9X|+,UC\"u1ӆ:L#Ff*6aBdVOZۮd|/RR,86)Qy|NQ 1$]{o0s_~ MyUՉgK~ THeSjt[k]˒wCRoywN3 =LcDd |V  c $A?"H2)~0b)PuԲ`!)~0b)Pu@`0[xڍJPs6 5 8Q]pu7iԂM+4 .wp!M0sz4*rs'@Ic rSdYeDZ!X+K4e&.'j~l G*F_[AtTׁ&xc? !M𪍰'N' ژ$fߤNJt6[cȯ.O9IuSe>!|-)bd#]mЩ_'X'Nug#᫼aY0u ^2ns{^i(t!~Fx}~NfO'#gs÷Jz҈yFtƌ>7]c3Dd dJ I C AE? "H2]|妇.y)\qج`!i]|妇.y)\v^ 7xڅRMK@}3~$SAPO7EE -^‚b4hs? ?ɋqwآV L rBc Y`-`"E!]Z>E<5y“ζ+DaH>ً:p4 *`(%FD7޿DaCqk989= xO*)^^7z>,[5S?7fo3>s'(FSSaܑEzG Ԓ9њ qxߛ;D!cg\e8_j{?(Uⲫx>_;.Dd <J J C AF? "I2؟ps~/nT l `!d؟ps~/nT l` h2xڍQ=KP=V NA\EG:#-ْi,KцV9cjI]_&JKB dOaS:w6:q~ T;?UHU061܅$NqJv忴á7azWt߃~_Ozp|r7: rΧ?&Vk'mO3Mƽ5/Yf6ӂ_O,>~*0Ͽϟg3,};7[.=ۈD덨3J :kbDd J M C AI? "L26z1zI_T`!6z1zI_T @PfxڍK@Ɵwb *"A\tqr\ڡ?J@iK?Y;dApt^ 3]{rG3%ȓR-2JDz8=Czf)Ryh3Ub_nU#GʶwRt 8$fuzVŹmnam*I / 5"~IXSIY釮\K]I/OwyYcvcnXs,Nu6f (0T;t#>M3O!8׬ Fw0Έ>AT|އHyԧ*aMMc}y6y|}TmDd t #V N c $AJ?"M2l%wa4] L`!l%wa4] D qwxڍR=KP&6 :Ht:($.iD (4miO'c:.߽y 5w^="; RjĻZ('Ha:fg_[3j\]Wy `O񳚓:Pt%F^.RwP'k}F=7.M1y5h}E"{l$;Ɉq5ɃZV1k ?D1ƍ*0؇/Gܩػ|= ϡ`=(Pmqcd$m]ohMR੉U]ï[N &!pyFBuop蟇~='{滑CӝEEzVA:Itտ#}* x_vOs=QLV璴:q2\"卅'3|m(S<08cdصru:iEv'a^ѵ_&ZQJDd J P C AL? "O2ɷ)F[F{`!ɷ)F[F{4 hNxڅ1KPwIBA⢠[LNP0A t\ѡ]1wy-Tp_#TF)騰ٹ:HN,-Zm&]q]-zyLGz6t~ЈppB*ܓP0,0Rnu! zߕ I; 9ԭ1qQ5'uVX*:qnj:E3d. =_9uf|AAk@'+Li}IgB7hN7R^$+E]L_fKk3'EZ~'L9o-Dd #V Q c $AM?"P20fSXsߩ=H`!0fSXsߩ=!xڕS=KA$&DsTDB XY 6HN=Q%!94R 6b+,l$;{%!}~0 dQYdLD:#"lJD֐yvuqB;ű̀C"P9솏O1nU*?9BP MsRr+Zg]wR}>HVêp:1%Q8$z1DsȨ9=2L'ZbeX=fNf$Z;LxiF6bSt{WNݪXzIr..~ؠc42?b$i?'7,φ\: Q7%~PJp]ӸD{?xcx*!u0_s@ BDd J R C AN? "Q2Sͱ9`]Q"d/`!'ͱ9`]Q"dê`PHxcdd`` @bD"L1JE `x0&Yjl R A@1 V UXRY`V! v \q.PHq%01ܐ3%`4'$37X/\!(?71j"#d <1\ĀEp!`;m# `p121)W2䡻v UXRY`V! v \q.PHq%0`rd}`d++&1HBM`8&ca30b$`,F0V ~ 6FPrAÌ `OB``cI)$5!ݍ`uJ+Dd @J T C AP? "S2/$.qےei`!a/$.qےeV(8 /xڅQJPIkI((\N.>"mfiŠ&Ul# gpt=M/^8Ckk*Yk %L$,ŲKbc]^y ~;J^R'!PEiS%Dp4/*N΄Hwh0hYx߄I2L&]nxUH.s{絎#+]RZ6Jei~l2a^LWWӸ6xz^]Kni|jpDsSđ*Zļ/z,0&L7A6%?&WSZh3@penR~N oDgE炦[.pG6M `pn`dbR ,.IeC#3X!|UDd J V C AN? "U2Sͱ9`]Q"d/`!'ͱ9`]Q"dê`PHxcdd`` @bD"L1JE `x0&Yjl R A@1 V UXRY`V! v \q.PHq%01ܐ3%`4'$37X/\!(?71j"#d <1\ĀEp!`;m# `p121)W2䡻v_M8ſT1*ٌw +x? 72JGxS7+zsԹ“:Չ..Nֺg!ybhz zBD,Av/φqt~Kج]KDd lJ Z C AU? "Y2&VK+9uޔ ue`!&VK+9uޔ u` OxڅRJ@_b`-CE|lQL <}z%^ 35H||79BArH /B ReY%l95Vk c<聗v9 輗^\#r|u\\}^D-*RA7yw F' \nZ JYDa0 bز +9Vi#@2 תߚw#xKw>iE g)ʕ&TG+ϯ:X_>n]=^5=+zDnٞ}]JDd hJ [ C AV? "Z2vK,VLPu}R`!JK,VLPu}@4 |xcdd``bd``beV dX,XĐ Ɂ A?d- πjx|K2B* R.ͤ XB2sSRs.Jsi#l G$e, >Ćl``ܦK =`_I9 pB3W'b ~ y XtO?8l!uF)3=xbdbR ,.IeCw#3Xs7O?WDd lJ \ C AW? "[2`Jl/d5/`!`Jl/d5/  [xڕRMK@}3Ik ~BAPPkozHQ0Ђ/=3 P\d%TЫvuDR!βL*۴"]&]W瑃ڂXEAU>TѫCxpq:Xh-AՐȰ/1xhQE%ޑ2y*IbͲ*lzC~'IߋSƛY(+k$s*PNܤ] ڳyJc07egyzOn݇gߌ'JZNsK$F{ I͎gjBDd lJ ] C AX? "\2 ؎W[tQ;m-`!x ؎W[tQ;m-` FxڍR=KAD,v XڦK$eN9Qp/Br,?!Xa/kϙٽ".a7*t_;F#g*%EQHHJuck7` &AH8lNuϚ8 xv9na/%7*O-̏+`ŬUgsQ~/ézܩ8:=p{@^_:Qa6%eZ^w1Dd J ^ C AY? "]2wI D r1o]`!gwI D r1V  5xڅQ=KP=&J BA R?J@ז6\G8#2\]h|5C-py\ιlh|%m56$BeDhI"L]穮WV^ 2#BST{*-N0œp%`7 6uowa(Vw()8'OϦzu>8S?. X/:3},Dg:/^KF5X%{?@k̦R'QS*,@sٛxr7fzKj7ǑBv Gx?&W_0Dd 4|J _ C AZ? "^2>{g~n`!f>{g~p `X04xڅRMK@}3I&A< ^)bmM+)Hn OzgOɡayl>XY-m>D!.R"!=&[/KWִbiBC][4$&R .#TgC|1VIQ9}kUᗶ齱eCH3aO8Gmp~wS15 !j*w5c,+ee:gG{(C`_+*/B[Лļ/i-s\XBDd \|J ` C A[? "_2|ن ^(`!x|ن ^(撠` `00FxuRJ@IS B W/=Q7O ƝqBk7f"8lub!( x 5IeGC6kzQ+h؊zaM#([E?QGO$8%uP}Fm)+%Zcƞe k쟦^K÷q{nE}Uc73scBu/,90 /gd 8v1JfzͷGpt]y2FӇ5AofYCq]gDd lJ a C A\? "`2Ы/…bHY`!Ы/…bHY(kxڍJAgb\s AH#ւID %9}(>O ot0(ץf_wBYZ |S=c&(}X3(*c%ݜq#*gYV ~h꿥` ,pȉ\0tcȝ1#]+ zh_6]Nݷ\L3W5oǕ%L2B"b pޥ#y?g\]Dd lJ b C A]? "a2 P5} j#Yg`! P5} j#Y @PaxڅR=KADKLrĀwQtil"BV939Dco-޼BA dҴ7kҎ!*K*r@cz~R<.d=j!Q4 eDd hJ c C A^? "b2b^d-) 8@>`!6^d-) 8@@|xcdd`` @bD"L1JE `x Yjl R A@2 N*{ UXRYvo&`0L` ZZ`v.PHq=`dauAJD& Ma?k` 2/ccjbuA X4E`#XgXZX_Z!0F{;o 8V+ #!l;Hq..ЄW啖8.BG"SV]#^VDL@9zȊo$w&py5ʫwVFaj~Hj4#kHo7"*S VYඞw%jFt!uSP)z~s~,37svҾ!)Kr@BB^EKSDFcidsDd lJ e C A`? "d2>Kq 3-n|'`!>Kq 3-n|'"`wxڅRJAˏH!h0yD94]bai!'U|Tܙ;S7;|3+ `\ iE g)ʕ&TG+ϯ:X_>n]=^5=+zDnٞ}]J Dd [J g C Aa? "f2ֺ(|s^`!Vֺ(|s"`S@2$xڅQN@ } W l #(H\J-/э'A}GE}96"`lm/a#D y5[9 utݥU/$?7g67̉Qu]2Fz 9}A}rz!za*~R($vti~R䃉+Ѝ*|fv;|Y Oq;* W|4sg*ʝM`QSj=f6!Lf*ABK'}5$t0nd KқDd lJ h C Ab? "g2&)M @vl`!&)M @vlP`"xڅRK@~Z`5tmՀ )H7''ܜgqpBM`w%R \}w3"ЗCG| cbabĊLeMkfy8y( LwF=螀29*,Qǻ&xZiԝJ<#Gx&)'ռ CFCXv )gF9.xj5̘syDd | J i C Ac? "h2NF?PQBR"`!NF?PQBR"` h{d}xڍRJAKwzjIai~Z¤S xA &6Fݜη7- \"`ddy$Q'!$INKzLhyHeEVr3 $[?>;c9 ώݓ9d2dtU4InF9]E*1+Ԟ F3:j;:W;8lAͪ LZ>nV F{"YEzt3-NõU۔4H)ݼdUa:a{_gE!>>5شCƧ#5A=׻~ޘP u#/dhqL{d=xObjInfoEquation Native |_1091283101F`d^`d^Ole v i2 =2 2a " 2s  FMicrosoft Equation 3.0 DS Equation Equation.39q`mI\yI  FMicrosoft Equation 3.0 DS EqCompObjfObjInfoEquation Native $_1091283112  F`d^`d^Ole CompObj fObjInfo Equation Native tuation Equation.39qXII v cm,f2 =2 a cm s  FMicrosoft Equation 3.0 DS Equation Equation.39q_1091283287F`d^`d^Ole CompObj fObjInfoEquation Native x_1091283479F`d^`d^Ole CompObjf\yI܌I v cm,f = 2 a cm s  FMicrosoft Equation 3.0 DS Equation Equation.39qII v cm,f = 2(1.21 m/s 2 ObjInfoEquation Native _1091283434 F`d^`d^Ole )(2 msin10 o )=5.23 m/s FMicrosoft Equation 3.0 DS Equation Equation.39qII CompObjfObjInfoEquation Native $_1091365567F`d^`^Ole CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qِIԂI Work all  except mg =KE+GPE cm FMicrosoft Equation 3.0 DS Eq_1091284614%"F`^`^Ole CompObj!#fObjInfo$uation Equation.39qڄ`mI\yI 0=12I z  z2 ()"mgh FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1091284911'F`^`^Ole CompObj&(fObjInfo)Equation Native _1091285783H,F`^`^Ole ڜ`mI\yI 0=1275mR 2 () z,f2 "mgh FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj+-fObjInfo.Equation Native _109128596841F`^`^h`mI\yI gh=710R 2  z2 FMicrosoft Equation 3.0 DS Equation Equation.39q``mI\yI R 2  Ole CompObj02fObjInfo3Equation Native |z,f2 =v cm,f2 FMicrosoft Equation 3.0 DS Equation Equation.39qXXInI v cm,f = 107gh _10912860786F`^`^Ole CompObj57fObjInfo8Equation Native t_1091365661>;F`^`^Ole CompObj:<f    #$%(+.1478;>?BEFILORUX[^_befilmpstwz} FMicrosoft Equation 3.0 DS Equation Equation.39qِIԂI Work all  except mg =KE+GPE cm FMicrosoft Equation 3.0 DS EqObjInfo=Equation Native _1091365691@F`^`^Ole CompObj?AfObjInfoB Equation Native  _1091334294/EF`^`^uation Equation.39qlI8I Work all  except mg =KE cm +RE cm +GPE cm FMicrosoft Equation 3.0 DS EqOle CompObjDFfObjInfoGEquation Native uation Equation.39qٴInI 0=12mv cm,f2 +12I cm  cm2 "mgh FMicrosoft Equation 3.0 DS Eq_1091334452CRJF`^`^Ole CompObjIKfObjInfoLuation Equation.39qItI mgh=12mv cm,f2 +1225mR 2 () cm2 FMicrosoft Equation 3.0 DS EqEquation Native _1091334565OF`^`^Ole CompObjNPfuation Equation.39qٰII mgh=12mv cm,f2 +210mR 2  cm2 FMicrosoft Equation 3.0 DS EqObjInfoQ!Equation Native "_1091334665MWTF`^`^Ole &CompObjSU'fObjInfoV)Equation Native *P_1091334712YF`^`^uation Equation.39q4II  cm = z FMicrosoft Equation 3.0 DS Equation Equation.39qOle ,CompObjXZ-fObjInfo[/Equation Native 0lP$InI R 2  z2 =v cm2 FMicrosoft Equation 3.0 DS Equation Equation.39q٤ADGJMPSVY\_behilopqtwz}uation Equation.39q֐`mI\yI =M total Volume total =dmdV FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native  D(ЩInI dm= dV FMicrosoft Equation 3.0 DS Equation Equation.39qp`mI\yI I cm =_1092288134F`^`^Ole  CompObj fObjInfoEquation Native _1092288406\BF`^`^Ole CompObjfx 2"L/2+L/2 +" dm FMicrosoft Equation 3.0 DS Equation Equation.39qP`mI\yI  l =mL=dmdxr      !"#%$&)'*P+,.-/013245678:9;<=?>A@CBDFEGHIKJLMNQORSTUVWXZY[\]_^`abdcegfhijklnmopstuvwxyz{|}~"HP\Y`4}^L$*ջ(֨'G~`=keDd J j C Ad? "i2xqՃ<_`!xqՃ<_ 8>hixڅJ@ƿTۤ E/RE/!b-jo=z|}^<D`ܙ{hLXv!؀@"V衈dTEM˲SRX^A_Ы9 Kp5PwĦ߽ZВ~TW#[A; nf7p&qNeg@[v e}{<&IU놗Dcp1YRmbEJ/ ٘)wythw-/[x f3,_ UOSRr§+آ=N"+,JJU{n1%vDnpDd TJ k C A? "j28P N/7t`!8P N/7  ȽXJtxcdd``a!0 ĜL  312Ec21BUs30)0)0Qcgb  P#7T obIFHeA*CT f0 Pe<Dd J l C Ae? "k28DrDl"z`!r8DrDl"` h@xڕRKPJ Aं檃Q# ) ťwPL8}wGhNHcU5X<i8slӪFv&Ox'emT\67x/3n7˚e8@_|Gdë$΂68J22YޘRY\Y܁rB/pWN ]?$}U{Lpa(YDLar^ÞOSMJ-N{stO\9MP?PiMx-{1 ]|!0 z`rDd J m C Af? "l2oHT߯oژg- `!oHT߯oژg-$ FvxڕRK@~%i`P ttpBW"8V(ТtP?Qtr_#`w~}w.5Fˑ%l Qy8V-\Sm„4@z,:đ^yGO88h/n^tXF}&lBfl1Ƅ}'9]3J>ý}žj|G!Ok0F;aTFWG.s{?e נ+xQB+X҇$heO;97Yb7Io2>䘺opb|ݳO&x +T@CVWnAn-+~mDd J n C Ag? "m2VYNP $2`!*YNP $>``(V'PxڅSKQ7o׻ޏ6.3K i2 ݻIa&U@JmB@KBlE/ `yoަ];73{O !UCP Ð=3b=&:Q^N}ɁAHIW@Y?G&mb20vpJ5 z&bYhֱ%둲 w1ðbt^vϩh,Hx V !4,7f"OTMٰTV I2L!,%?cg,-6Rn(3x7KkMq"vrX<{>Hj+|T47/> wPD G+3xh[Fg f*Y|y%?ATo~6擶E }iqKjT9Nde;}g@'h ρrCܽO蒇|b^r=բlpDd TJ o C A? "n28P N/7`!8P N/7  ȽXJtxcdd``a!0 ĜL  312Ec21BUs30)0)0Qcgb  P#7T obIFHeA*CT f0 PeKDd D |J p C Ah? "o2f 䪁f`!f 䪁fz`k0OxڍRJ@ĶI -x1xDloV 赱&m-#xѧcx<+g&҄ag] h"EoRd Q"46.KdGf;*ϫzj[/`RNy/d@Tďo:Q0 Tvn`0!lv]KJw"ݮZ7cZhuu/=>uZ\w_ 6!g2@H>}@ۮY^hՈ0j"_le[gTy_=wzb5X>l660 aa5׭d5?[{͏6^Iz0 ?wyۙ ׬?; =t\v5|vgrö`13ufJՌRis_rsB7r Xݩ،ehDd @lJ s C Ak? "r2k;>F۷JT M `!k;>F۷JT M plxڕR=KAM.*R1A"IgDH J,޿ baaem3"1 ;oB@ݠ^` x$G = Q2HqKf%S"s4eMUo3 G:zs+ͱ5YoۍYk |a(Ju49cUEzyj3Q^&T\_T-I PFBZe3\\v!eQKdI||iM;dm3WLVyD{|#c<^o;| &:{>0iri+ߕwȏEb}t&=*kCDd J t C Al? "s2lʹy@?8tK`!ylʹy@?8tK %hGxcdd``gd``beV dX,XĐ ISRcgb sznĒʂT ~3);a#L ! ~ Ay +@.c0 g`zՙQ囘Q!.`H K8@zlI΅2ʋUPw͌P0jà wcl2(ZP 27)?O0`)ψU?{0Bk.pzG*Lv0o8121)W2䡻A`{Dd |J u C Am? "t2GJhj}QA`!GJhj}Q(` `PxڅR=KA};{KC ՑʨJ[-L~@F!AI'ZH~+Vϝ́']7y], /^:YQAQ̼ҙ2 Ѱ.O5捫h65D\_၊n < drf׬3{' ^Eݱ$$hBE i{{* anǰ)Yµm9h C𫭰 'v'1m2^T24/1L աe;H}ǝe?Ư w~\J?"vldl=ԍw$5;7駸h'qbr1@~?Q GZwQ!IR~$kKDd D |J v C An? "u2j起B'o"= `!j起B'o"= z`k0OxڍRJPMTZ(.| 趱&Mͮ'ѯpg\+g&n09s<Z`#3{:B4M%QmUTO=5mT)'C^B|;teo2<"w8{Ve-ڮҺk}qYϬPORiuóv^{_>j%ɹT7~WQY,+ L/<;*?}`:|n?x얬=Ykx&^ߛ%w+@Wy z'*h<\fDd h|J w C Ao? "v2iz, 0Kǻ`!iz, 0Kǻ@`(0jxڍJAg$w FB QZBIư. &Jm#SX>(x\,ewaK! 7#(""^Y 4Hıخu:D '=rZ_4Gg@ /E\"9dr ?ꝸEwag@)}SL+Ebw_Cѩ^όS=+U)?CT*o^۾ZW3hlXVnx8ɚe?~ؿc*s3װ7{d:;t~wjPN8 |Kq >|WӄF7|(k%*=k #Dd hJ { C As? "z2b3R%#NfċX9> `!63R%#NfċX9@ |xcdd`` @bD"L1JE `x Yjl R A@2 ^T 7$# !L aA $37X/\!(?71]46zh R[2+6L@(\5Ld^9/ᣛǀ<ype78l@``CI)$5!ݍ`uAx)Dd |J | C At? "{2Yf (@g"`!_Yf (@n@`0= 0-xcdd``6gd``beV dX,XĐ )Ɂ]Rcgb x @P5< %! `fRvF+B2sSRs.si#/ l Fp>?3*?G PM LYAM΅á jr?B&#1jFV28ߍɠhBܤ.](Rj~L W\G LPtp `dbR ,.IeCw3XX[Dd T lJ } C Au? "|2Ssjӥ"$`!Ssjӥ"ׂ xڕSKBQ><5}i $ATQCKAiCApkƆۂhƈ(uwQC} Ā?^6#($۶1 }uD$¢q_`&0n肍NDc=NKFЬY _ |OR(N=Jq̉h^(8d%[|.f0E [K`vr=\>HY*;+@oF?Ǐ/8M&S*d^*~x"I()営hNH\K[={qNCWKtUѴ :Ax9kdw {snOKгTj9&A),\-Dd @J  C Ax? "2Ɗ׿'6k.`!cƊ׿'6^ p@C1x}QKPiŊCppSI\:؀sĀijl[\\2!38}}Cm 肜:E8Q+Hyke WCX[kSF`w80`r8LǍ*9mɹ+/52:4< G\mkR](R7_U ?M>V7(ah//jܯIlCX:fA:za0r8 :iLxY'Q|iuqI7Dt! 7=-YDd J  C Ay? "2_{m׎K;0`!3{m׎K@2hxmQ=O@ }~)M+ءH01t@@i+n։? C vf/'/{w $ =iQK"˲tBNR.WDxG8CiM4iU%M;UMq/O+Kd?FsQ]IȰ`Ɗ:k}%+糇*]F^&y/W"VsH}V,3Ágo>]!%Lgx9>9+Dd J  C Az? "2O@g2-1pɥi2`!aO@g2-1pɥn h/xڕRJ@IA)B)œAX%`Rm?_ᡟO^plTELv̛xG@ 5g#D Yk56"2457aAxތp]NnX>F,)*(|xx2}Il3(["&[%Hc418Ff68m sb S^VUxPfUɬ!5}5R? F?xGgX@*DezİM_uH3i'gYpuDd \lJ  C A{? "2 M,1;ͅy4`! M,1;ͅyV` 0yxڕJAg/xhH!*MM'-E9Q%.X)E'+ ͢[f3,BEq ;IҊM!p_;Mø =Y^vJX 0 G@XbYkKh 4:Gީ/|d-F*BcZ7`!uX63>c@(xڕ?K@{F Nhт"8N:U* UZn͵tsqp88uRG]NJ{!1mb)"&!J RQ%}%h1v4ޛ1 B:)c mbmP-x%#-3N"RQ tXk?BSJUw*^ :2BL5^ qVs.Sf%UG :djYI4=sD!NHܠ5<ýcUedz8it?1?ϠA8GrfsOe"8^ف?:iA?E~ă_mRB{|k)h>A>c7;SMW/A\ߓKg=8t.ojA9|&jxDd 8 lJ  C A}? "2BH6t:%:`!BH6t:%V|xڕ1KP.icA+S( *mw7C**-lAtIpppg? w>$UｻBC'+VʈEr)IrLiҫ5愵25 KpzkX"F4Pp-A*T\G@@ya]O {fq`'Nj,98xsEngΕk²VPJoFs%mM*ÝUT 'yoJ_:OI9FrTykr)%1Gq9?:i4 2̧c|du8N:ak lIO#Qw{at͖xslDd 4lJ  C A~? "2Ls_Q^4{4<`!Ls_Q^4{46 XpxڕRJA%rՑB􂅤B*QOT@H| JUsgEQvo`!f '"BfdD}p `X04xڅR=K@}3.A*XXR;EFz`r' H +O\me/wMp7oΆuQleL$⪪$Kc2}|peMy*6Q& +EU75>NnXN`" 9DH`3(M,y /i!%JԾCD\5~8YxlIyJ249N!$,`PW{4tQzwlк9βVs2xK~{7\%ADd \|J  C A? "2X x-N) rYA`!wX x-N) rY` `00ExڅR1KPIS NAwRAFZK+)Hn`|wM"4{}!X9lM,b!)sx v84lik܄]9\ߴ.8}N#([ETxgѣw1Qa +yKvdW3^^55|X@?Zx% ߨ%mM=~+O6DŽ5Uxn0V'7rowtb_KI-dEN6ʛA MfxU!_Dd lJ  C A? "2R T]C`!R T`xڕK@߽Hcmq*D- 8nC)ULP98:Cw88[㽻"\>wraк?`h@"|"cbbeM";8l[31=VJ1[a}ls>PFȋ&g +-/E~bΗL`VVpj.ԃ, ]oeKw,CHOh0$,MΧ9ٓmQRޱ-EEIxPK8ۂMK[O<7?y|xp&?8 sN h'G&hzAUβo}LDd |J  C A? "2xF+c.{]J`!UxF+c.{4` 0#xuQJ@}3M . (*?xԃV(Bܼ {‹w*N֥27d h؀VL4f &bЦ29vZxd;жɞLc 2p2^q'ZBP<,=!ɾ^5QJ^y|5}S$ߗEis,l~B[? ~u%-K}Fx"" yGbicXQ7Dd J  C A? "2Lfu`ȶ|JuL`!mLfu`ȶ|Jڎ  h;xڅRJAf.^K G@JhI:ʉ gGXo`a'N6KH 1~7;K( uoX.gCmgexVW!'mlB"$)տnf节oN# FܼQi/9KIZm]/}#5+< si#;F3_Cwaq-?1ަLNQUz~Oc3E+hKg7a2yܙ1^V#S@~]:kDd lJ  C A? "24-NA}43N`!4-NA}43 xoxڕJAgf$wUbl-K0yD9%B^ KK+k Ծ]A ;-$0ۛ"ǔ6s q,+(+$:OW?*'I[FUu m|pNi8QLVjs-?Jۏ1:gA#s.,ckyanZ=ZwDn1M}oa 8iTC,/dw,;lGKޔ'N./@VE^*jn/_HlgDd J  C A? "2p;1E,a>aQ`!p;1E,a>*` .hkxڕJAًIr)(P ;-ca9%bKBRZاX־;sw &Je۝fg6`V2z("!Eі%sD5uͮ/W3=|E-^6ƽГ>KBFk?i]0*q^gJ ;p1NJ~L(̯F*ÔEm7reZ?ܦ9 xx@Orr1{xԿWg+F3CöTP,LKpe gfŞR(f'3GzQ'!'(KJmǃa+Dg>zKp&SDd lJ  C A? "2ƕf`yFxS`!ƕf`yFx` WxڕR=KAK?@8RXybR %E`6 KE~?ZsfnsȁvwAX=AFg$*$D=AJTO6qYO=C:[G,B*I2s,1::dxO2d4F3:)K4Ѯ-~K3#gsw~İfT*j/xe*7 SE#W~nʿc)nB+N)SY=xˑg?& kХ;[rnsd Nr kE+.c_;ij'j7EoiHhAA\R=b#Dd lJ  C A? "2,9.dnV`!,9.dnZ`GxڕJ@ϜIoiUA]VlwJIe AЅBwʅqL:"&9߹̟?)~ A^1R(#C,d2 *[L Y؝ќ１Ɋh9> !E,b&`m{_&'"S!!Q-5mot5]?܎Su/5ܙAR<#۷ds9!H.R|KRK*"=??.)YW1>".A+*e Kɑ7dU=sorBs56ɟgK.H>⿃]y3ⴁllpEU ϥIg~fiCN;Czȩo{Dd |J  C Am? "2GJhj}QX`!GJhj}Q(` `PxڅR=KA};{KC ՑʨJ[-L~@F!AI'ZH~+Vϝ́']7y], /^:YQAQ̼ҙ2 Ѱ.O5捫h65D\_၊n < drf׬3{' ^Eݱ$$hBE i{{* anǰ)Yµm9h C𫭰 'v'1m2^T24/1L աe;H}ǝe?Ư w~\J?"vldl=ԍw$5;7駸h'qbr1@~?Q GZwQ!IR~$k˚Dd h J  C A? "2ՕʤT~Me<[`!ՕʤT~Me<@0JxڅJ@g6iڤbE<B[l'oz} -4؀"}DIDD|Oy+^ƝI6a|3ɿlL HC 0 y[<gERWsS*h(6TGؑ98tAovXpBH_ؽߟ:GKxc II].eoD ?T K;)BPJn=j\79b_N|UaGf̟MFɅhfVj?"d)u_F\W܆݀5H{!]6]o`Щ &;|z_:Ea8iQЯ$W܋gV{{ՎzXCF4V|->,7 Y{/Pi4ZDd D J  C A? "2tȢKzz#Wh]`!tȢKzz#WhkxڅRN@ݖBK*hb%`Q8h?` Gƻ݄3m7 fv;y tpX2b#U 0!*48<|>" !x*g1J[L>1:k`F dNB *ܡ}٧9E%5sxW| Ex ^W{(;;=O+޺7~a1"iMh;"jB)γVS%Β_Rh)ڏzT?{nAst+]IM+[ _Mû0b_{)"iG:_&Nud\trCFU2'2a[g/2h"j1m,z8߈_M wk pDd \J  C A? "2LMavQ!C6``!LMavQ!CD` 0txڕJ1g~n[\A<B{,x*k-b ړ} /g!cfvt7?3A0% :%lAf:γ(0ΉI^^ YsZJhJ9= [\F>zcV&f!pB[o^OUs5ϘhyWN-nÊVDǶH.&&ĕV ĕp'K\'$uNFh+Tr:J/J*si9YHV?>g?|MSN,QІ;[ʐI}s|.K"d'zR|TÅ4kb6=:q͒>;yu2Dd hJ  C A? "2&8p1k=zpb`!h&8p1k=z @x|6xڍRJ@fCAJ{,TO*>@ThAr#=|<{tPovogK0_M[jc"y.CږHn3^7!C{o"ԅ2U<{g osW2\F0~T4N^ۉSn-FDܫad Y,ӜR9Wxn8)8t20 VaOiRow-Y`,'YQ->q QNT>zeqDʻEh#@CлbeYV5; vZHDd hJ  C A? "2Ɓo4 d`!~Ɓo4  @|LxڅR=KA}3|x HZ "Qn!IF'X@He3w"'aμ};;2` _Z1F8.iɔfWy^UmVXXbGk>i i3 *zBpGr:&֏(i][ݨs0L~1HH/4k^8;'8X8Ie=6g+LG%Iw7PۋMnR*tKzx}[Np3'Y]}'J:r(*zY#x8 s /:l#Dd hJ  C A? "2|:PA_nm/ X g`!P:PA_nm/ N` @|xڍQJA}39c*j*-!) n"$ +,+k+uvr".̛yv=!s"1& ^3!SYU_7؅M0_%zrݻ4˚β͸ρT|*nf|=e UI_[⏒7궂+dX|ŰU2iAf`r ?k>GrzJђ*(TѫֺS_RER!%01ty/fR |TwPoPH$:['N^5MuJ6AлHqx݄i2A(#>Y-Y_5ߨV?Hsa_V2Y9cV&=di6  t4aލ%Մ^?u WFfJEkPS޳+YhFK~| YDd t@J N C AM? "K2i$,WlX0E}`!=$,WlX0   xcdd`` @bD"L1JE `x؅,56~) M @ ?`c`8 UXRYvo`0L` ZZPv.PHq%0a9"% ƷbsMda"`? I 4} \wD?#\>f" Xc_g[.hqC `p<021)W2䡻$&K YDd J J C AI? ObjInfoEquation Native l_1092288501F`^`^Ole  FMicrosoft Equation 3.0 DS Equation Equation.39q4InI dm= l  dx FMicrosoft Equation 3.0 DS EqCompObj fObjInfo Equation Native P_1092288564 F`^`^Ole CompObj fObjInfo!Equation Native "uation Equation.39q؀IxI I cm = l x 2"L/2+L/2 +" dx FMicrosoft Equation 3.0 DS Equation Equation.39q_1092288627F`^`^Ole %CompObj&fObjInfo(ؔIlI I cm = l 13x 3 () "L/2+L/21 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native )_1092288879 $F`^`^Ole ,CompObj-fObjInfo/Equation Native 0_1092289038F`^`^Ole 3ؤ4II I cm = l 13L 3 8"("L) 3 8() FMicrosoft Equation 3.0 DS Equation Equation.39q\yII I cm =CompObj4fObjInfo6Equation Native 7x_10924661602U!F`^`^112 l L 3 FMicrosoft Equation 3.0 DS Equation Equation.39q0`mI\yI  l =)mLOle 9CompObj ":fObjInfo#<Equation Native =L_1092289179)&F`^`^Ole ?CompObj%'@fObjInfo(B FMicrosoft Equation 3.0 DS Equation Equation.39qP`mI\yI I cm =112mL 2 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native Cl_1092310960+F`^`^Ole ECompObj*,FfObjInfo-HEquation Native It_109237614680F`^`^Ole KX`mI\yI I cm =r 20R +" dm FMicrosoft Equation 3.0 DS Equation Equation.39qLIԂI   =mCompObj/1LfObjInfo2NEquation Native Oh_10923761825F`^`^V=dmdV) FMicrosoft Equation 3.0 DS Equation Equation.39q(IpI dm= dVOle QCompObj46RfObjInfo7TEquation Native UD_10923762273=:F`^`F^Ole WCompObj9;XfObjInfo<Z FMicrosoft Equation 3.0 DS Equation Equation.39q<II dV=(2r)dr L FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native [X_1092376276?F`F^`F^Ole ]CompObj>@^fObjInfoA`Equation Native a`_1092376324~ DF`F^`F^Ole cDIܝI dm=  L(2r)dr FMicrosoft Equation 3.0 DS Equation Equation.39qpxII I cm =LCompObjCEdfObjInfoFfEquation Native g_1092376356IF`F^`F^  2r 30R +" dr FMicrosoft Equation 3.0 DS Equation Equation.39qؼ@I I I cm =24L  r 4 (Ole jCompObjHJkfObjInfoKmEquation Native n) 0R =12L  R 4 FMicrosoft Equation 3.0 DS Equation Equation.39qDInI   =mR 2 L_1092376403GNF`F^`F^Ole rCompObjMOsfObjInfoPuEquation Native v`_1092312075jSF`F^`F^Ole xCompObjRTyf FMicrosoft Equation 3.0 DS Equation Equation.39qLII I cm =)12mR 2 FMicrosoft Equation 3.0 DS EqObjInfoU{Equation Native |h_1092319599XF`F^`F^Ole ~CompObjWYfObjInfoZEquation Native |_1092314187]F`F^`F^uation Equation.39q`IԂI I cm =x 2+R"R +" dm FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj\^fObjInfo_Equation Native ؘ`mI\yI  A =mA=dmdA   thus, dm= A dA FMicrosoft Equation 3.0 DS Equation Equation.39q؀|I4I dA=2(Rs_1092314329[ebF`F^`F^Ole CompObjacfObjInfodEquation Native _1092314761gF`F^`F^Ole CompObjfhfin)dx , from geometry FMicrosoft Equation 3.0 DS Equation Equation.39q(`mI\yI x=RcosObjInfoiEquation Native D_1092314853`ylF`F^`F^Ole  FMicrosoft Equation 3.0 DS Equation Equation.39q<`mI\yI dx=Rsin  d FMicrosoft Equation 3.0 DS EqCompObjkmfObjInfonEquation Native X_1092315220qF`F^`F^Ole CompObjprfObjInfosEquation Native 4uation Equation.39q`mI\yI dm= A dA= A 2Rsindx= A 2R sin(Rsin)d=2 A R 2 (sin) 2 d_1092467163vF`F^`F^Ole CompObjuwfObjInfox FMicrosoft Equation 3.0 DS Equation Equation.39qIԂI I cm =x 2"RR +" dm=2 A (R  cos) 20 +" (Rsin) 2 dEquation Native _1092316185o{F`F^`F^Ole CompObjz|f FMicrosoft Equation 3.0 DS Equation Equation.39qT`mI\yI sincos=12sin2 FMicrosoft Equation 3.0 DS EqObjInfo}Equation Native p_1092318421QF`F^`F^Ole CompObjfObjInfoEquation Native _1092317438F`F^`F^uation Equation.39qؤIԂI I cm =12 A R 4  0 +" (sin2) 2 d FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native ؜`mI\yI (sinax) 2 +" dx=12x"14asin(2ax) FMicrosoft Equation 3.0 DS Equation Equation.39q_1092318489F`F^`F^Ole CompObjfObjInfoEquation Native \_1092317879F`F^`F^Ole CompObjf@II I cm =12 A R 4  0 +" (sin2) 2 d=12 A R 4 2"14sin(4){} 0 FMicrosoft Equation 3.0 DS Equation Equation.39qX~IPI I cm =4 A R 4 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native t_1092318570F`F^`F^Ole CompObjfObjInfoEquation Native `_1092318647F`F^`F^DII  A =mR 2 FMicrosoft Equation 3.0 DS Equation Equation.39qL(InI I cm =14mR 2BOle CompObjfObjInfoEquation Native h_1092321288F`F^`F^Ole CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q8`mI\yI I cm =mR 2 FMicrosoft Equation 3.0 DS EqEquation Native T_1092331317F`F^`F^Ole CompObjfuation Equation.39q`mI\yI =mV=dmdV  , and solving for dm, dm= dV= c dx dz FMicrosoft Equation 3.0 DS EqObjInfoEquation Native _1092331420F`F^`F^Ole CompObjfObjInfoEquation Native l_1092331593V.F`F^`F^uation Equation.39qP`mI\yI I cm =R "2 +" dm FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native hL`mI\yI R "2 =x 2 +z 2 FMicrosoft Equation 3.0 DS Equation Equation.39qt`mI\yI I cm =x 2 +z 2 ()  _1092331721F`F^`F^Ole CompObjfObjInfoEquation Native _1092370884F`F^`F^Ole CompObjf  #&'*-0347:=@CFILMNORUVY\]^adefilmpsvy|+" dm FMicrosoft Equation 3.0 DS Equation Equation.39q`mI\yI I cm =x 2 +z 2 () "a/2+a/2 +" "b/2ObjInfoEquation Native _1092371817F`F^`F^Ole  +b/2 +"  c dx dz FMicrosoft Equation 3.0 DS Equation Equation.39q`mI\yI I cm = c "b/2+b/2 +" x 3 3+z 2 x[CompObj fObjInfo Equation Native  _1092372425F`F^`F^] "a/2+a/2 dz FMicrosoft Equation 3.0 DS Equation Equation.39qظ`mI\yI I cm = c "b/2+b/2 +" a 3 12+z 2 aOle CompObjfObjInfoEquation Native []dz FMicrosoft Equation 3.0 DS Equation Equation.39qظ$InI I cm = ca 3 12z+az 3 3[] "b/2_1092372617F`F^`F^Ole CompObjfObjInfoEquation Native _1092373163F`F^`F^Ole !CompObj"f+b/2? FMicrosoft Equation 3.0 DS Equation Equation.39q؄`mI\yI I cm =112 ca 3 b+ab 3 ()ObjInfo$Equation Native %_1092373301F`F^`F^Ole ( FMicrosoft Equation 3.0 DS Equation Equation.39q,ԨInI =mabc FMicrosoft Equation 3.0 DS EqCompObj)fObjInfo+Equation Native ,H_1092373380F`F^`F^Ole .CompObj/fObjInfo1Equation Native 2uation Equation.39qt0nItI I cm =112ma 2 +b 2 () FMicrosoft Equation 3.0 DS Equation Equation.39q_1092400348F`F^`F^Ole 5CompObj6fObjInfo8<`mI\yI I=R 2 +" dm FMicrosoft Equation 3.0 DS Equation Equation.39qHIԂI I cm =Equation Native 9X_1092406009F`F^`F^Ole ;CompObj<fObjInfo>Equation Native ?d_1092406656LF`F^`F^Ole AdI all +" FMicrosoft Equation 3.0 DS Equation Equation.39qXIԂI dI cm =12dm x 2CompObjBfObjInfoDEquation Native Et_1092408708F`F^`F^Ole GCompObjHfObjInfoJEquation Native K@ FMicrosoft Equation 3.0 DS Equation Equation.39q$IԂI I cm =dI all +" =  +" 12x 2 dm=  +" 12(R 2 "y 2 ) (R 2 "y 2 )dy_1092408958F`F^`F^Ole PCompObjQfObjInfoS FMicrosoft Equation 3.0 DS Equation Equation.39qؤInI I cm =12(R 2"R+R +" "y 2 ) 2 dyEquation Native T_1092409163 F`F^`F^Ole WCompObjXf FMicrosoft Equation 3.0 DS Equation Equation.39q}IHI I cm =12(R 4 "2R 2 y 2"R+R +" +y 4 )dyObjInfoZEquation Native [_1092469253tKF`F^`F^Ole _CompObj`fObjInfobEquation Native c_1092410353F`F^`F^ FMicrosoft Equation 3.0 DS Equation Equation.39qIԂI I cm =212 0R +" (R 4 "2R 2 y 2 +y 4 )dyOle gCompObjhfObjInfojEquation Native k FMicrosoft Equation 3.0 DS Equation Equation.39qd؜`mI\yI I cm =R 5 "23R 5 +15R 5 [] FMicrosoft Equation 3.0 DS Eq"L2χA3 bBrq`!χA3 bBrq@ `\]xڍR=KA};QSFZ[AQ ll h*SZŁ[dsDw,o㽙Y,(0RveO.RJnq,7kj^n6H9< ̌=[Ӄ]Ԧ;kJ2dVI/; W&lUp4EIњm:rYę`ľ ;3$eǠdO2Xs̞U}4 LL9i^ _]+>(;~\ߓǮgE立ѭyks0qe I)$x=QZWuOfoϹ%B\u>=ʋTDKN4Fg<Dd dlJ K C AJ? "M2:Gx%ā;0|[z&`!r:Gx%ā;0|[f^ @xڍQJ@Mc ?/)G UHŀ&[<  @ |xcdd``gd``beV dX,XĐ ɁI]Rcgb 6g@P5< %! `fjvF ,L ! ~ AyP9Wc%+)0 1Q?!5&2p0epAALbQIhdBܤ.i` _~LK1f {.p=5SQF&&\8.]3U%ѽpr!L 9Fwqt{~, ]sDd lJ H C AG? "P23 AYeJ=`!3 AYeJ=@ pxڕS;KA<$ Q$Jb-X 0MC\"&  H*[ 6V=w6ofvvBAïfId۶L5;B-]HESFKxNlmTQ?&>@@~`Vޟ{}ejzZУtxNLt$7^S'䞚K>n[f9bV%+_TV'Ҟ6,̢GOkܠcyk2ё~9~Ǔx$8#G:j5Kneӝq/hj )G(y`ۯ=gڟ;xG9:Ιǎу] KIJg"hVc2 /Dd ` J E C AD? "Q2@z2Gxw2;̠y`!@z2Gxw2;̠@`\xڕ;KAg.KbR*B-lZhDA@K|F;`'X[hmR1๳$Qػ 3߻AX>_!bĖ-!JӋғ%q ,p#tI D,/ *uL&Wo"zOR|e=jV.".Kr }6;5SvJk@w3k '9])rV0ɯ圭+~|GGZ>'?"W.[JOv%w O)$(}n4dvJ0wސtHuub0WgMUg+epr5?5_{JDd <lJ F C AE? "R2n+c|(G仉.`!n+c|(G仉` Nxڕ=KAߙͷ?@ENA-L~@(1IgE ;%ݙ#qܥ@j)2ّ[fB=$j; I5wǽT: ^5OiN6Jn&J:73C^'4b4Ad4Khg|&Dd J G C AF? "S2^=@=d)u<dx`!\^=@=d)u<"*xڅQJ@}3IԦcU/)U?z*VlA⩟Я|J?BZ 2۷m`57D!J;#&7/bxe0:vQ KMM ljupޥC >2d5[&l'O-vLg} )N a m=sXŤO1*"BO&4;nm#gǟўN/?77 U1VzW#f eOLDDd l:3V D c $AC?"T2 4\e}Mv`!n 4\e}M  <xڕJAƿyN᮷RAEL\g%y T>Gj+kϝݍEvf7Kj+ixmLd2eY6LyW瑊ۜ[:JiB^|`tݿhש-d!aE7%ޒIb]8Dd J ? C A>? "U2pw0X=E)v`!npw0X=E)n  `\<xڅRJA};wg.N4VI>T\gi/aqUz{ k+sgw`rǰfgY*(*[#$(sllr5*qdE""h3=X{=ۦr8:e$nE397(>Jt苒 ڵeSS{k=0ǔ!14d Փ5-c3Z:g_Eg0v&dBY8sӞeN6Ǝxm#Wb-R g0mB0%2w='c?axJyK膾BqwUX쫌aHvb Jl6q/!P*Zi ǟ Qȶj1MR1j  K!Dd @J B C AA? "X2 ?v_K`!W ?vD  %xڅQJ@}3Ic Bb (RlJ -mAr')%Ż`:;73o]Sxi8sӞFNxum#7$-2[Vq՟wq o~x,'S跮7"rdK2.{%Nƨ1щP\Dd XJ > C A=? "Z2뚩sg;Q*`!뚩sg;Q*  7`\`xڕR=KA;s]#Nш`+(jA/ I@ҥ֟a?Jl8w6 ;ݷ;oޛE(X@P(l=Qv8u\MBєW&e+WyXIkNi<S֨Ca4#p`(ePE.l5}\sU첉?;H-Ƈ)(,ŎQD_S":T78^#_2ifn8:]@ZA p8 L/~g({U3{ɉ]6 ^-gv Lj?Hѳ)vͫr1#A*4QA7ђo{siDd lJ = C A<? "[2 \a'FlmA`!\a'FlmAPtxڕS;OP>- $1N A"DGVufI&!1l NFwg'M $lN΢{nK}q )I/d| cb8b%Jc_8fb@=6w^I\q"]iZusxGWA_Jd\eny|JpV&V?{AuЊ-sv AYus!,d:ۀ^WҋkF"OLyґi4T/̊n*7:zX>Nu?a[d[Lg_eG[JFIbg|Gb.ΎefSXƍ9|%qB'>y ͦ܃vVb+%xީΪūX w__? bD]zJoO\/Akeb4LF{Y[1Dd tlJ ; C A:? "\2@/$ z,Nbo`!g@/$ z,NbB  5xuQN@sb"aʤS4t %*>/((wT avNZjvgg <2F+<ʮY^y_2՗9\2~#G=pzޟc|R!;J[+ g/~EO&ꎒa7`v`?ju85O Q>W? N~/TںQ+7KF(T:?vx@5Ϋо}#] QWjZd'H~&%ODDd dV < c $A;?"]2ۦ熁p}zv¦`!nۦ熁p}z^ <xڕJAƿ˙N!VI*+(\2`uO"O3hZsgb =nf;G( !˷Vf ֘H3Yiv4Sgʫ+<2ֱ҆LxjU Hm}%;$10-d!;o%}69WO~q1fFع3 Vt{\B{Qp*^N`~R#yחȴR#sNz/jZ8韚i*_G'0 .A.US.IMok:ԃKĹ5 ޵ a:Ma*r%_3s{ 3%Y)ȔaW6cuڻB9! NZ9Ihsޚ89LM<={>D֝ 'QhB/c+pDd lJ 9 C A8? "_2p^D S;@J`!p^D S;@txڅRJAM$H% ڧH&BtW-VsgH.BnB? tH^F ardKrdETqo*%[EAHMhhQ[PKذڣSWאFA7n ^AI^Y{8voűIf_+Hci{@}hDrS`Dc5cn3 t/_{`p";0}&Xf% vg;O a2PYw4˗"6b /7iw65)“eR`T飙Xb̛*`zO$lSDd @J 3 C A2? "`2!Ӯ}|@[7`!!Ӯ}|@[7+ WxuRJ@}imz DA{ (x`mK5 ~xz?ówŸ;,h40d̛3;K(!P_^ZI(/'MqD$Y9llGueWg45$ıFU`, )\+8>=qn}aΝi}L0w%?8)cϙ0Ri+j3\Tx(-;])ƲCKZf6gy=]R}^m_on$(8 /anO>.(k-Qo_6l[KԚ. Őͨ?td1] Dd `J 4 C A3? "a2lw[R&)tNH `!@w[R&)tNX Hxcdd`` @bD"L1JE `x02 Yjl R A@1 > UXRY`V! v \q.PHq%0crd}`d++&1L`8&af1baV.5Z 8߇įV!ӟv-4̸ $Lv0o8&+KRs VjJDd +J 5 C A4? "b2z|MV`!N|M& MHxcdd``fd``beV dX,XĐ N! KA?H: ֵ5rC0&dT20UfjvF+ss . ri#  7nl(9 ,@sB2sSRs&2H&`IA 8ߊ tn;$N(?* v>p2pAØ `A``cI)$5!ݕ`uJMDd hJ 6 C A5? "c2/ ˙}Cn0>%b /`! ˙}Cn0>%b ,@hD|xڕMKAǟRw-[*{a7IЭQ($[>W?A.~ Huxewh 7\뻣2? |!'\7>oꙢR YI'ݤY| H~z9JE^3IYיgg?:4٣I3-U, ֕G g9ӟdoRd=QQ98ܳI [|ѩ^)_"䫸Wq#EXQ6u;Μ."U]0@Sa9߲zP0ܗ_mCVKeÄߣ">R {>Dd lP 8 S A7?"e28lm``yvι`!n8lm``y> <xmQJA};krˁC ; Z| H:?"aaagH>ABT2QWͳ S"S*vQgyYic#߽+*]M} #빞` AL9Py,)IP#ŋk2 a<"%+ 9@EʌbY" Ϡ+MҤ7$Dd l lV  c $A?"2O\bQT(;rz٥N`!O\bQT(;rz٥DxڅRK@~bkqt(u `RMpMM] #]\Qx~h{{fY/d (gM ObQ7$ qǭ\ ( k^{9fwEէp$ւ}7m TGNw;_A r@3H#Ș0 PD"S,Fy6&]>.[eJcZ&^7 bi&c4WfFcQb [йEz9Ŷ j^i;gXDݮ)ʹdYs|z?u?NR=v=뚯9sI!%#[޽k 3TV)yrx^2Կ\ʜv}DISg&e"e͠cg!z{ͯn{09lEz ;[kٹipan {{`H؀]uۮ&D:!.B'=&^♮ɯ[n[ +B x*ݳhgpl|poܡ}c5:aQrH33y*0s!,nm7_3X{s] ecTѿJgvNSgQ(_'&{2V9G1jFoiYEDd lV 0 c $A/?"h2OLWHT1w`!oOLWHT1 =xڕJAƿ=s xLa`k%Z$yD9P!AGHJ VًEvfݷ3K(fU5'VfybL'y6I^q]_̓$fimB3Fbm7RSTpXr|"G_apf,ޣg~q[R͌u(B dpY̳E;E|NOצ4S#sAjgۚ4>,j|ӷ"lq*"(f蜵ypF#}\mA;h(^Dd $|gV / c $A.?"i2p[g93 6LM`!D[g93 6`HD 0xڅJAK.갰S0m")>a'Eݒ5zw `<3O( +2FV<3l<8 dW/@K|ǭ.m kX.v%#j&Jxlc PXܲC>CڮSZ+C+[v . w.qF(Zp=)rma v$/}':<Mb~wtw}k"KHJ牤a>T[hz&e<­jy`Vq=s&*2ܢxk9%VE)Z9g>0s gjlPDʐK ;cT+<C ݯ-2hZw5.Dd <J . C A-? "j2ր/Ć{vl%`!dր/Ć{v` @C2xڍRJ@}3Ik\DA<,x(Y`ZɭGԏ$^Eθ 5a{3!4d|{H+ykeֵ-.xmb26 晍 `j-i?OH|]'8<(NmNQTWI?)"S J!B.Ϣ ˾&J J?|rcy}8?X&:YuAv辵sgٳ!ļI`Xu)_'Dd hJ  C A? "2{|XTRգDce`!]{|XTRգDc4 @ |+xcdd``bd``beV dX,XĐ ɁICRcgb v@P5< %! `fRvF+B2sSRsx\ F\_AZT~8#F6H2@l`6-`\Sc+|7?kd@ I9 0@ga)ՆSl 5%l^&3!pB``㑑I)$5!ݍ`uONDd lJ  C A? "2WZ="%3a˄v'<`!WZ="%3a˄v'< Xh@CRxcdd``gd``beV dX,XĐ n! KA?H @P5< %!@qL3>d& MρH(3a) R#TL A,@! ~ Ay p@Lbtq8Rra|nUpyQVwc=6+'d?(0bٜ ˽1~!" @Jpa*xFpV v l0 `pddbR ,.IeCw=3XqDd 0 ovJ  C A? "2O3YyƏE+i`!#3YyƏE:`\xcdd``nab``beV dX,XĐ K A?d-bad1P5< %! K@*3 P 27)?"](6ϴD8}`^RM35cd++&1b#ԤLAR 00?dz1 W+|72\&Ï~, =IP~' ;ObˆW`~Ir.z1F M pV~T~> $9>< PXct@xY~3+$("L7yEsB kFtZ/I܋Op>/D03H6 $9>< PXct@xY~3+$("L7yEsB kFtZ/I܋Op>/D03H6iW|M| 'ep͝{EmӃo&břjӇ&Dd < J + C A*? "m2<­)sL`!­)sLv`Hb`\xڍK@߽U?@BAupAEVpM[ -H'E]!?qSx)\r{?GDi/z"Ez:oȘacbda5.X7CSX=;Pib'KJe@p-21d~tԠ^eLZ){kKnvfUL/I^(6c{ٍh7#x"o"},A19ŎԢ/0Dd lJ ) C A(? "o2\CJ5bvbyQ?`!\CJ5bvbyQ`Gxcdd``$D@9@, fbd02,(1dbfaz'd3H1ibY;Xa3PT obIFHeA*P - :@@ l.#t&c 0 Ri`@3W&0@22pi#xVv 1Hj7~br|!65wY{bAPJxodU\yB15I@e&ESpJ=W"|Ex t S@ك?:#EtNlؐTtDRt={4pxDd J ' C A&? "q2pJUq /ľj_%LZ`!DJUq /ľj_%@0= @Cx}QJ@}3IM QPO7W`m~b@m$=!$^3&"nvfv{B.:^,D![,s \z\E{[-+ĕxjh n|*$o 6"x%3lPɾ쇦< dٱq2Dy`\5Q-8n܆̍*̓嘗UNx2s;D\\TnfI{[2X{o |TBF ͝"?ךcr-}~u^yHB>ߡdY?~s T|ү!ٹkIXW%;zm5\TÁ~Aix~׳9Dd < J # C A#? "u2X%S^ 0`!X%S^ 0`Hb(+xڕS1K@~iZ5h :"UIth;9jA$.N":dr8:T4޻6)׼}wA@{e@O8R󁌉Y6.frTv;uI<(�>űǣ3>l8MB$a**@; z2B1${ xQ4ƣK~=fGGJNiVBݩdyav}jAE R)ovlX7sYl +AߵI%TL)||".qxkpM(MX7u+߲;s!e?:WƻcRj6G$+>oemNE~ ԙg<#tv#G1ӂ@1Lm5 應Dd 3 J " C A"? "v2NΉWPwz[`!NΉWPwz6  (+xڍK@߽6E*utEYZmAh" 8trprVhwMK.>}_A@D>{yG Cu$3id;^V_[8 KA`prg`J,+NP#-^abyMFV[:oX߱ekxh=YT$+=~>*ySd)P5,Ns-H\;YjVB)AAźqL5~dO-5SmSBIVHq%/|1{sVn:SqgeYԦĞxߟ@_ZgT"EĽ%st/n2C^sʕ {5 aDd J ! C A!? "w2 3=Xxm" cAVL'+#}[{(H`V6\g^vR]قE@RSi/-}Ai'~%\R%Չ7EH%]w;)\\CfTg, S89M)9FAׯp=33g 24پNlUk6ɷ&V#Y&|[.>>-j>G}O ;ͤ]q#NKp\؊k!ѠDd  J  C A? "x2}!U'`^& .yL`!}!U'`^& .yL@Rxڕ;KAgb]^Ԁ )l$B`Ebi჈tJ""V)Dğ VFX;xI{;ܼe`O  e"I>1a躮L1I#SApwҸ؜Tk5 ipeŵrxZ=I1dQLIv"Z[P_;Wu֋1=cōkL73_v%k{okx"м Ni>SoaU7xNX'Eg3D3\Ng=+ U 5;1{OM)_rƓ<ݧ#\y@.g:V`^RwC dL_1092411822F`F^`F^Ole nCompObjofObjInfoquation Equation.39qdT`mI\yI I cm =815R 5I FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native rp_1092411997F`F^`F^Ole tCompObjufObjInfo wEquation Native x`_1092412069 F`F^`F^Ole zdD`mI\yI =m)43R 3 FMicrosoft Equation 3.0 DS Equation Equation.39qdL8II I cm =25mR 2CompObj  {fObjInfo}Equation Native ~h_1092413706F`F^`F^Ole CompObjfObjInfoEquation Native ` FMicrosoft Equation 3.0 DS Equation Equation.39qdDIԂI dI=14dm R 2 FMicrosoft Equation 3.0 DS Eq_1092413904F`F^`F^Ole CompObjfObjInfouation Equation.39qdd`mI\yI dI=14dmR 2 +dm x 2 FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1092414519F`F^`F^Ole CompObjfObjInfoEquation Native _1092414055 F`F^`F^Ole d،`mI\yI I cm =14R 2 dm+x 2 dm()  +"W FMicrosoft Equation 3.0 DS Equation Equation.39qdX`mI\yI =mV=CompObj!fObjInfo"Equation Native t_1092414111%F`F^`F^dmR 2 dx FMicrosoft Equation 3.0 DS Equation Equation.39qd<8I؎I dm= R 2 dxOle CompObj$&fObjInfo'Equation Native X_1092461067-*F`F^`F^Ole CompObj)+fObjInfo, FMicrosoft Equation 3.0 DS Equation Equation.39qؠIԂI I cm =14R 2 +x 2 ()  +" R 2 dx  FMicrosoft Equation 3.0 DS EqEquation Native _1092461141/F`F^`F^Ole CompObj.0fuation Equation.39qظ0II I cm =R 2 (14 "L/2+L/2 +" R 2 +x 2 )dx FMicrosoft Equation 3.0 DS EqObjInfo1Equation Native _1092461369(<4F`F^`F^Ole CompObj35fObjInfo6Equation Native _10924614519F`F^`F^uation Equation.39qذIԂI I cm =2R 2 (14 0+L/2 +" R 2 +x 2 )dx FMicrosoft Equation 3.0 DS EqOle CompObj8:fObjInfo;Equation Native uation Equation.39qظ`mI\yI I cm =2R 2 14R 2 x+x 3 3[] 0+L/2 FMicrosoft Equation 3.0 DS Eq_10924616307A>F`F^`F^Ole CompObj=?fObjInfo@uation Equation.39q؜$InI I cm =2R 2 R 2 L8+L 3 24[]O FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native _1092461784FCF`F^`F^Ole CompObjBDfObjInfoEEquation Native T_1092461911HF`F^`F^Ole 8~I8I =mR 2 L FMicrosoft Equation 3.0 DS Equation Equation.39qxpIlI I cm =CompObjGIfObjInfoJEquation Native _1268566748MF`F^`F^14mR 2 +112mL 2 FMicrosoft Equation 3.0 DS Equation Equation.39q$·hd KE Total =12m 1 v 1Ole CompObjLNfObjInfoOEquation Native 2 +12m 2 v 22 ... FMicrosoft Equation 3.0 DS Equation Equation.39q$ d 'v cm_1268569105RF`F^`F^Ole CompObjQSfObjInfoTEquation Native <_1268568447nWF`F^`F^Ole CompObjVXf FMicrosoft Equation 3.0 DS Equation Equation.39q$(dJ 'v e FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoYEquation Native 8_1268580123_\F`F^`F^Ole CompObj[]fObjInfo^Equation Native H_1268580180saF`F^`F^$,hd 'V cme FMicrosoft Equation 3.0 DS Equation Equation.39q$d  + e v =+ cm v +'V cmeOle CompObj`bfObjInfocEquation Native _1268570630fF`F^`F^Ole CompObjegfObjInfoh FMicrosoft Equation 3.0 DS Equation Equation.39q$  + e KE () total =12m ii=1n " (+ e v ) i2Equation Native _1268570575PdkF`F^`F^Ole CompObjjlf FMicrosoft Equation 3.0 DS Equation Equation.39q$ d + e KE () total =12 i=1n " m i (+ cm v ++ e V  cm )ObjInfomEquation Native _1268572481iZpF`F^`F^Ole     !$'(+./012369:;<?BEHIKLMNOPRSTUVX i2 FMicrosoft Equation 3.0 DS Equation Equation.39q$Nd + e KE () total =12 i=1n " m i (+ CompObjoqfObjInforEquation Native j_1268580238uF`F^`F^cm v2 +2+ cm v + e Vcm ++ e Vcm2 ) i FMicrosoft Equation 3.0 DS Equation Equation.39q$»pD + e KEOle  CompObjtv fObjInfowEquation Native  () total =12 i=1n " m i ('v icm ) 2 +'V cme m ii=1n " 'v icm +12'V cme m ii=1n "_1268575284zF`F^`F^Ole CompObjy{fObjInfo| FMicrosoft Equation 3.0 DS Equation Equation.39q$—edT + e KE () total =+ cm KE () total +'V cme m ii=1n " ('v cm ) iEquation Native _1268574652F`F^`F^Ole "CompObj~#f +12M total ('V cme ) 2 FMicrosoft Equation 3.0 DS Equation Equation.39q$ˆhd M total V cm a"m ii=ObjInfo%Equation Native &_1268575266}F`F^`F^Ole )1n " v i FMicrosoft Equation 3.0 DS Equation Equation.39q$& + e KE () total =+ cm KE () total +'CompObj*fObjInfo,Equation Native -_1268575253F`F^`F^V cme M total 'V cmcm ()+12M total ('V cme ) 2 FMicrosoft Equation 3.0 DS Equation Equation.39qOle 4CompObj5fObjInfo7Equation Native 8+$d + e KE () total =+ cm KE () total +12M total ('V cme ) 2 FMicrosoft Equation 3.0 DS Equation Equation.39q_1268576273xF`F^`F^Ole =CompObj>fObjInfo@$chd + cm RE =12I cm  2 FMicrosoft Equation 3.0 DS Equation Equation.39q$‹hd + e KE =Equation Native A_1268576604F`F^`F^Ole CCompObjDfObjInfoFEquation Native G1Table$SummaryInformation(J12M total 'v cm  2eOh+'0   @ L X dpxRotational DynamicsgkappNormal George Kapp133Microsoft Office Word@t@      !"#.%&'()*+,-/0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrsuvwxyz{|}~݈jw&C䵈FDd lJ  C A? "y2''y4u ^`!|''y4u   JxڕR=KA}3 xbREQ -p] ubHﰶVܙ=S\,y; V 'oĒl0'I77u=?2m>5$Bo͞lSQwtڎň|cȩj9eɖl\I3or\Va+U!h(,Z(ݘG#7\Q5'";S4ӛST4'>DOc~xG;W8VϺY\mQ"z2?dQDd  J  C A? "z2KD WRy`!KD WRy UxڅJAƿ%?*SPS"I\ $\gi%)B >@j+kٍ%o~7@ @atdJ AireWlresiWzVt_Rӄ@w=`1E)5h&8z|kw!=+ҝRr2nܠ Y73P9{e ʧuދ[]Tx}CpЩYd?V}JK-}X8%KG˧;#etw!@Hv̟t,?tH0j|!=N56 3_ 40(&'^DDd lcKV   c $A ?"{28v:0E[7v`!n8v:0E[7x <xڕRJA}3{z x:R) -|Q܋\ү*`% fwfx*#Uz1& qY٢5l3y"Oz?8)D3#n봗WI;X lDT -0uolю3ߙi6̞ǁmnN}XLWXS|+VO_ʮe #43F(ѓ b^?Ȧ5)}!a7Dd J  C A? "|2|Qo!ybO|NX9`!PQo!ybO|N `\x}QJA}3{buX)h/(D<@.ץol,,+ zݥ]y{v#х5kI0!eYr@ۆ2mnx.%ٞJB(u!ًĉLE]tpOch看3ge<}A7< ݗ7*do89*Y2.it5!N*%ҵdqڑ6.y[t]Ϡ= 3~5Vn?O eM7Dd <lJ  C A? "}23yL?!o+uS `!m3yL?!o+\` ;xڅQ=KA}3{&3Ց΃" Z$.⁂w $  K+B ?Zswvs!0w7IsVt4d&,&mi39*]g54E5~٫q&NӫA1N5/q&6t{oJ}$tQo s;6G'xYݐ; ?)9gAE?Du8s'ZԭzN\9=Scό2C> ߤ_d,}/Lw/lg7I{ ê |3EVfUIΏJG4#Ar <ܚȄSkt%C:ܮ xV?;e~6F}nvW6G 5A")G,-~RUDd J  C A? "2N` հo `!N` հov@jxcdd``$D@9@, fbd02,(1dbfar`9/d3H1ibY;3PT obIFHeA*[~1AtF]F"L LѬ 0R_ 0@penR~T/#6b> R~}L`\| ߡ[(# ~* W?@|m8¶xAv/Hސbr<I!.gE)0~77Ȍ !D6|\ލD?!L@6%碛@0F4!|Q%ɍkKYAL>|0ߐ fl04- 0y{9I)$5!`u!5SDd TlJ  C A? "2H3Z]‚eGY`!H3Z]‚eG WxuR=KA|:R)^N-` ^L@,-,`ga! uϙٽ+N]yf߻]D(H"'!j)MSͬfme%Nr\%H'x&|܍[S=D\#Z`B!'>O@ I4Qr,쉇/:6K.A>a~P_9Uqp۲pveݏ p^?QQGU@~J>"Ak!w`;y) `qAɯ~g]R;4طw=Q{FE2GQ fG~Y*Dd hJ  C A? "2X(/I&rb9Lh`!`X(/I&rb9L^ @%|.xcdd``gd``beV dX,XĐ ɁISRcgb @P5< %! `fjvF ,L ! ~ Ay -^ 1q-`,a9"Ʒbs!&*ȀD`?~)H b#܍XAE, 8||I9s0]X/0y+Ǩ-4n(Lv0o8+KRs Vf\eDd J  C A? "2aH|<`ӊ`!aH|<`ӊ@(xcdd``~$D@9@, fbd02,(1dbfar`y d3H1ibY;x7$# - :@@ l.#t&hVKrf7S? 01d++&1BMadF\ L 'TyAz'9A 8"qb]?]%ɹ&1 $&IW43P!|Pc kKYAL>|0khI9 8\ Erb> R ~_ Rʿ W1B꧲!p3Dpαe.p>G `pbdbR ,.IeCf:)܊pDd , J  C A? "2MXhF(e:~߇u`!MXhF(e:~߇6 `\xڕ/A߼]v[*.MBM' @&$~"n 7ÞH%("ךvLllf}f{};bFt&cH); 3}A3Yʻ7viFڠ| G'ُEYaza{y^Mx Zẅ { j!gG{$H̯Li75.!䘳=;$yb+z&ռq @Ao`L)@K\F)fJq<qʸRZtkk;|FA>1B/FWW#$sNsK;[ >z' ;gq*x2og@!٣*{z''*>trcw=s9SW 6mul!UNLg|bz]r\`O[2Dd J  C A? "2y? 4խ>)`!y? 4խ>x`\xڕK@߽6M 8?ZVWA?@SVl q&/xr=lK^}{};B6{yGC #)dV\=݋ǹ56g[dOrAZjI4 'pMukOg[,;c'n`^fa`oзIvUV|"+Άy P)8 P|_"ўڠG挪-ȌJMAw8#N)%=8U#łL-~d]E9+xV~/hNݽG{nAϝi Sqd>{kMƷTz z^tfLyLqď9m 7!&jJхR&M}Dd J  C A? "2#W$FLڟqv!`!W$FLڟqv!:8xڕ;KAgN|BTc|0bJDKKXB}bcc&gID!X#}L~LaEڎ4 R*ʹ(2amf~-(wϡ4OA5퍼|Rq n~l+ 9iU󙭸"W=p^W\2GT~u>۶-_R`]Թw!|t0mC\>Ay,ht^Dd J  C A? "25w\EчK`!5w\EчK`\xڅ;KAgμ_hbuD X H%`K t)V X\ie'&๳Y ػ p`<15Ï2AƄ³g  /ىrkp) bn2AlyThԄk X+Ys*w>ve9ID~mv KuBzwM}ŷLqWK~;x$^|* &Y况_nY]B˒Q(? SSM1S)w#;daJZ◕%1w&6Sș<#KN{a~Ӟa❱)IO2@=c3MF'q¤z@}/僬{0dhmj2D>>2Dd $lJ  C A? "2^lj`L We;p;"`!h^lj`L We;BHD 6xuQKP%6 qZptum'T(؂dO\?١C7Wg5C}ww3hΎ `kx.CEQhdliMƲWhyVB2~e-{s,'pz> sD >̐Ҏ{A9ޑ'zEL:aboǿsNVܳ83+8E!*Oہ:ӋVD5<~.Q),R[[S~ßzbz-·t0[3Ē!j4I5:PgDd pl53V  c $A?"2;eшDG0Im$`!;eшDG0I _xڕR=KA}3{yC ՑFl+AP$?( ^"$"RZWXYXXXJ;{wEDԅ7o>v..IP93l$G_& q$٠eTR9|AiAKXAbh<ң;%NAw_DԂwT$:Y%?+#-jtnL&6~H#hQ7я;O:E㺖kٮmxxK2| dw{QtJMiDd ( lJ  C A? "2 x:^$wtUP &`!x:^$wtUP @1xڝSJQ>h:XV`D4m0Q4k#H z.ڹ \Zm (- rùRf HmN]hƞbTJ06sZ Qp)' nv`PPE;>zW=2&twuѐ{ 1x% ρL2m,Cٯ^_}.vaSψ:uCᚆbRda{Wv(n)3E#:,/8lS;xOtx8] 3lpF?j#3:?2TmYB ~Eekf*4j%-^TgU 85p? gkaB1<91KߓDd J  C A? "2RބQRW@jٸ.|)`!&ބQRW@jٸВR@Cxcdd``> @c112BYL%bpu 1,KDd J   C A ? "2PQ?,b0U|,l+`!$Q?,b0U| Ƚ@Cxcdd``> @c112BYL%bpufK#F&&\= s: @> 1,iP?Dd J   C A ? "2ԧK0}S/`!uԧK0Ԓ``hn :CxڝR=KP=Ƃ!S"q8`JDgC&;.R'()I4^R4I=<֦ PëBeEBJ]:h_F:]k|F/%0-~g`zwgꈦ⹪"VcUtGco9>t6vIF|"u/ss혏u\@c[i3Gk}a@RY]O4`!GEe˧>RY]" `Pxڥk@߼IM6ؘz(!zE-gϮi..9{O xDP$Kł i+źT8$mCZ&_}χ@Σg@ s 0,X_sJ:<8 !#e/)FGM1:.7LW#-QpBh06I#JzP0ȴ`}oj:fQ+8x5F0uj)8քF4|2H5wkLG}hqGko%:?֩@G#wYOۜr?]y*X 1-%M( d,PA?Dd J  C A? "2t}uU. S}7`!ut}uU. Sw)`\CxڥkA߼Y&4Q[RڃqORz(+MU*ʚu5cw{3g8 ^||eQSu[jꢮ)e]%(bb$.v<8(njxFo>5??R{-VaGzvx/o(~q߻Z8eZGr[cn8η?$h~xEgxUcO_v[s3ӛ=&C\IO3KxS҂<#qz0OyAD m@g2~Dd 8J  C A? "2K毦{+[:`!K毦{+[`p/PxڥMhAn6ta 1hЃOR@7Q {=?NA'/=z&m⡕j\cwI?Pa.t=KsC>y-!6UʛP8 1CYstV%:unZN܂6gM.jmllP3BU'rb}:^,Poш۸4%%?^d>y3M֟Ѹ_gl*+z2hUĬzo?S N]0E+7O"%ǿy|tqhϔv}c]%NBQ~j) :;>ͨ4fF*;cGf}+1]:>+xH0z'Ƈ|vu8wX}]CQ7bDd J  C A? "2H ie\1[>`!H ie\1: !`Y3PfxڭMhQ罷cm6nE"[rzŋƒh rO~PTOsBRA"XK*1ydž]331H~! "$#X1h4T4z:. gѾX'A`IKcHOIH|v?WXv/)rwsid)}~NטsJna9^8'<7?W*\)Ɗ5Tf}F>SꏿU﫺N?ߍ)#؛nˤPw{!t@}  A%>Nf3;qeP3=%(nx&LpBq [yAT-Ճڃm\LE{x%'5_5\e'j'}#]<Ӆj$?5_?X.:ocio`xcdd`` @c112BYL%bpu30 Bl,4=tĤ\Y\0d.P"CDHg!t?0e 1Dd ,J  C A? "2\ _ލR7{oFD`!g\ _ލR7{!`4P5xڭOhp%6+.f:2\Aqzޱ tUzRu~G7b]7YV \m)TVz%Jd-ܯZfҷ .8\>H|Dc2%x̘Y`ufiwAu&dM1i*Ǐ=nczOsi͒S3ܷw>3qΏ]$H+N~g^m\;9] ̿>Qɰvޟ8暟~8s/_5v]gH8yםf0y y;C;o`H̝7_0y̝-)z[+Ԫs$ۙf<\7ԔPyݽeջ{Ȧ6j_L6M{[ 6PdO u$d@9`,&FF(`TŁSRcgbR x@0@0&dT20\`b e-fI?DFz&br<'5li? f228ըb^?cS,s!"#`!=@Q&=I9 il\ > t7&= L s+Ã"`*bEkr.Y~XA&xB=:ro+x'%AFغrAS08 .v0o8_021)W2ԡRM4~Dd lJ  C A? "2.T,וX-L`!.T,וX-r Xh@2xڥR=KA};wI]XXvJL/ؤ R(' 9Oa2Zig% O#DO,D u%Eday˼yoa  NG1f <^]?)FG1@#qgM&8'IPvJ a~bc8uy֊dT: ҮcWfNyFvʼɤVָ`ھmzco9FcBsr%jRv?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~$'5:CDEFXY`jpods?@Aef1345789:@A& ' ( 0 1 2 3 4 5 6 9 ] ^ ` a c e g h i , - :;<=>BCEcd-.mpqr C$<i/01""" " " """"""w""O##$$%%n&H''(())%************H@H Normal CJOJQJ_HaJmH sH tH J@J Heading 1$$@&a$56CJ\]:@: Heading 2$@&5\F@F Heading 3$$ @&a$>*DA@D Default Paragraph FontVi@V  Table Normal :V 44 la (k@(No List 4 @4 Footer  !.)@. Page Number4@4 Header  !DC@"D Body Text Indent 8^82>@22 Title$a$5\2B@B2 Body TextCJHR@RH Body Text Indent 2 ` (+1:@CFMVZ^bfjmquy| "&*-169=BGJMQUZ^cgkosx| "%(,047:CFIMRX\_closy*+,EFGHINOQWY[abcihkmostu~!$%./0124?@ABCKLMNOPQRTXZhij**++++E+,,,- -).*.B.C.z..../0;0S0k0000111 1 1 111111111111111222222222223333)4*4+4 77n888<9=9>9A9E9G9J9N9P9S9T9V9X9Y9Z99<:::;);*;;;<.</<<<<<O=P=h===>J>>D?@@@cBBBCbCzCDDDDkElEFFG*GGGHHI4IIIJJJJKKKKKKKL}L~LLLLLM7MdM|MMMMOOOOfQRSSSSSTTTTTTTTTTTTTT6UNUUUV7VeV}VWWWWWWWX@ZXZ[-[[[[[Q\R\U^___``` ` ` `````w```aaKa4b5bMbNbcccXdddfgpggggh.h/hhjHkIkakbkkk*l+l,l;l>lmoooooooooooooooooooooooor sss%t=tUttttt.u/uw+wCw[wpxxxxxxx y$y'z?zzzzz{{{{{{|| }<}x}y}j~~e3ˀՁց9Qi #;iɇʇ։&>VߍG_wُ0HI!"#dh7Oݕ-.Fvʘ˘̘͘ΘϘИјҘӘԘ՘֘טؘ٘ژۘܘݘޘߘ™Ù1f%Pcv !-./012345\)z͢ܢ }~VZ\]jkmnoqrsuvwxyzåDצ7oէ֧ ! Sk۫%='?ЮѮҮ#$%&)*-24<@ADGIKLOPQRϱ Og0͵$%=>gh"%'+-./0234Zrn!":R]u߻klWʿ4Jb~23)*B !RS|89QR} "#+,34;<CDFGLMUV[\^_abhiqruvyz}~  !$%)*-.23569:=>ABEFHILMQRTUXY]^bcefhilmpquvyz~  "#&'*+./4578:;=>@ACDGHKLOPRSUV[^_abdehimnstwxz{~000000000(00>0>0>0>0>0>0>0>0>0>0>0>0>0>0>0>0>0>0>0>0>00090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909 09 09 09 090909 09 09 09 09 090909090909090909090909090909090909090909090909090909090909090909090909090909 09 09 09 09 09 09000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000@0h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h00h000j00j00j00j00j00(fh00h00@0@0j00fj00h00h00hiA0h00iA0h00iA0h00/A0h0 0 hA0h0 0 <'A0h00iA0h0 00h0?0@40h0A0Bl0h0C0D0h0E0F@0h0G0H@0h0I0J A0h0K0LDA0h0M0N|A0h0M0N|A0h0M0N|A0h0S0T$B0h0U0V\B0h0W0X00h0Y0Zh0h0[0\0h0]0^ت0h0_0`0h0a0bH0h0c0d0h0e0f0h0g0h0h0i0j(0h0k0l`0h0m0n0h0o0p0h0q0r(0h0s0t`0h0u0v0h0w0x0h0y0z0h0{0|@0h0}0~x0h000h000h00 0h00X0h00 0h00 0h00 0h000h00 h0h00 0h000h000h0000h000h000h000h00(0h00`0h000h00Ж0h000h00\;0h00@0h00@0h00<0h00<<0h00t<0h00<0h00<0h00=0h00T=0h00=0h00=0h00=0h004>0h00l>0h00>0h00>0h00?0h00L?0h00?0h00?0h00o0h00o0h00o0h00(o0h00`o0h00o0h00o0h00o0h00@o0h00xo0h00o0h00o0h00 o0h00Xo0h00o0h00o0h00o0h008o0h00po0h00o0h00L?0h00 o0h00 P o00h00P o0h01 o0h01!o0h01 L!o0h0 1 !o0h0 1 "o0h01#o0h01<#o0h01t#o00h01#o0h01$o0h018$o0h01p$o0h01$o0h01 $o0h0!1"%o0h0#1$P%o0h00?0h01$o0h00LiA0h00 JJJMN&*2=ECCH6OPR&V]`acyjl2tzH|x2̌4ҙ<0lҬh!0&JO[^ڋI"GPmrtvwy{}~ 4#`-;ADeHOYjaglz`ČGTĴǹw% M[jT4x ;Pnpqsuxz|Oo$8:i}))),--*.>.@.z... / /"/;0O0Q0S0g0i0k000000000011333 ::::::;%;';;;;;<<<*<,<W<k<m<t<<<<<<<<<P=d=f====J>^>`>>>>D?X?Z?@@@BBBbCvCxCDDDG&G(GGGGHHHI.I0IIIIIIIJJJJJJKKKBKVKXK_KsKuKKKKKKKHL\L^LM3M5MdMxMzM|MMMMMM6UJULUUUUV3V5VeVyV{VWWWWWW@ZTZVZ[)[+[[[[w``````aaaa0a2a3aGaIaaaa5bIbKbXdldndtdddfggpggggggh*h,hWikimijjjIk]k_k&r:rRTƍڍ܍ G[]_suɎˎҎi}ُ0DFh|~7KMݕ.BDǝ۝ݝ)=?auwzɢˢܢ EY[~ *>@|צ35Oceo(<>Sgi۫%9;SgiĬجڬ';=Znp̮ήʱ̱!57Oce,.͵%9;h|~̶Znp"68"68:NP]qs߻ 8LNl;>RTƿȿ9MOXln02J^`~*>@\prDXZ |9MO}u::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::  "-@GM!!8@3fJnP(  r  6?0<P TB  c $D NB  S D N  3  TB  c $D N  3  TB  c $D  NB  S D  N  3   TB  c $D  N  3   TB  c $D T  C  r  6?0<P Z  S  ?  (b  TB   c $D NB ! S D Bb " 3 G4HKI-B # 3 ?<2 $ # NB %@ S D TB ' c $DTB ( c $D*TB )@ c $D)N * 3 ( N + 3    N , 3  -  (b 5 $TB 6 c $DjJ NB 7 S DjJ&(b 8 TB 9 c $DjJNB : S DjJ6r >  S!TB ? c $DjJ#NB @@ S DjJ"B2 C 3 GnMHI!,TB D@ c $DjJ+N E 3    N F 3  %  N G 3  N H 3  N I 3 ' (2 J 1HB K C D22 L s "ZCAZENHQԔ `T+TAZ`T+TAZ`T+TAZ`T4HB M C D3N N 3 . N O 3 / NB P S D0N Q 3 5 (R R =< S # <NB T S D>>TB U c $D9N W 3 : TB X c $D8N Y 3 ; TB Z c $D7N [ 3 6 ZB \@ s *DoB<2 ] # DZB ^ s *D?NB ` S DjJ@T a C aA N b 3 E N c 3 C ZB d@ s *DjJL<2 e # KZB f s *DMNB g S DjJJT h C hI T i C iH TB j c $DjJON k 3 P T m C mN TB n c $DFN o 3 G ZB q@ s *DjJVTB r c $DjJQN s 3 T N t 3  R  N u 3 !U !B2 v@ 3 G0H&I)SB2 w 3 [TB x c $DdTB y c $DXTB z@ c $DcTB { c $DoZTB | c $DobTB }@ c $DoaN ~ 3 "` "N  3 #W #N  3 $h $N  3 %Y %N  3 &_ &N  3 'g 'N2 @ S GDH!I%fHB @ C D^HB @ C DeHB  C D]HB  C D\TB  c $DojTB  c $DonTB @ c $DomN  3 *i *N  3 )k )N  3 (o (H2  C ln   *$hBC DEF?0y2P<vvpd d  x * {4 ^ +,mll4<g$   g F q0  Z  | v ^x T ,4 {6{h@)XlbC4Oi   } d\XJY3P E 5(m0;_,P=)TTO>,p)|%x7C h|  pdRT@                                       `#" wTB  c $DtTB @ c $D~TB  c $D|N  3 + +N  3 ,z ,N  3 -s -<2  # {N  3 .y .NB  S D}TB  c $DjJvN  3 /x /N  3 0r 0N  3 1u 1(b  qN  3 2p 2x  HBu C DEDFYTM25M0 }  V t    _ _ @ V e    y (B 55 :" F bk6CignS>XuYbh$ $ l  0 R5 E\ @   C  X   ! s^ *  0 I: :A )N ` y    T> u i "l 7 4@,&iA@*[X՜.+,0 hp  Washtenaw Community Collegeuih Rotational Dynamics Title  FMicrosoft Office Word Document DocumentSummaryInformation8QDCompObjWq K t` #  o r@ Ip`SC&g&|iT7vTId:1  XH:>X4v*%.OqWI2;Ydh@                                                `ZB  s *D TB  c $D <2  # TB  c $DjJTB  c $DjJN  3 3 3N  3 4 4N  3 5 5NB  S Dg NB  S Dg N  3 6 6(b  N  3 7 7  NBu C DEDFYTM25M0 }  V t    _ _ @ V e    y (B 55 :" F bk6CignS>XuYbh$ $ l  0 R5 E\ @   C  X   ! s^ *  0 I: :A )N ` y    T> u i "l 7  K t` #  o r@ Ip`SC&g&|iT7vTId:1  XH:>X4v*%.OqWI2;Ydh@                                                `#" TB  c $DNB  S DTB  c $D TB  c $D NB  S DN  3 8 8N  3 9 9N  3 : :N  3 ; ;N  3 < <N  3 = =ZB  s *DjJ<b  # <b  # HB  C DHB  C DHB  C D<2  # TB  c $DZB  s *DZB  s *DjJZB  s *DjJTB  c $DjJT  C > >T  C ? ?T  C @ @T  C A AT  C B BT  C C CN  3 D D(2  NB @ S DTB  c $DT2  c $PGH(*Ib'333N  3 E EN  3 F FN  3 G GN  3 H HN  3 K KN  3 L LN  3 M MNB @ S D(2  NB @ S DTB  c $DT2  c $PGH(*Ib'333N  3 I IN  3 J JN  3 O OTB  c $D(2  (2  HB  C DHB  C DHB  C DHB  C DHB  C DHB  C DHB  @ C DHB   C DHB   C DZB   s *DoTB   c $DjJZB  s *DjJTB  c $DjJZB  s *DjJN  3 N NN  3 P PN  3 Q QN  3 R RN  3 S SN  3 T TN  3 U UNB @ S DT  C X XT  C W WN  3 V VTB  c $DZB @ s *DZB   s *DN ! 3 Y YZB " s *DZB #@ s *DN $ 3 Z ZN % 3 [ [6 &  HB ' C DHB ( C D( ) H2 * C 3fH + C TB , c $DTB - c $DN . 3 \ \N / 3 ] ]N 0 3 ^ ^N 1 3 _ _N 2 3 ` `N 4 3 b b(2 5 H2 9 C 62 :  <2 ; # NB = S DTB > c $DN ? 3 c cN @ 3 d dN A 3 e eN B 3 f fN C 3 g gT2 D c $? `B E 0DTB F c $D HB G C DHB H C DNB I S DTB J c $DT K C hK hT L C iL iT M C jM  jT N C kN  kT O C lO  lT P C mP  mT Q C nQ  nT R C oR oTB S c $DZ T S pTGqH  pH2 U C ?TB V c $D TB W c $D T X C qGPH_  qNB Y S DN Z 3 r rHB [ C D%HB \@ C D$HB ]@ C D#HB ^@ C D"HB _ C D:HB ` C D9HB a C D&HB b@ C D@HB c C D5`B d 0D1 !`B e 0Dg  `B f@ 0Dg N h 3 s sN i 3 t tN j 3 u uNB n S D-TB r@ c $D'NB t@ S D,TB u c $D4TB v c $D3TB y@ c $D2HB z C D8HB {@ C D1TB } c $D7TB ~ c $D6TB  c $D0TB @ c $D=HB  C D>T  C vG HxK  vN  3 w) wN  3 x+ xN  3 y? yN  3 z( zN  3 {< {N  3 |* |N  3 }/ }N  3 ~; ~(b  TB  c $DN  3  TB  c $D TB  c $D H2  C ? R`B  0DZNB  S DUTB  c $DQN  3 P N  3 O N  3 W T  C GxHS B N  3 M N  3 N TB  c $D L(b  KN  3 ] Bb  3 G0*JN  3 F N  3 D TB  c $D IZB  s *D HNB  S DGNB  S DEN  3 X T  C GH C <b  # xYGfZb  s *G0*eZB  s *D dTB  c $D cNB  S DbTB  c $DaN  3 l N  3 ^ N  3 ` N  3 _ N  3 i N  3 h T  C G7H~ g NB  S DjN  3 k T  C aG pHĕ  aT  C GyH"  HB  C D.H2  C VBb  3 G0*SZB  s *D3f ANB  S D[TB  c $DT`B @ 0D3fYN  3 \ (2  \  3 "` B S  ?LX  $%5:;<=>( ) * + , - . 6 7 ^ a c e >?@"""" """***************** 1111111111111111111222222>9?9A9B9C9E9G9H9J9K9L9N9P9Q9T9V9SSTTTTTTTTTTTT_____`````` ` ```ooooooooooooooooooooottttttt#$%&'()*+,-./0123456789:;def\]^_`VZ]^_`abcdefghkos&'*+-./02456789:<=>ABDEGILM·÷ "#%'()+0 t ;t ,Ntdqdt\ q\ At\ +\ tHt@ ,t ,t@ y,tHytd dyt\ \ -t\ \ t ytdd t dt<t"\ Pt!ktHt$st% s,t#@ 't+f t'tH\ etEhTt:xxt9xxt8\ tG4;t6t>24jt@rt?t5htF ;t7StIx &t*t)?t(tD,HtC,t, @\tN,HtOHtP,""tJ tK,tM,6"tL#tQ%t[etZItX,I,tUlltWl<5tY,tSltRxtTHt^'<Yt`Yata$ t\pYYtct]P|tb tn@ L,to@ tiP` |thp@ tgPPtetd,tfhjtmHjdtj,jPrtktr -5ttHdtvx*ts,5tu 4Ptq@ 44t,tyH7H?t4t{44 tw\ 4 tV4VtVtH t4t~t}xHEt|H4tzHtxHltHt,Ht,'Ct ,t,t,/,ttxHt\ ,;t,ttt,tPOlkt4ttt7 t t,t4t<X tP<tln<ntnPnt\ xt@ n\ t:tddr td4t;t L: t &t rtdV t4td4t>Pt4At'Ctddt0tdtXt,,Zt xt;tdUtt*t,tttt,,*t'Ct9tHIt PItH5Qt< ! t@ ety <ItTI t y ItP PtpPtp Pt tOktxQtd4tp7@ tt n\ tP $t8t &tX (#"t"!tX X t(#tlX t(#RtH!;$Wtxt W"tH!H! t\H!tx$ t $stxH$ t $ tHTpt$ H$ t,HtP,tHddt$ P t p\ t ,t ,,,t4 4,t 4,l,t d  t  @ t @ t4t\ @ t@ 4t @ t x t t ,t P,t, !,t X VX ^t!X ?#t tC _t$  t t@ t%\ t$t# t"4t&L#Xt(PPXt' Xt ,Vt-  t,  t+ t)< t4 &t*Yt1!t2pt0t/4t.,t5 pt tTT+ttC,tB? [t@t?wtA\ [,wt>dwt=Hwdt;Ht:c t9 4 tT'pt^ztl !tT xtJ  tHddM tD tR O tPtQwtO3 tNHcdtMxH3 tK tL3 tGdtSd^^tI^d^tF tE^^tY%tW tU@ qtZ%dtXH tVtH ti\ , tjp@ ;thctf cxtexcPctdx[ xct^OHOt]H+ 4Ot\+ \ Ot[\ + 4+ ta44tr4 tp txtdt\ xcttxctnxt4G4ttH/Ht{/Hty,/tv/H/tu /d/tcH/4S t~~t},~,tz,~~t`HfHt_ftxfHttOHt,t tbHt:t,t,t t  t t t p t x Ht@ xxtPldt dt! $tt  tP l t  tX Httt<t X t  <qtUpt= t=(#=t==tt X ,tP[ wt t7 !S tP[l[tH[P[tHH+tH7 HtP tKttdtt<otUtdtoj$;oj\"oj3ojt3oj3oj~"oj43ojt# oj43!oj3"oj\3#oj3$oj :%oja>>>D?[?@@G)GGGHHJJJKKKBKYK_KvKKKKKHL_LdM{M|MMMM6UMUUUV6VeV|VWWWW@ZWZ[,[[[w````aaaJaaa5bLbXdodtddfgpggggh-hjjIk`k&r=rrrrssstt%tzzzzz{{{{{{Q|h|||||| } };}j~~2ʀ%9PQhiʅЅ "#:iLj1H&=>UƍݍG^_v̎Ҏiُ0Gh7Nݕǝޝ)@axz̢ܢ E\~¥צ6Of(?Sj۫%<SjĬ۬'>ZqϮͱ !8Of%<h̶Zq"9"9:Q]t߻8Ol;>Uɿ3Ja~:B:=wgv35F: = ^a]f3= _a$;z~ ''4(:()),---*.A.z..;0R0S0j0k000000001E2G2c2k2333333G4J466\7`7 : :.:4:::;(;;;;;<<-<T<W<n<t<<<<<<P=g=====J>a>k>>>>>D?[???@@BBBBpDuDEEG)GCGNGGGHH4I9IIIJJJKKK?KBKYK_KvKKKKKKKHL_L7M9MdM{M|MMMMMMTN^N2O5OoOyOIPKPQQTT6UMUUUV6VeV|VVVVVWWWWsXxXXX@ZWZ [[[,[[[[[[[^%^|^^__w````aaaJaaaa5bLbHcMcccXdodpdsdtddddfgpggggh-hhhjhhijjIk`kbkgkyl{lBmFmoptppqq&r=rrrrrsssstt%tziznzzzzz{{{{{{{{|O|Q|h|||||||| } };}H}U}z}}j~~2ʀˀ̀%ׁ9PQhiʅЅ "#:;>iˇLj1H&=>UčƍݍG^_v̎Ҏgiُ0GIe<DOWh7Nݕ{ǝޝ5[)@ADaxz̢͢Тܢ @E\~ӣ¥(*z|צ6Of֧%(?ASjkn۫%<FSjĬ۬'>UZqϮҮͱ !8NOf[`%<hǶ̶ķڷZq"9"9:Q]tux߻8Ol¼;;>UɿʿͿQT3Ja~FRT !DEGKVZ\]_`ip .1:<MPY\^a5689;<>?ABDFHJLNSTVZ[]bcnrtvxy{}333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333:B$;,-*.A.z..;0R0S0j0k0000000133 : :::;(;;;;<<-<W<n<t<<<<<<P=g===J>a>>>D?[?@@G)GGGHHII/I0I3I4IJJJKKKBKYK_KvKKKKKL_LdM{M|MMMM6UMUUUV6VeV|VWWWW@ZWZ[,[[[__w````aaaJaaa5bLbXdodtddfgpggggh-hjjIk`k&r=rrrrssstt%tzzzzz{{{{{{Q|h|||||| } };}j~~2ʀ%9PQhiʅЅ "#:iLj1H&=>UƍݍG^_v̎Ҏiُ0Gh7Nݕǝޝ)@axz̢ܢ E\~¥צ6Of(?Sj۫%<SjĬ۬'>ZqϮͱ !8Of%<h̶Zq"9"9:Q]t߻8Ol;>Uɿ3Ja~$;,-*.A.z..;0R0S0j0k0000000133 : :::;(;;;;<<-<W<n<t<<<<<<P=g===J>a>>>D?[?@@G)GGGHHJJJKKKBKYK_KvKKKKKHL_LdM{M|MMMM6UMUUUV6VeV|VWWWW@ZWZ[,[[[w````aaaJaaa5bLbXdodtddfgpggggh-hjjIk`k&r=rrrrssstt%tzzzzz{{{{{{Q|h|||||| } };}j~~2ʀ%9PQhiʅЅ "#:iLj1H&=>UƍݍG^_v̎Ҏiُ0Gh7Nݕǝޝ)@axz̢ܢ E\~¥צ6Of(?Sj۫%<SjĬ۬'>ZqϮͱ !8Of%<h̶Zq"9"9:Q]t߻8Ol;>Uɿ3Ja~ ={\Zn.H .%ʩoPU%U:-g(\ZnkN…*RrN\Zn2'ZUp%k> tdDaruPh^`.h pp^p`OJQJo(oh @ @ ^@ `OJQJo(h ^`OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h PP^P`OJQJo(oh   ^ `OJQJo(h^`.h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L.h hh^h`OJQJo(h 88^8`     OJQJo(oh ^`OJQJo(h   ^ `OJQJo(h   ^ `OJQJo(oh xx^x`OJQJo(h HH^H`OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh pp^p`OJQJo(h @ @ ^@ `OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh PP^P`OJQJo(h^`.h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L.h^`.h   ^ `OJQJo(oh   ^ `OJQJo(h xx^x`OJQJo(h HH^H`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh pp^p`OJQJo(h @ @ ^@ `OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh PP^P`OJQJo(h ^`OJQJo(h pp^p`OJQJo(oh @ @ ^@ `OJQJo(h ^`OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h PP^P`OJQJo(oh   ^ `OJQJo(h^`.h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L.h88^8`.h ^`OJQJo(h L ^ `L.h  ^ `.hxx^x`.hHLH^H`L.h^`.h^`.hL^`L.h ^`OJQJo(h ^`OJQJo(oh pp^p`OJQJo(h @ @ ^@ `OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h ^`OJQJo(oh PP^P`OJQJo(h^`.h^`.hpLp^p`L.h@ @ ^@ `.h^`.hL^`L.h^`.h^`.hPLP^P`L. kNaru%k.%H  tPU%2'ZUrN={g(                                                                                                              @ @ u7k#,/8~8(xNyEmz4MR%|6m~si@::R::t  # $ & ' ( * +23=>BCVW[\^_ablmqrz{|}rrrp@pp,@ppp8@p$pL@p(p*pX@p2ph@p6p8pt@p<p|@pFp@pRp@pXp@pnp@ptp@pxp@p|p@pp@pp @pp4@pp<@pp`@pp@pp@p@pp@ppp@UnknownGz Times New Roman5Symbol3& z Arial9MT ExtraK,Bookman Old Style?5 z Courier New;Wingdings"1h[hff iuiu4dhh 2QHP? (x2Rotational Dynamicsgkapp George Kapp<         MSWordDocWord.Document.89q