ࡱ> Ktb ` ] o ^ bjbjT~T~ 66D% //$P7<{ !0+GL"iLiLLX^`]z_z_z_z_z_z_z$$}\zQhvVXhhz//iLL zIyIyIyh/"iLL]zIyh]zIyIyQrIyL^23ukJ IyIzz0{Iy"v"Iy"Iya(cIyHe<f;aaazzCxaaa{hhhh"aaaaaaaaa :  Physics 103 General Physics I Notes J Kiefer 2013 Table of Contents  TOC \o "1-3" \h \z \u  HYPERLINK \l "_Toc248049723" Table of Contents  PAGEREF _Toc248049723 \h 1  HYPERLINK \l "_Toc248049724" I. Introduction  PAGEREF _Toc248049724 \h 3  HYPERLINK \l "_Toc248049725" A. Dimensions and Units of Measure  PAGEREF _Toc248049725 \h 3  HYPERLINK \l "_Toc248049726" 1. Physical Dimensions  PAGEREF _Toc248049726 \h 3  HYPERLINK \l "_Toc248049727" 2. Systems of Standard Units  PAGEREF _Toc248049727 \h 3  HYPERLINK \l "_Toc248049728" B. Scientific Notation and Significant Figures  PAGEREF _Toc248049728 \h 5  HYPERLINK \l "_Toc248049729" 1. Significant Figures (Digits)  PAGEREF _Toc248049729 \h 5  HYPERLINK \l "_Toc248049730" 2. Scientific Notation and Significant Digits  PAGEREF _Toc248049730 \h 5  HYPERLINK \l "_Toc248049731" C. Vectors  PAGEREF _Toc248049731 \h 6  HYPERLINK \l "_Toc248049732" 1. Coordinate System  PAGEREF _Toc248049732 \h 6  HYPERLINK \l "_Toc248049733" 2. Definition of Scalars & Vectors  PAGEREF _Toc248049733 \h 7  HYPERLINK \l "_Toc248049734" II. Kinematics  PAGEREF _Toc248049734 \h 11  HYPERLINK \l "_Toc248049735" A. One Dimensional Motion  PAGEREF _Toc248049735 \h 11  HYPERLINK \l "_Toc248049736" 1. Variables of Motion  PAGEREF _Toc248049736 \h 11  HYPERLINK \l "_Toc248049737" 2. Motion with Constant Acceleration  PAGEREF _Toc248049737 \h 12  HYPERLINK \l "_Toc248049738" B. Two Dimensional Motion  PAGEREF _Toc248049738 \h 16  HYPERLINK \l "_Toc248049739" 1. Projectile motion  PAGEREF _Toc248049739 \h 16  HYPERLINK \l "_Toc248049740" 2. Uniform circular motion  PAGEREF _Toc248049740 \h 18  HYPERLINK \l "_Toc248049741" III. Dynamics  PAGEREF _Toc248049741 \h 21  HYPERLINK \l "_Toc248049742" A. Newtons Laws of Motion  PAGEREF _Toc248049742 \h 21  HYPERLINK \l "_Toc248049743" 1. First Law  PAGEREF _Toc248049743 \h 21  HYPERLINK \l "_Toc248049744" 2. Second Law  PAGEREF _Toc248049744 \h 22  HYPERLINK \l "_Toc248049745" 3. Third Law  PAGEREF _Toc248049745 \h 23  HYPERLINK \l "_Toc248049746" B. Case Studies  PAGEREF _Toc248049746 \h 24  HYPERLINK \l "_Toc248049747" 1. Weight & Friction & Tension  PAGEREF _Toc248049747 \h 24  HYPERLINK \l "_Toc248049748" 2. Circular Motion  PAGEREF _Toc248049748 \h 29  HYPERLINK \l "_Toc248049749" 3. Sliding & Pulling  PAGEREF _Toc248049749 \h 31  HYPERLINK \l "_Toc248049750" 4. Restoring Forces  PAGEREF _Toc248049750 \h 34  HYPERLINK \l "_Toc248049751" 5. More than One Object  PAGEREF _Toc248049751 \h 35  HYPERLINK \l "_Toc248049752" C. Relative Motion  PAGEREF _Toc248049752 \h 38  HYPERLINK \l "_Toc248049753" 1. Reference Frames  PAGEREF _Toc248049753 \h 38  HYPERLINK \l "_Toc248049754" 2. Accelerated Reference Frames  PAGEREF _Toc248049754 \h 40  HYPERLINK \l "_Toc248049755" IV. Work and Energy and Impulse and Momentum  PAGEREF _Toc248049755 \h 43  HYPERLINK \l "_Toc248049756" A. Physical Work  PAGEREF _Toc248049756 \h 43  HYPERLINK \l "_Toc248049757" 1. Definition of Physical Work  PAGEREF _Toc248049757 \h 43  HYPERLINK \l "_Toc248049758" 2. Two Dimensional Case(s)  PAGEREF _Toc248049758 \h 46  HYPERLINK \l "_Toc248049759" B. Conservation of Energy  PAGEREF _Toc248049759 \h 47  HYPERLINK \l "_Toc248049760" 1. Conservative & Non-Conservative Forces  PAGEREF _Toc248049760 \h 47  HYPERLINK \l "_Toc248049761" 2. Potential Energy Functions  PAGEREF _Toc248049761 \h 48  HYPERLINK \l "_Toc248049762" 3. Mechanical Energy Conservation  PAGEREF _Toc248049762 \h 50  HYPERLINK \l "_Toc248049763" 4. Gravitation  PAGEREF _Toc248049763 \h 51  HYPERLINK \l "_Toc248049764" C. Conservation of Momentum  PAGEREF _Toc248049764 \h 54  HYPERLINK \l "_Toc248049765" 1. Momentum & Impulse  PAGEREF _Toc248049765 \h 54  HYPERLINK \l "_Toc248049766" 2. Collisions & Explosions  PAGEREF _Toc248049766 \h 56  HYPERLINK \l "_Toc248049767" 3. Center of Mass  PAGEREF _Toc248049767 \h 60  HYPERLINK \l "_Toc248049768" V. Rotational Motion & Oscillatory Motion  PAGEREF _Toc248049768 \h 62  HYPERLINK \l "_Toc248049769" A. Rotation of a Rigid Body  PAGEREF _Toc248049769 \h 62  HYPERLINK \l "_Toc248049770" 1. Rotational Kinematics  PAGEREF _Toc248049770 \h 62  HYPERLINK \l "_Toc248049771" 2. Rotational Dynamics  PAGEREF _Toc248049771 \h 64  HYPERLINK \l "_Toc248049772" 3. General Motion of a Rigid Body  PAGEREF _Toc248049772 \h 70  HYPERLINK \l "_Toc248049773" B. Oscillatory Motion  PAGEREF _Toc248049773 \h 76  HYPERLINK \l "_Toc248049774" 1. Simple Harmonic Motion  PAGEREF _Toc248049774 \h 76  HYPERLINK \l "_Toc248049775" 2. Pendulum  PAGEREF _Toc248049775 \h 79  HYPERLINK \l "_Toc248049776" VI. Wave Motion  PAGEREF _Toc248049776 \h 81  HYPERLINK \l "_Toc248049777" A. Waves  PAGEREF _Toc248049777 \h 81  HYPERLINK \l "_Toc248049778" 1. General Wave Properties  PAGEREF _Toc248049778 \h 81  HYPERLINK \l "_Toc248049779" 3. Stretched String  PAGEREF _Toc248049779 \h 85  HYPERLINK \l "_Toc248049780" 4. Pressure waves--Sound  PAGEREF _Toc248049780 \h 87  HYPERLINK \l "_Toc248049781" VII. Thermodynamic  PAGEREF _Toc248049781 \h 89  HYPERLINK \l "_Toc248049782" A. Heat Flow  PAGEREF _Toc248049782 \h 89  HYPERLINK \l "_Toc248049783" B. Laws of Thermodynamics  PAGEREF _Toc248049783 \h 89  I. Introduction We observe the motion of objects in the universe around us, and we experience the passage of time. Those objects move in three dimensions of space. Mathematically, we can treat time as a fourth dimension, or axis, giving rise to the term space-time. We observe further that sometimes an objects motion changes. We want to understand how an objects motion changes, meaning what causes the change in motion. A. Dimensions and Units of Measure 1. Physical Dimensions The dimension of a physical quantity specifies what sort of quantity it isspace, time, energy, etc. a. Fundamental dimensions We find that the dimensions of all physical quantities can be expressed as combinations of a few fundamental dimensions: length [L], mass [M], time [T], and electric charge [Q]. For example, energy  EMBED Equation.3 . The physical quantity speed has dimensions of  EMBED Equation.3 . b. Operational definition Ultimately, the fundamental physical dimensions are defined operationally. That is, we prescribe how the physical quantity is to be measured. The length of an object is defined thusly: (i) lay a meter stick next to the object, (ii) count the marks on the meter stick between the two ends of the object, (iii) that number is the length of the object. 2. Systems of Standard Units In the example of an operational definition of length, we used a meter stick. Of course, this refers to one of the widely accepted units of length, the meter. To facilitate communication, we all agree to express our measurements in terms of a standard set of units of measure. A complete set of units would include a standard unit for every physical dimension. a. International system (SI) The units of the fundamental dimensions in the SI are dimensionSIcgsBrit[L]metercentimeterfoot[T]secondsecondsecond[M]kilogramgramslug These fundamental units are defined operationally. The meter is defined as the distance that light travels in a vacuum in  EMBED Equation.3  of a second. In turn, a second is defined to be equal to the time required for 9192631770 cycles of microwaves whose frequency is just right to excite a particular atomic transition in Cesium. The kilogram is still based on an actual prototype, a chunk of platinum-iridium alloy kept hermetically sealed in France. The SI units of other physical dimensions will be introduced as we go along. They almost all are named after a person. b. Unit conversions Now, the amount of a physical quantity remains the same, no matter what system of units is used to obtain a numerical measure of that quantity. For instance, we might measure the length of an (American) football field with a meter stick and a yard stick. Wed get two different numerical values, but obviously there is one field with one length. Wed say that  EMBED Equation.3 . In other words,  EMBED Equation.3   Suppose we wish to convert 2 miles into meters. [2 miles = 3520 yards.]  EMBED Equation.3  The units cancel or multiply just like common numerical factors. Since we wanted to cancel the yards in the numerator, the conversion factor was written with the yards in its denominator. Since each conversion factor equals 1, we can do as many conversions as we pleasethe physical measurement is unchanged, though the numerical value is changed. c. Abbreviations The names of the standard units also have standard abbreviations. The abbreviation for meter is m, for second s or sec, for kilogram kg. So each time a new unit is introduced, so will be its standard abbreviation. d. Dimensional analysis We can check for error in an equation or expression by checking the dimensions. Quantities on the opposite sides of an equal sign must have the same dimensions. Quantities of different dimensions can be multiplied but not added together. e.g., a proposed equation of motion, relating distance traveled (x) to the acceleration (a) and elapsed time (t).  EMBED Equation.3  Dimensionally, this looks like  EMBED Equation.3  At least, the equation is dimensionally correct; it may still be wrong on other grounds, of course. B. Scientific Notation and Significant Figures Scientific notation is simply a way of writing very large or very small numbers in a compact way.  EMBED Equation.3  1. Significant Figures (Digits) Instruments cannot perform measurements to arbitrary precision. A meter stick commonly has markings 1 millimeter (mm) apart, so distances shorter than that cannot be measured accurately with a meter stick. We report only significant digitsthose whose values we feel sure are accurately measured. There are two basic rules: (i) the last significant digit is the first uncertain digit and (ii) when combining numbers, the result has no more significant digits than the least precise of the original numbers. A third rule is, the exercises and problems in the textbook assume there are 3 significant digits. Therefore, we never include more than 3 significant digits in our numerical results, no matter that the calculator displays 8 or 10 or more. 2. Scientific Notation and Significant Digits The uncertainty in a numerical value may be expressed in terms of a tolerance, as  EMBED Equation.3 . Alternatively, the uncertainty can be shown in scientific notation simply by the number of digits displayed in the mantissa.  EMBED Equation.3  2 digits, the 5 is uncertain.  EMBED Equation.3  3 digits, the 0 is uncertain. Examples  EMBED Equation.3  2 digits  EMBED Equation.3  2 digits  EMBED Equation.3  2 digits C. Vectors Mathematics is a system of logic that we use to discuss physical phenomena. A system of logic consists of entities and of a list of rules that govern how the entities relate to one another. A general term for this is group theory. Of course, this is a very general definition. In physics, entities or concepts are represented mathematically by the mathematical objects scalars, vectors and sometimes matrices. These mathematical objects are defined by the rules that govern them. A particular physical quantity, such as velocity, may be represented by a mathematical vector. We often say then that velocity is a vector quantity. In the same vein, temperature is a scalar quantity. 1. Coordinate System We measure locations in space relative to a coordinate system. Firstly we select the origin of coordinates, and then the directions of orthogonal axes. The choice of coordinate system is entirely at our convenience. However, like a consistent choice of units, we need to be consistent with our coordinate system. Since the directions shown by orthogonal axes are mutually perpendicular, components along different axes are independent of each other. a. Cartesian  b. Polar  c. Three Dimensional The three dimensional Cartesian coordinate system is comprised of three mutually perpendicular, straight axes, commonly denoted x, y, & z. The spherical polar coordinate system is comprised of a radius and two angles, as shown in the figure. Notice how the polar coordinates are defined in terms of the Cartesian system. Any point in space can be uniquely specified by listing three numerical coordinates. 2. Definition of Scalars & Vectors a. Scalar A scalar is a mathematical entity that has one property, magnitude, only. Temperature, mass, speed, and energy are scalar quantities. Scalars obey the familiar rules of addition, multiplication, etc. b. Vector A vector is a mathematical entity that possesses two properties, which physically we call magnitude and direction. Displacement, velocity, acceleration, force, and momentum are vector quantities. Vectors are symbolized graphically as arrows, in text by bold-face type or with a line/arrow on top. A vector may also be represented by an ordered list of numbers, which are called the components: [t,x,y,z]. Written this way, mathematically the vector would be called a row matrix. A column matrix is a vertical list of components:  EMBED Equation.3 . Superscripts and subscripts and implied summation. . . . . . . . c. Unit vectors A unit vector is a vector of magnitude 1. E.g.,  EMBED Equation.3 , where  EMBED Equation.3  is the magnitude of the vector  EMBED Equation.3 . Often, the magnitude of a vector is indicated by the letter without the line on top:  EMBED Equation.3 . d. Components The directions defined by the Cartensian coordinate axes are symbolized by unit vectors,  EMBED Equation.3 .  EMBED Equation.3  e. Adding vectors The sum of two vectors is also a vector.  EMBED Equation.3   EMBED Equation.3  Graphical method: Vectors are represented by arrows, drawn to scale. Place the tail of the 2nd vector on the head of the 1st, preserving the relative orientations. The resultant vector extends from the tail of the 1st to the head of the 2nd vector.  Component method Decompose both vectors along the same set of coordinate axes. The components are scalars, and so can be added in the usual way of scalars.   EMBED Equation.3  f. Multiplying vectors scalar (or dot) productresult is a scalar; the operation is symbolized by a dot.  EMBED Equation.3  The angle  EMBED Equation.3  is the angle from  EMBED Equation.3  to  EMBED Equation.3 . Note:  EMBED Equation.3  and  EMBED Equation.3 . As a matrix multiplication,  EMBED Equation.3 . The magnitude squared of a vector,  EMBED Equation.3 . Vector (or cross) productresult is a another vector; the operation is symbolized by a cross,  EMBED Equation.3 .  EMBED Equation.3  Since the result is a vector, it has magnitude and direction. The rule for evaluating a cross product is as follows:  EMBED Equation.3 , direction perpendicular to both  EMBED Equation.3  and  EMBED Equation.3  according to the right-hand-rule. Alternatively, the cross product can be evaluated by use of a determinant, thusly:  EMBED Equation.3  Note: This must be done in Cartesian coordinates. Evidently,  EMBED Equation.3 . g. Multiplication of a vector by a scalar A vector may be multiplied by a scalar. This affects the magnitude of the vector, but does not affect its direction. The exception to this rule is multiplication by 1. That leaves the magnitude unchanged, but reverses the direction.  EMBED Equation.3  II. Kinematics Kinematics refers simply to the description of motion. We will be studying the mathematical representation of the motion of a point mass. A point mass is an idealized object having no extent, nor volume, nor internal structure. A. One Dimensional Motion The motion is constrained to lie along a single straight line: back & forth, up & down, left & right, etc. The coordinate system consists of a single line, with components on either side of an origin. 1. Variables of Motion We describe the motion of a particle with four motion variables: time, displacement, velocity, and acceleration. We will be looking for a mathematical means of describing the relationships among these variablesan equation of motion. a. Displacement The displacement vector,  EMBED Equation.3 , points from the origin to the present location of the particle. If a particle is at  EMBED Equation.3  at time  EMBED Equation.3  and at  EMBED Equation.3  at some later time  EMBED Equation.3 , then we say the change in displacement is  EMBED Equation.3 . Likewise, the elapsed time is  EMBED Equation.3 . b. Velocity The average velocity during the time interval  EMBED Equation.3  is defined to be  EMBED Equation.3 . Its the time-rate-of-change in the displacement. In terms of vector components, wed write  EMBED Equation.3 . The instantaneous velocity is defined to be  EMBED Equation.3 . c. Acceleration Similarly, the average acceleration is  EMBED Equation.3 . The instantaneous acceleration is  EMBED Equation.3 . d. Motion graphs    2. Motion with Constant Acceleration a. An Equation of Motion To be specific, lets say that the motion is along the x-axis. Then we will deal with x-components only. To develop equations of motion, we begin with the definition of acceleration.  EMBED Equation.3  We might solve for  EMBED Equation.3  as a function of elapsed time and of  EMBED Equation.3 .  EMBED Equation.3  If we say that  EMBED Equation.3 , and  EMBED Equation.3 , and  EMBED Equation.3 , then  EMBED Equation.3 . Or we might begin with the definition of velocity. . .  EMBED Equation.3  But because the acceleration is assumed to be constant, the average velocity can be written also as  EMBED Equation.3 . Set em equal  EMBED Equation.3  Next, we might substitute for the velocity component. . .  EMBED Equation.3 . Finally, we could eliminate  EMBED Equation.3  from the equations by solving the definition of the acceleration for  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3  We have four possible equations, each involving three of the motion variables. They are valid in cases of constant acceleration only. Example: a train traveling on a straight and level track  time, t (seconds)acceleration, ax (m/s2)0 - 10210 40040 - ?-4We might compute the total displacement from time t = 0 until the train comes to rest, given that the train starts also from rest. Firstly, we set up the coordinate system, in this case just the x-axis. We have three different accelerations, so we find the displacement for each segment. Segment 1: We are given the acceleration, elapsed time and initial velocityvxo = 0 m/s. So we choose the equation of motion  EMBED Equation.3 .  EMBED Equation.3  Segment 2: To find the displacement during the second segment, we need the velocity component at the end of the first.  EMBED Equation.3  Then,  EMBED Equation.3  And, of course  EMBED Equation.3 . Segment 3: For this segment, we know x2, vx2, vx3, and ax, but not  EMBED Equation.3 .  EMBED Equation.3  b. Vertical motion in a uniform gravity field A freely falling object is one on which only the gravitational force is acting. Such an object experiences an acceleration toward the center of the Earth. The magnitude of the acceleration due to gravity varies slightly from place to place, and with altitude. However, we will use the average magnitude g = 9.81 m/s2 and pretend the acceleration is constant and uniform. Example: A hot air balloon is rising at a constant speed of 5 m/s. At time zero, the balloon is at a height of 20 m above the ground and the passenger in the balloon drops a sandbag, which falls freely straight downward. What are the height of the sandbag and its velocity as functions of time? How long does it take for the sandbag to reach the ground? Firstly, set up the coordinate system and draw a picture.  Secondly, select the appropriate equations.  EMBED Equation.3  t (s)y (m)vy (m/s)02050.2520.92.550.521.30.11.020.1-4.82.010.4-14.6 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 We could answer the second question numerically, by continuing the table until  EMBED Equation.3 . Thats perfectly legitimate. However, in this case we have an exact expression for the magnitude of the velocity at y = 0 m.  EMBED Equation.3  Notice that we had to decide, based on our physical intuition, what algebraic sign to attach to the velocity component. Finally, we use this final velocity to solve for the elapsed time.  EMBED Equation.3  Now, there is an alternative, one-step solution.  EMBED Equation.3  We cannot have a negative elapsed time, physically, so we ignore the negative root. B. Two Dimensional Motion The general approach is to decompose the vector equations of motion into component equations. We obtain thereby a system of simultaneous equations to solve for the motion along the coordinate axes. 1. Projectile motion The motion of a freely falling body in two or three dimensions is often called projectile motion. a. Set up At any instant, the projectile has displacement  EMBED Equation.3 , velocity  EMBED Equation.3 , and constant acceleration  EMBED Equation.3 . Using the coordinate system shown in the figure, well have  EMBED Equation.3  and  EMBED Equation.3 . We are assuming that the projectile is initially at the origin of coordinates, and that there is no other force, such as air resistance, acting.  The equation for the velocity vector is  EMBED Equation.3 . Decompose  EMBED Equation.3  The components of the initial velocity are  EMBED Equation.3  The equation for the displacement vector is  EMBED Equation.3 . Decompose  EMBED Equation.3  b. Example  i) The maximum height occurs when  EMBED Equation.3 . So we choose the equation involving y, vy, and ay, and solve for y.  EMBED Equation.3  ii) The elapsed time to reach that maximum height is  EMBED Equation.3  iii) We might ask, at what time(s) is the projectile at a specified height, say y = 25 m?  EMBED Equation.3  The significance of the two roots is that the projectile achieves the height of 25 m on the way up and on the way down. iv) The velocity components at t = 0.910 s and at t = 5.61 s.  EMBED Equation.3  time (s)velocity (m/s)0.910 EMBED Equation.3 5.61 EMBED Equation.3  c. Example   EMBED Equation.3  2. Uniform circular motion a. Curvilinear motion Envision an object having, at the moment, a velocity  EMBED Equation.3 , subject to an acceleration,  EMBED Equation.3 . We might decompose the acceleration into components parallel to and perpendicular to the velocity vector. The parallel acceleration component affects the speed of the object, while the perpendicular component affects the direction of the velocity vector, but does not change its magnitude.  b. Circular motion Uniform circular motion refers to motion on a circular path at constant speed. While the magnitude of the velocity is constant, the velocity vector is not constant. The same is true of the acceleration vectorits magnitude is constant but its direction is not. However, the acceleration is always directed toward the center of the circular path. The component of acceleration parallel to the velocity vector is zero. The acceleration component directed toward the center of the circle is called the centripetal acceleration. What acceleration is necessary to cause uniform circular motion? Let the origin be at the center of the circle, as shown.  Consider two successive displacement and velocity vectors.  EMBED Equation.3  By the definition of uniform circular motion,  EMBED Equation.3  and  EMBED Equation.3  and  EMBED Equation.3 . In the limit as  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 .   Representing these differences graphically, we can see by similar triangles that  EMBED Equation.3  The magnitude of the centripetal acceleration always  EMBED Equation.3 . The centripetal acceleration is the perpendicular acceleration component for an object executing curvilinear motion since we can suppose the object is following a circular arc at any time, but perhaps with inconstant radius of curvature. III. Dynamics A. Newtons Laws of Motion In effect, Newtons Laws define force and give two of the attributes of matter. 1. First Law An object in uniform motion remains in uniform motion unless it is acted upon by an external force. In this context, uniform motion means moving with constant velocity. a. Force In other words, if the velocity of an object is changing (whether magnitude, direction, or both), then a force is acting on the object. A force is an external influence that changes the motion of an object, or of a system of objects. We find that there are four fundamental forces in nature, gravity, electromagnetic force, and the strong and weak nuclear forces. All other types of forces that we might give a name to are some manifestation of one of the fundamental forces. Force is a vector quantity having magnitude and direction. The dimensions of force are:  EMBED Equation.3  The SI unit of force is the Newton (N); the common British unit is the pound (lb). Note that for historical reasons, the gravitational force may be called the weight. b. Inertia & mass Two of the attributes of matter are i) resists changes in its motionmatter has inertia, and ii) a force acts between any two pieces of mattermaterial objects or particles exert forces on each other. The quantitative measure of inertia is called the inertial mass of a particle. The SI unit of mass is the kilogram (kg). The inertial mass of a particle is measured by observing the acceleration that results from a known applied force. It is observed that if the same force is applied to two objects, then  EMBED Equation.3 . So, we see that inertial mass is a relative thingwe have to pick some object to define to be 1 kg, and compare all other objects with that standard. 2. Second Law The Second Law makes quantitative the relation between applied force and the resulting change in motion. a. Statement The change in motion of an object is directly proportional to the net external force. Symbolically,  EMBED Equation.3 . If the mass of the object is unchanging, then  EMBED Equation.3 . This is the famous eff equals em ay. Note that the acceleration results from the vector sum of all forces acting on the object. b. Momentum The quantity  EMBED Equation.3  is called the momentum. Like velocity, momentum is a vector quantity. c. Equilibrium Should the vector sum of all forces acting on an object be equal to zero, then  EMBED Equation.3  and the object is said to be in static ( EMBED Equation.3 ) or dynamic ( EMBED Equation.3 ) equilibrium. d. Free body diagram(s) To help us avoid confusion when identifying all the forces acting on an object or particle, we draw what are called free body diagrams. A free body diagram is a sketch of the object only, with arrows indicating each force acting only on that object. For instance, heres a block resting on an inclined surface. We sketch the incline and the block, showing coordinate axes and the angle of incline and whatever might be given, such as initial velocities, etc. Then, to one side, we draw a second sketch, which includes the block only, as if it were isolated.  3. Third Law For every action, there is an equal and opposite reaction. In the old days, the word action was used in the sense that we now use the word force. a. Action & reaction Force is an interaction between two material objects. There is a gravitational interaction between the Earth and the Moon. They exert forces on each other of equal magnitudes but opposite directions. b. Example(s) i) Consider a block resting on a table. The forces acting on the block are its weight,  EMBED Equation.3 , and the contact force,  EMBED Equation.3 , exerted by the table on the block. In turn, the block exerts a contact force,  EMBED Equation.3 , on the table. A contact force is the force one object exerts on another by virtue of being in contact with that second object, i.e., touching. It is the force with which two objects are pressing together. In fact, on the microscopic scale, two objects are not really touching each other.   EMBED Equation.3  and  EMBED Equation.3  form an action-reaction pair, because they are acting on different objects. We would never add  EMBED Equation.3  and  EMBED Equation.3  together. On the other hand,  EMBED Equation.3  and  EMBED Equation.3  are both acting on the block. We add them to obtain the net force on the block. The reason this is confusing is that  EMBED Equation.3 + EMBED Equation.3  may = 0 and  EMBED Equation.3 = -  EMBED Equation.3 . ii) Consider a block being dragged along a frictionless surface.  Decompose the forces acting on the block. x:  EMBED Equation.3  y:  EMBED Equation.3  Notice that in this case, the reaction of the block on the surface is  EMBED Equation.3 , but  EMBED Equation.3 . B. Case Studies 1. Weight & Friction & Tension a. Weight & scales Weight is the term we use to refer to the force of gravity near the Earths surface, or near a planetary bodys surface, or near a moons surface, etc. In such a situation, the magnitude of the weight is  EMBED Equation.3 , and its directed downward.  We do not measure weight of an object directly. Instead, we place the object on a scale. The number we read off of the scale is actually the contact force exerted upward by the scale on the object. If the object is in equilibrium, then we infer that the weight has the same magnitude.  EMBED Equation.3  b. Friction When two objects slide against one another, there is a resistive force, called friction. On the microscopic scale, it arises from the electromagnetic forces that act between molecules and atoms. Macroscopically, we observe that the frictional force  EMBED Equation.3  is different if there is relative motion between the two objects, or the two surfaces. If there is no relative motion, then we have static friction,  EMBED Equation.3 , where N is the contact, or normal, force and  EMBED Equation.3  is the coefficient of static friction. If there is relative motion, then we have dynamic or kinetic friction,  EMBED Equation.3 , where  EMBED Equation.3  is the coefficient of kinetic friction. In either case, the direction of the frictional force is opposite to the motion (kinetic) or opposite to the pending motion (static). Its direction is always tangent to the surfaces at the point or area of contact. The value of the coefficient of friction depends on the nature of the two surfaces at which the friction is acting. See the table in the textbook. Notice that the static friction does not have a single value, but can be any value from zero (if N = 0) up to some maximum, at which point the objects start to slide, and the friction becomes kinetic friction. Notice, too, that  EMBED Equation.3  in every case. i)  Decompose the forces acting on the block. x:  EMBED Equation.3  y:  EMBED Equation.3  Lets say F is given, then there are two unknowns, N and ax. Fortunately, we have two equations. We can solve the y-equation for N and substitute that into the x-equation.  EMBED Equation.3  ii) Now suppose a second block is resting atop the block which is resting on the floor.  It is important to see that the applied force  EMBED Equation.3  is acting only on the lower block (m), and not on the upper block (M). Lets call the blocks 1 (upper) and 2 (lower). A free body diagram of block 1 might look like this:  On the other hand, a free body diagram of block 2 will look like this. (The block 1 is still present to show the origin of the contact force between the blocks).  EMBED Equation.3  is the contact force exerted on block 1 by block 2. So, the contact force exerted by block 1 on block 2 is  EMBED Equation.3 . The floor exerts a contact force on block 2,  EMBED Equation.3 , but not on block 1, cause block 1 is not touching the floor! There are frictional forces between the blocks, and between block 2 and the floor, but they are not the same.  We apply Newtons 2nd Law to each of the blocks. (Notice the action-reaction pairs.) Block 1 x:  EMBED Equation.3  y:  EMBED Equation.3  Block 2 x:  EMBED Equation.3  y:  EMBED Equation.3  The frictional forces would be found by setting  EMBED Equation.3  and  EMBED Equation.3 . The coefficients of friction are  EMBED Equation.3  between the blocks and  EMBED Equation.3  between the floor and block 2. We see, for instance, that  EMBED Equation.3  and that  EMBED Equation.3 , but  EMBED Equation.3 ! And so on. c. Tension Tension is a force that tends to pull apart. In our common introductory examples, we consider a cord or rope, but more generally any material object, such as a rod or beam or wooden board can be subject to tension. The ideal cord is massless, non-stretchable and perfectly flexible. This means that it can sustain tension, but cannot resist compression along its length. It means also that the tension in the cord is the same throughout its entire length. i) a mass hanging at rest from a cord   EMBED Equation.3  Decompose (we have only the vertical direction, anyhow.) y:  EMBED Equation.3  ii) a mass hanging from two cords  Again, we only have one dimension. But, we have two objectsthe hanging mass, m, and the pulley, M. As before,  EMBED Equation.3 . For the pulley,  EMBED Equation.3  y:  EMBED Equation.3  We have made use of the fact that the tension is the same T2 on either side of the pulley. iii) a rope stretched across a chasm  The forces on the hanging mass are  EMBED Equation.3  Decompose x:  EMBED Equation.3  y:  EMBED Equation.3  2. Circular Motion Earlier, we described uniform circular motion, defining the centripetal acceleration, etc. Now, we examine how applied forces produce circular motion. We saw that for circular motion, the radial acceleration had to be equal to  EMBED Equation.3 . That is the a in the F = ma! Any number of forces may cause the centripetal acceleration. a. Ferris wheel Imagine one of the cars on a vertical Ferris wheel. Consider the forces acting on the car at some arbitrary position on the circle. The velocity vector is always tangential to the circle, but may not be of constant magnitude.  A free body diagram for the car--  Rather than x- and y-components, we have radial and tangential components. Radial component:  EMBED Equation.3  Tangential component:  EMBED Equation.3  At the highest ( EMBED Equation.3 ) and lowest ( EMBED Equation.3 ) points, we have special cases. In those cases, the tension is entirely radial because there is no tangential component of the weight.  EMBED Equation.3  We might be given v, r, and m and be asked to find the Tr. b. Banked curve Consider an automobile driving around a banked curve. In cross section, it looks like an object on an inclined plane. Viewed from above, its an object traveling on a circular arc, perhaps with constant speed.  In connection with potential sliding up or down the incline, the friction is static friction if the car is not sliding. Decompose: x:  EMBED Equation.3  y:  EMBED Equation.3  Make the substitution for N in the x-component equation:  EMBED Equation.3  Now, for instance, say  EMBED Equation.3 ; what must the angle  EMBED Equation.3  be if the car is not to slide? We are looking for the limiting case. Notice that the mass of the car appears on both sides of the equation, so it will divide out.  EMBED Equation.3  c. Merry-go-round, or Orbit Consider a satellite moving around the Earth in a circular orbit at constant speed. According to Newtons Law of Gravitation, the Earth exerts a force on the satellite equal in magnitude to  EMBED Equation.3 , where G is the universal gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the distance between the satellite and the center of the Earth. In Newtons Second Law  EMBED Equation.3  Notice that the mass of the satellite divides out! We might solve for v in terms of the orbit radius, r.  EMBED Equation.3  The larger the orbit, the slower the satellite moves. 3. Sliding & Pulling a. Inclined plane Consider an object, such as a box or trunk, resting or sliding on an inclined plane surface. That is, the surface is straight and flat, but inclined at an angle to the horizontal.  We apply Newtons 2nd Law to the mass, m.  EMBED Equation.3  It is convenient to orient the coordinate axes parallel and perpendicular to the inclined surface, because we expect the mass to slide along that incline. Decompose x:  EMBED Equation.3  y:  EMBED Equation.3  What are the components of the weight this time?  EMBED Equation.3  The components of the weight vector form a triangle similar to the triangle formed by the inclined surface. What if there is friction? What if there is an additional applied force?  We apply Newtons 2nd Law to the mass, m.  EMBED Equation.3  It is convenient to orient the coordinate axes parallel and perpendicular to the inclined surface, because we expect the mass to slide along that incline. Decompose x:  EMBED Equation.3  y:  EMBED Equation.3  Had there been friction, the  EMBED Equation.3  would have been parallel to the inclined surface, and we would use the y-component equation to find  EMBED Equation.3 , whence  EMBED Equation.3 . b. Pulleys & cords Imagine an object hanging from an ideal cord that is wound around a set of two pulleys. We assume the cord is perfectly flexible and does not stretch, and has no mass of its own. This means that the tension within the cord is the same along its entire length.  Newtons 2nd Law applies to each object. Well just need vertical components, so for the pulley  EMBED Equation.3 . For the hanging mass,  EMBED Equation.3 . In other words, we find that we need to pull on the cord with a force (T2) of only half the hanging weight (W) to support it. Thats why we have pulleys. They provide mechanical advantage, similarly to the use of a lever. 4. Restoring Forces a. Spring If a spring is stretched by an external force, the spring exerts a force in the opposite direction tending to restore it to its original length. For instance, when a mass is hung on a spring, the spring is stretched by the weight of the mass. The spring exerts a force upward on the mass. On the other hand, when an external force compresses a spring (shortens its length), the spring pushes back.  A linear restoring force is such that  EMBED Equation.3 . This is the magnitude; the direction is always such as to restore the spring to its resting length, the length of the spring when no external force is acting on it. The proportionality constant, k, is called the force constant. Applying Newtons 2nd Law to the figure above, with the hanging mass in equilibrium,  EMBED Equation.3  Notice that we have taken the +x direction to be in the direction of increasing spring-length. So, x is positive downward in this case. Often we find it convenient to place the origin at the end of the unstretched spring. Therefore, xo = 0. b. Pendulum  If a mass hung on a light cord is pulled to one side, and released, it will tend to swing back toward its starting point. One way to analyze the pendulum is to regard it as a case of circular motion. We have radial and tangential components:  EMBED Equation.3  radial:  EMBED Equation.3  tangential:  EMBED Equation.3  5. More than One Object a. Inclined plane and pulley  We will ignore the mass of the pulley, and also assume that the pulley has no friction. The inclined surface also is frictionless. There are two bodies in the system, and we draw free-body diagrams of each of them.   Notice that a + direction of motion is indicated on the sketch. We are allowed to use different coordinate axes for the two bodies, for our own convenience, as long as we remain consistent with the (+/-) directions. For m1, there is only vertical motion.  EMBED Equation.3  For m2, there are x- and y-components.  EMBED Equation.3  x:  EMBED Equation.3  y:  EMBED Equation.3  What we appear to have is 3 unknowns and 2 equations: T, a1, and a2x. However, because the two masses are connected by a non-stretchable cord,  EMBED Equation.3 ! So, we can solve for T and for a1.  EMBED Equation.3  b. Atwoods machine Consider two masses hung over a massless, frictionless pulley by an ideal cord.  Once again we have two objects, with a free-body diagram for each object.   Newtons 2nd Law applies to each object. m1:  EMBED Equation.3  m2:  EMBED Equation.3  Notice the + direction is defined to be clockwise around the pulley. Too, since the pulley has no mass nor friction, and the cord is ideal,  EMBED Equation.3  and  EMBED Equation.3  (magnitude).  EMBED Equation.3  2 unknowns & 2 equations. . . .  EMBED Equation.3  and  EMBED Equation.3  C. Relative Motion Query: To what extent does it matter, physically, where we stand while we observe the motion of an object? 1. Reference Frames We always measure the position and velocity and acceleration vectors relative to a convenient coordinate systeman origin and coordinate axes. The coordinate system is called a reference frame. a. Position vectors Consider a point in space. Its position is the vector that runs from the origin to the point.  The two position vectors,  EMBED Equation.3  and  EMBED Equation.3 , point to the same point, yet they have different magnitudes and directions. The origin of the primed reference frame is a point in space, too, and has a position vector,  EMBED Equation.3 , relative to the unprimed reference frame. The rule for vector addition tells us the relation between  EMBED Equation.3  and  EMBED Equation.3 . Further, it may be that the primed frame is moving with relative velocity,  EMBED Equation.3 , with respect to the unprimed frame.  EMBED Equation.3  Notice that the relative velocity is measured in the unprimed frame of reference. Further, well label the reference frames as the S-frame and S-frame rather than spell out the words primed and unprimed. b. Relative velocities The time rate of change in position is the velocity.  EMBED Equation.3  At this point well assume that  EMBED Equation.3 , whence  EMBED Equation.3  (  EMBED Equation.3  These three equations together constitute the Galilean Transformation of coordinates.  Heres an example. Let the S-frame be fixed to the Earths surface while an S-frame is pained on the wall inside the cabin of a jet liner flying overhead. Wed say that the S-frame is moving with velocity  EMBED Equation.3  with respect to the S-frame. Now, as shown in the sketch, we choose the coordinate axes, both sets, to make things as simple as possible. For instance, the orientations of the two frames are the same. We often imagine that the two origin coincide at time t = 0. We get a simplified sketch of the situation. Keep in mind that whatever object is located at the point P is not attached to either coordinate system. It moves through space, and we are observing that motion from one reference frame or another.  2. Accelerated Reference Frames a. Inertial reference frame An inertial reference frame is one in which Newtons Laws of motion are validobserved accelerations result from observable physical forces, such as a spring, or gravity, or tension in a cord, etc. b. Accelerated reference frame  Non-inertial reference frames are accelerated reference frames. That is,  EMBED Equation.3 .  EMBED Equation.3  i) Consider a railroad boxcar. Inside, from the ceiling, hangs ball of mass, m, on a cord. Imagine two observers, one standing outside the boxcar and the other standing inside the boxcar. Both apply Newtons 2nd law to the hanging mass. As seen by the observer inside the car:   EMBED Equation.3  Decompose x:  EMBED Equation.3  y:  EMBED Equation.3  Now heres the thing. We normally think that the direction that an object hangs on a cord defines the vertical. The gravitational force is presumed opposite in direction to the tension. As seen by the observer outside the boxcar:   EMBED Equation.3  Decompose: x:  EMBED Equation.3  y:  EMBED Equation.3   Now we bring the two views together by noting that  EMBED Equation.3 . Evidently,  EMBED Equation.3  The two observers interpret the weight differentlydifferent in magnitude and in direction. c. On the rotating Earth Consider a plumb bob hanging at rest, and let the origin of our coordinate system be at the bob, so  EMBED Equation.3 . The real physical forces on the bob are gravity and the tension in the cord. We have also  EMBED Equation.3  and  EMBED Equation.3 . As seen in the rotating frame:  EMBED Equation.3 . Since we use plumb bobs to determine the vertical direction, wed say that  EMBED Equation.3 . However, since the Earth is rotating,  EMBED Equation.3  does not point toward the center of the Earth. As seen in an inertial frame, Newtons 2nd Law appears as  EMBED Equation.3 , where  EMBED Equation.3  is the centripetal acceleration, directed toward the Earths rotational axis. Now,  EMBED Equation.3 , so we can write  EMBED Equation.3 , or  EMBED Equation.3 . The deviation from the true vertical (pointing toward the center of the Earth), is a function of the latitude,  EMBED Equation.3 , since  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Our scales actually measure  EMBED Equation.3 . However, since the Earth rotates slowly,  EMBED Equation.3 . The same would not be true on a planet that rotates much faster than the Earth. See for instance the science fiction novel A Mission of Gravity by Hal Clement. IV. Work and Energy and Impulse and Momentum A. Physical Work In Physics and engineering, the word work has a specific definition. Its basically equal to a force times the displacement that occurs while the force is acting. 1. Definition of Physical Work a. Path integral Imagine a particle moving along a curved path through space. As it moves, an external force acts on the particle. The vector  EMBED Equation.3  is an infinitesimal displacement along the pathits always tangent to the path.  EMBED Equation.3   EMBED Equation.3  Work has dimensions of  EMBED Equation.3 . The SI unit of work is the Joule (J).  EMBED Equation.3 . The British unit of energy is the foot-pound (ft-lb). The British Thermal Unit (BTU) is also used. Since the scalar product is involved, the result is a scalar.  b. One dimensional cases i) constant force  EMBED Equation.3  For instance, friction acting on a sliding block:   EMBED Equation.3  Notice that if  EMBED Equation.3  or  EMBED Equation.3  or  EMBED Equation.3 , then W = 0. No work is done. ii) varying force We suppose that Fx = constant during an infinitesimal displacement,  EMBED Equation.3 . A tiny bit of work is done by the force  EMBED Equation.3 . The total work done during a displacement from x1 to x2 is the sum  EMBED Equation.3 . If we graphed F vs x, wed see that this is the integral of F(x) from x = x1 to x = x2.  In the limit as  EMBED Equation.3 , the sum becomes an integral  EMBED Equation.3  As an example of a force that varies with x, consider a spring.  The spring exerts a force on the block (mass = m)  EMBED Equation.3 , where x is the displacement of the block from the equilibrium position (at the unstretched length of the spring).  EMBED Equation.3  This is the work done by the spring on the block. c. Kinetic energy At the same time,  EMBED Equation.3 . During the infinitesimal displacement, the elapsed time is  EMBED Equation.3 . We assumed that the force is constant during that  EMBED Equation.3 , so  EMBED Equation.3 .  EMBED Equation.3  What we see is that there is a quantity that changes while the work is being done. We call that quantity the kinetic energy. It depends on the mass of the object and its speed. The equation  EMBED Equation.3  is called the Work-Energy Theorem. When a force does work on an object, it changes the objects kinetic energy. [Check the dimensions of K.] d. Power Power is the rate at which work is done. If W is the work done during an elapsed time,  EMBED Equation.3 , then the average power during that interval is  EMBED Equation.3  The SI unit for power is the Watt:  EMBED Equation.3 . A kilowatt is 1000 Watts. The electric bills often mention kilowatt-hours. Thats one kilowatt times one hour = 3.6x106 Joules. Imagine a locomotive engine dragging a train along a straight track at a constant speed of 20 m/sec. Lets say the engine exerts a force of 105 N and pulls the train 100 m. The locomotive engine expends  EMBED Equation.3   EMBED Equation.3  2. Two Dimensional Case(s) a. Projectile A free-falling projectile is acted upon only by gravity. As the projectile falls, gravity does work on it, changing its kinetic energy.   EMBED Equation.3  We might solve for v.  EMBED Equation.3  Notice that this gives us only the speed; we have to decide on the direction based on our physical intuition. Notice that all that matters are the magnitudes of the initial and final velocities, not their directions! Energy has no direction; its a scalar. b. Inclined plane An object on an inclined plane is acted upon by a number of different forces, gravity, friction, the normal force, perhaps a tension in a cord or some other applied force. Each and every force does work on the object as it moves on the inclined surface, from xo to x.   EMBED Equation.3  The total work done by the forces is just the sum of these  EMBED Equation.3  This total work in turn equals the change in the objects kinetic energy that took place while it slid along the surface.  EMBED Equation.3  B. Conservation of Energy 1. Conservative & Non-Conservative Forces a. Conservative forces The work done by a conservative force is independent of the path taken from the initial position to the final position. On the other hand, if the work done does depend on the path taken, then the force is a non-conservative force, or a dissipative force.  b. Potential energy For a conservative force, the work done during a displacement depends only on the coordinates of the beginning and end points. We define a function, U, such that (for a one-dimensional case)  EMBED Equation.3  The function U is a function of x, and is called the potential energy function. To obtain U for a given conservative force, solve the expression for the work done by the force, W, for U2, and let U1 be zero. Because we are interested only in the change in potential energy, we can set the zero anywhere we like.  EMBED Equation.3  c. Dissipative forces A dissipative force is a force like friction. The work done by a frictional force depends on the distance an object slides from point 1 to point 2. If a longer path is followed between points 1 and 2, then more work is done. We cannot define a potential energy function for a dissipative force. 2. Potential Energy Functions a. Spring   EMBED Equation.3 , where x is the change from the resting length of the spring-- EMBED Equation.3 . The work done by a spring is  EMBED Equation.3  According to the definition of a potential energy function,  EMBED Equation.3 , so evidently,  EMBED Equation.3  b. Uniform gravity   EMBED Equation.3  We identify the potential energy function for a uniform gravitational field to be  EMBED Equation.3  c. Working backwards We defined the potential energy in terms of work done by a force. The potential energy, then, is related to the integral of the force over a displacement. The converse process is to derive the force expression by taking the derivative of a given potential energy function. Thus, the potential energy function for an elastic, or spring, force is  EMBED Equation.3 , where x is the displacement from the equilibrium position of an object. Take the derivative. . .  EMBED Equation.3  Or, take the potential energy for uniform gravity (up is positive).  EMBED Equation.3  We can extend this to two or three dimensions, with some hypothetical potential energy function, U(x,y,z).  EMBED Equation.3  These three expressions are summed up in one by using whats called the gradient operator.  EMBED Equation.3  Qualitatively, the force is in the downhill direction of the potential energy function. 3. Mechanical Energy Conservation a. Mechanical energy The mechanical energy is defined to be  EMBED Equation.3 , the sum of kinetic and potential energies. Of course, we add them only at the same position or the same time. Now, if only conservative forces are acting, then  EMBED Equation.3 . Evidently,  EMBED Equation.3  We have what is called a Conservation Principle or Law. If only conservative external forces are acting, then the total mechanical energy of a system is conserved. Dissipative forces, such as friction, do not conserve the total mechanical energy. In the particular case of friction, some of the mechanical energy of the system is dissipated, or lost in raising the temperature of the system. b. Examples i) gravity near the Earths surface  EMBED Equation.3  Notice that energy is a scalar quantity, so it will give us information about the magnitude of velocity, but not the direction. ii) spring  EMBED Equation.3  4. Gravitation a. Universal Law of Gravitation  EMBED Equation.3  The direction is along the radius, r, and always attractive. G is the universal gravitational constant,  EMBED Equation.3 . The gravitational masses of the two objects are m1 and m2. It turns out that the gravitational mass is the same as the inertial mass, though that need not have been so.  EMBED AutoSketch.Drawing.7  Two objects exert a gravitational force on each other, equal and opposite.  EMBED Equation.3  Near the surface of the Earth,  EMBED Equation.3 . b. Potential energy The work done by the force of gravity while one mass is displaced from r1 to r2 is  EMBED Equation.3    SHAPE \* MERGEFORMAT  For a system of objects, the total potential energy is the sum over pairs:  c. Orbit Consider two objects in orbit around each other, such as the Earth and the Moon. The total mechanical energy of the system is  EMBED Equation.3   d. Escape velocity Heres a question: How fast must an object, starting on or near the Earths surface, be moving in order to escape from the earths gravity? We use the conservation of mechanical energy to answer the question. i) What constitutes escape? Well say that if the object just comes to rest as  EMBED Equation.3 , it has escaped. ii) Well assume that once started, the object coastsno rocket engines, etc. iii) We have, therefore,  EMBED Equation.3 .  EMBED Equation.3  Any object, starting at the Earths surface with a speed of  EMBED Equation.3  will escape from the Earth. Notice, too, that the direction of the initial velocity does not matter (as long as its not straight down or something like that). A black hole is an object that is so dense that the escape velocity exceeds even the speed of light. Therefore, even light cannot escape, and the object would be invisible to an external observer. C. Conservation of Momentum 1. Momentum & Impulse a. Impulse Recall the definition of acceleration:  EMBED Equation.3 . Multiply both sides by  EMBED Equation.3 . . .  EMBED Equation.3  On the left hand side is what is called the impulsethe force multiplied by the elapsed time. On the right hand side is the change in momentum during the same elapsed time. Extended to two or three dimensions, the impulse is  EMBED Equation.3 , and momentum is  EMBED Equation.3 . Analogous to the Work-Energy Theorem, we have the Impulse-Momentum Theorem:  EMBED Equation.3 . Impulse and momentum are vector quantities. If the force varies with time, we have to integrate over time.  We define an impulsive force to be a force that acts for a very short time, and is much larger in magnitude than other forces acting. Usually, we do not know the exact time-dependence of the impulsive force. Therefore, rather than integrate the force over time, we measure the  EMBED Equation.3  that results from the impulsive force. Example: A person of 70 kg mass approaches the ground (wearing a parachute) at  EMBED Equation.3 . If he/she lands stiff-legged,  EMBED Equation.3  in the space of  EMBED Equation.3 . What is the average impulsive force on the person? (That is, average F over 0.002 s.) We have one-dimensional motion, with downward being (+).  EMBED Equation.3  If the parachutist allows his/her legs to bend on landing, the  EMBED Equation.3 and  EMBED Equation.3  This is the reason we want our car fenders to crumple when the car runs into something. The crumpling has the effect of prolonging the collision thereby reducing the average force exerted on the car. The deformation of the fenders also dissipates some of the kinetic energy. Example: What is the force exerted on a wall when a ball is thrown against it?  We dont know  EMBED Equation.3  during the impact, nor even the duration of the impact for that matter. We can evaluate  EMBED Equation.3 .  EMBED Equation.3  Lets say that m = 0.5 kg, vx1 = 40 m/s, and vx2 = -20 m/s. Then the change in momentum is  EMBED Equation.3  This is the impulse on the ball! The ball exerts an equal and opposite impulse on the wall. If the impact lasts  EMBED Equation.3 , then the average force on the wall is  EMBED Equation.3  b. Conservation of momentum Now, if the net external force is zero, then  EMBED Equation.3  The momentum of an isolated system is constant; its conserved. If we divide the impulse by the time, then we get Newtons 2nd Law  EMBED Equation.3  2. Collisions & Explosions a. Collisions In a simple, idealized collision, we imagine two objects which collide, exerting only an impulsive force on one another during the collision. Visualize two billiard balls. We presume that no other net force is acting on the two balls, so the system of two balls is isolatedthe total momentum is conserved.  EMBED Equation.3  Momentum is a vector; the total momentum is the vector sum of the momenta of the two balls. We distinguish between elastic collisions and inelastic collisions. An elastic collision conserves the total kinetic energy as well as the momentum. In an inelastic collision, the total kinetic energy decreases to some degree. In a perfectly inelastic collision, the two objects stick together, and become one. To sum up: for elastic collisions,  EMBED Equation.3  and  EMBED Equation.3 . For inelastic collisions,  EMBED Equation.3 , but  EMBED Equation.3 . The  EMBED Equation.3  stands for the change(s) in kinetic energy of the objects. The objects grow warmer, or possibly emit sound waves, etc. i) elastic (1 dimensional)   EMBED Equation.3  Because we can choose a reference frame to suit our convenience, we can usually set  EMBED Equation.3 . ii) elastic (2-dimensional)   EMBED Equation.3  iii) plain inelastic  EMBED Equation.3  ii) perfectly inelastic  In a perfectly inelastic collision, in effect  EMBED Equation.3 .  EMBED Equation.3  b. Explosions We might regard an explosion as a reverse perfectly inelastic collision. A single object separates into two or more objects. The conservation of momentum still holds.   EMBED Equation.3  c. Rockets Suppose the total mass of a moving object is not constant. Say the net external force acting on an object (such as a rocket or a rain drop) is  EMBED Equation.3 . Assume that during a short time interval,  EMBED Equation.3 , the  EMBED Equation.3  is approximately constant. Then the impulse delivered to the mass, m, is  EMBED Equation.3 . Further suppose that during that interval  EMBED Equation.3  the mass changes by an amount  EMBED Equation.3 . The change in momentum that results is  EMBED Equation.3 .  EMBED Equation.3  We want to rewrite this in terms of the change in velocity of the mass, m, and the relative velocity of the m and  EMBED Equation.3 . Namely,  EMBED Equation.3  and  EMBED Equation.3 .  EMBED Equation.3   EMBED Equation.3  The impulse, then, is  EMBED Equation.3 . We may as well just let m +  EMBED Equation.3  be m at this point.  EMBED Equation.3  Divide by  EMBED Equation.3 ;  EMBED Equation.3 . Recap:  EMBED Equation.3  is the velocity of the object (rocket or rain drop),  EMBED Equation.3  is the velocity of the  EMBED Equation.3  relative to the object, and  EMBED Equation.3  is the absolute value of the time rate of change in the mass of the object. Actually, we have to be careful of the directions of things. As derived here, if  EMBED Equation.3  is leaving the object, then the object is losing mass and  EMBED Equation.3  is in the opposite direction as  EMBED Equation.3 . Consider a rocket in the absence of gravity or any other external force.   EMBED Equation.3   EMBED Equation.3 .  EMBED Equation.3  3. Center of Mass a. Position   EMBED Equation.3  b. Velocity  EMBED Equation.3  The total momentum of a system of particles is equal to the total mass of the system times the velocity of the center of mass.  EMBED Equation.3  If no net external force acts on any part of the system, then  EMBED Equation.3  is constant, and so is  EMBED Equation.3 . The individual parts of the system may exert forces on each other, but those do not affect the motion of the center of mass. c. Acceleration On the other hand, if one or more external forces acts on the system, then  EMBED Equation.3  is not constant. The sum of all forces acting on all parts of the system is  EMBED Equation.3  However, because of Newtons 3rd Law,  EMBED Equation.3 .  EMBED Equation.3  Consider a projectile that fragments in mid flight.  V. Rotational Motion & Oscillatory Motion Until now, we basically ignored the fact that physical objects occupy volume, and have shapes. We treated everything as if it were a particlean object with mass but no volume or internal structure. A. Rotation of a Rigid Body A rigid body is an object whose shape and dimensions (length, width, height, radius, etc.) do not change. The distances between any two mass elements that comprise the body do not change.  When a rigid body rotates about an axis, all parts of the body rotate at the same rate. However, because different parts of the body are farther from or closer to the axis, all parts do not have the same linear, or tangential speed.  1. Rotational Kinematics a. Rotational variables In the rotational equations of motion, we always use radians as the angular measure, rather than degrees. Recall that the radian as an angular measure is the ratio between the arc length and the radius of the arc. By convention, Greek letters are used to represent angular variables. namedefinitionangular displacement,  EMBED Equation.3 angular velocity component,  EMBED Equation.3  EMBED Equation.3 angular acceleration,  EMBED Equation.3  EMBED Equation.3  b. Equations of rotational motion The equations for cases of constant angular acceleration are the same as those for constant acceleration. Just imagine all of those divided through by a radius.  EMBED Equation.3  Every point in and on the body executes circular motion.  EMBED Equation.3  Example: a rigid body, such as a disk, rotates about an axis perpendicular to the plane of the disk.   EMBED Equation.3  The average angular velocity is  EMBED Equation.3 . The angular acceleration is  EMBED Equation.3  [Since  is constant,  EMBED Equation.3 .] 2. Rotational Dynamics a. Moment of inertia A moment of ____________ is equal to __________ times a squared distance from a point, or axis. The moment of inertia is mass times radius squared from a point.  For individual, or discrete particles, EMBED Equation.3 . For an object having a continuous mass density,  EMBED Equation.3 , we have to integrate over the volume of the object.  EMBED Equation.3  The dimensions of moment of inertia are [M][L]2. Examples. . . .   Parallel Axis Theorem: The moment of inertia is just a sum of scalars. Therefore, we can divide an inconveniently shaped body into sections that are easier to handle. By a similar token, we can construct the moment of inertia of a body about an arbitrary axis from the moment about an axis passing through the center of mass of the body.   b. Kinetic energy Each mass element that comprises the rigid body is moving in a circle with speed  EMBED Equation.3 . We could add up a total kinetic energy for all the mass elements:  EMBED Equation.3  Notice that the angular velocity is common to all the mass elements. c. Angular momentum Consider a mass, m, with momentum,  EMBED Equation.3 . Relative to a point O, the mass has an angular momentum equal to  EMBED Equation.3  The magnitude is  EMBED Equation.3 , while the direction of the angular momentum vector is obtained with the right-hand-rule.  d. Torque The time-rate-of-change of the angular momentum is  EMBED Equation.3  Define the torque, or moment of force, about the point O  EMBED Equation.3   The magnitude of the torque about the point O is  EMBED Equation.3  and the direction is obtained from the right-hand-rule. e. Conservation of total angular momentum For a system of particles or of rigid bodies, the total angular momentum of the system about the point O is the vector sum of all the individual angular momenta, about the same point O.  EMBED Equation.3  If the system is isolated, so that the net external force is zero, then  EMBED Equation.3  is conserved. That means both in direction and in magnitude. A rotational collision:  A particle of mass m = 0.05 kg and speed v = 25 m/s collides with and sticks to the edge of a uniform solid disk of mass M = 1 kg and radius R = 0.3 m. If the disk is initially at rest on a frictionless axis through its center, what is its angular velocity after the collision? The total angular momentum of the system is conserved.  EMBED Equation.3 .  EMBED Equation.3  f. Work done by a torque  EMBED Equation.3  is the force applied a the point P.  EMBED Equation.3  is a short displacement.  EMBED Equation.3  is the position vector from the selected axis, O, to the point P.  EMBED Equation.3  is the angle between  EMBED Equation.3  and  EMBED Equation.3 .  EMBED Equation.3  The work done on a rigid body by an external torque is equal to the change in rotational kinetic energy that results.  EMBED Equation.3  The power expended is  EMBED Equation.3 . 3. General Motion of a Rigid Body a. Rolling along When a rigid body is both translating and rotating, we can divide its total kinetic energy into two parts.  EMBED Equation.3  Here, the angular speed is about the center of mass. b. Case studies i) Consider an ideal cord wound around a solid cylinder of radius R = 0.5 m and mass M = 10 kg. The cylinder is set on a horizontal axis and mass of m = 4 kg is hung on the free end of the cord. Whats the torque experienced by the cylinder and whats the acceleration downward of the mass, m?  First thing we do is assume that the cord does not slip on the cylinder. We have a system of two bodies, to which we apply Newtons 2nd Law. The mass, m:  EMBED Equation.3  The cylinder, M:  EMBED Equation.3  The tension would tend to cause a clockwise rotation, so it produces a negative torque. Because the cord does not slip on the cylinder,  EMBED Equation.3 . We have therefore two equations and two unknowns.  EMBED Equation.3   EMBED Equation.3  ii) Rolling down an incline   c. Gyroscope & precession  EMBED Equation.3  Notice that  EMBED Equation.3  is not down, but sideways. The spinning disk does not fall, but rather precesses in a horizontal arc.  4. Static Equilibrium a. Conditions for equilibrium of a rigid body For a rigid body to be in equilibrium, both the translational and rotational accelerations must be zero. Therefore, the sum of the external forces and the sum of the external torques must both be zero.  EMBED Equation.3  The torques must be computed about the same axis, but the choice of axis is arbitraryit need not be necessarily through the center of mass. b. Case studie(s) i) A uniform beam of length r = 4 m and mass 10 kg supports a 20 kg mass as shown. The beam in turn is supported by a taut wire. Whats the tension in the wire? The beam is in static equilibrium. Therefore, the sum of forces on the beam is zero, and the sum of torques exerted by those forces is zero. Fb is the weight of the beam itself.  EMBED Equation.3  To compute torques, we need to specify an axis of rotation. We choose one that will eliminate some of the unknowns, for instance the end of the beam where it joins the wall. The force  EMBED Equation.3  exerts zero torque about that point. Then the sum of torques reduces to:  EMBED Equation.3  We return to the force component equations in order to find the force the wall exerts on the beam.  EMBED Equation.3  ii) ladder leaning against a wall The mass of the ladder is M. The angle is . The length of the ladder is L.  EMBED Equation.3   EMBED Equation.3  The forces on the ladder are the normal contact forces and friction exerted by the wall and the floor and gravity, which acts at the center of mass of the ladder. Each of those forces exert a torque about some selected axis. Typically, the selected axis is either the top or bottom of the ladder, or both. It may be the ladders center of mass, as well. If, for instance, the wall is not smooth, then there are potentially 4 unknown force components, and therefore 4 equations are needed.  EMBED Equation.3  Since this is a two-dimensional situation, we indicate the directions of the torque by +/- signs. B. Oscillatory Motion Here, we speak of things that go back and forth in a regular way. 1. Simple Harmonic Motion Harmonic means there is a sine or cosine in it. a. Description Let y be the displacement of a particle from its equilibrium position. Then by definition, if the particle is executing simple harmonic motion its position as a function of time is  EMBED Equation.3  A is the amplitude, the maximum displacement either side of equilibrium. f is the frequency of oscillation, in cycles/second.  EMBED Equation.3  is a phase factor, which depends on the initial y at t = 0. Graphically,  We find sometimes other parameters useful, such as T is the period of the oscillation.  EMBED Equation.3 .  EMBED Equation.3  is the angular frequency, in radians/second.  EMBED Equation.3  [This was called the angular velocity, and had a line on top in the context of rotational motion. Here, omega is a scalar, not a vector.] b. Equations of motion  EMBED Equation.3   Notice that  EMBED Equation.3 ; the acceleration is proportional to the displacement from equilibrium, but in the opposite direction. This is exactly what we saw with springs,  EMBED Equation.3 . In Newtons 2nd Law  EMBED Equation.3  Evidently, linear restoring forces give rise to simple harmonic motion. c. Energy The total mechanical energy of a simple harmonic oscillator of mass, m, is  EMBED Equation.3  Given v, we can find the corresponding y; given y, we can find the corresponding v. Notice that the total energy is proportional to the square of the amplitude of the oscillation. d. Applying initial conditions to the equations of motion  EMBED Equation.3  Suppose that at t = 0, y = yo and v = vo. Then  EMBED Equation.3   EMBED Equation.3  Given the position and velocity of the mass at one time (usually t = 0) we are able to specify its motion exactly at any other time. 2. Pendulum a. Non-simple pendulum Consider a mass hanging on the end of a light string, cord or stiff rod. Two forces act on the mass, the tension in the string, cord, or rod and gravity. The string, cord or rod constrains the mass to move on a circular arc of radius equal to the length of the string, cord or rod.  The force of gravity exerts a torque on the mass, about the pivot point of the pendulum  EMBED Equation.3  Notice that the mass has divided out. Notice that the torque is restoring, tending to return the mass to  EMBED Equation.3 . b. Simple pendulum Now we make a simplification by assuming that the pendulum swings in a small arc. In that case,  EMBED Equation.3 , in radians.  EMBED Equation.3  This exactly simple harmonic motion, if we identify  EMBED Equation.3 . The frequency, or period, of a simple pendulum depends only on its length (and on g), not on how much mass is hung on the end. c. Physical pendulum Of course, all pendula are physical, except for politico-socio-economic pendula. The Physical pendulum is a pendulum in which the mass is not concentrated at the end of a string, cord, or rod. In the equation of motion, we use the moment of inertia. Assuming small angular displacement (a simple physical pendulum, as it were),  EMBED Equation.3  In this case, the length,  EMBED Equation.3 , is the distance from the pivot point to the center of mass of the pendulum. We can identify the angular frequency as before:  EMBED Equation.3 . VI. Wave Motion A. Waves A mechanical wave is a disturbance that propagates through a material medium. Electromagnetic waves are moving electromagnetic fields and do not require a material medium to propagatethey can propagate through a vacuum. All forms of waves have certain wave-like properties in common. [Propagate is a word meaning to travel, but it means more than that. It implies that the wave is traveling by reproducing itself.] Here, we will be discussing mechanical waves. 1. General Wave Properties a. Wave pulse  i) wave speed is a property of the medium. ii) shape of the wave pulse is unchanged as it travels iii) two or more wave pulses that exist at the same place & time in a medium addsuperimpose. Mathematically, we write the displacement of the medium at position x and at time t as  EMBED Equation.3  b. Periodic waves A periodic wave consists of a series of identical wave pulses, so the wave pattern is repeating. The easiest periodic wave mathematically is the sine wave (or cosine).  The amplitude is A, the wavelength is  EMBED Equation.3 ; the frequency is f; the wave speed is  EMBED Equation.3 . As the wave passes a specified point in space, the medium there executes simple harmonic motion.  EMBED Equation.3  The (+) sign yields a wave traveling from right to left; the (-) sign a wave traveling form left to right. We define the wave number it be  EMBED Equation.3 .  EMBED Equation.3  c. Transverse and longitudinal waves The direction of the disturbance of the medium may be either perpendicular to the direction of wave-travel, or it may be parallel to the direction of wave-travel. The former is called a transverse wave, while the latter is called a longitudinal (or compression) wave. For the time being, we wont concern ourselves with the distinction, but waves in a stretched string are transverse, while sound waves are longitudinal. d. Refection  e. Superposition If two or more wave disturbances exit at the same place in a medium and at the same time, they simply add to give a net displacement of the medium. It is entirely possible then, to have waves traveling in a medium that superimpose to give a zero net displacement of the medium.  i) standing waves Consider two sine waves traveling in a medium in opposite direction and having the same amplitude, A.. If their wavelengths are the same, they will superimpose to form what is called a standing wave pattern.  Every point in the medium oscillates with a constant amplitude. At some points, spaced one half wavelength apart, the medium does not oscillatethe amplitude is zero. These points are called nodes. Midway between the nodes are the antinodespoints whose amplitude of oscillation is a maximumtwice the amplitude of the individual waves. The standing, or stationary wave pattern does not move through the medium.  EMBED Equation.3   ii) beats Consider two waves traveling through a medium, in the same direction but slightly different frequencies. They will superimpose to form a disturbance whose amplitude rises and falls periodically. These patterns are called beats. The frequency of the beats is equal to the difference in the two original frequencies. The effective frequency within the envelope is  EMBED Equation.3 , the average of the two separate frequencies.  EMBED Equation.3  f. Energy transport While the medium in which the wave propagates does not flow from one place to another, the wave disturbance nonetheless carries energy from one place to another. Each mass element, dm, of the medium executes simple harmonic motion. K is the restoring force constant. Its related to the frequency by  EMBED Equation.3 .  EMBED Equation.3  Over one cycle, the cosine-squared and sine-squared average to  EMBED Equation.3 . The total mass of the medium spanning one cycle (or one wavelength) is  EMBED Equation.3 , where  EMBED Equation.3  is the mass per unit length of the medium.  EMBED Equation.3  In terms of the wave speed, c,  EMBED Equation.3 . The energy flux is the power transported through the medium by the wave:  EMBED Equation.3 . The intensity is the power per unit area through which the power is transported:  EMBED Equation.3 , where  EMBED Equation.3  is the mass per unit volume.  3. Stretched String a. Tension One end of a stretched string is fixed, as if tied to a wall. That end cannot be displaced. If the free end is disturbed by an outside force, then a pulse is generated which then propagates along the string, until it encounters the wall. The fixed end cannot move, so a reflected pulse is created which travels back along the string, but in reversed orientation. Regardless of the exact shape of the pulse, it travels with the same speed. We observe that the wave speed in a stretched string is  EMBED Equation.3 . By extension to three-dimensional media,  EMBED Equation.3 . We can derive the wave speed in the string by applying Newtons 2nd Law to a short segment of the string.  EMBED Equation.3  But also,  EMBED Equation.3  This is the wave equation for a wave travelling along the x-axis with speed  EMBED Equation.3 . b. Normal modes Because of the boundary conditions imposed on the string, only certain waves are able to persist in the string. In particular, since the ends of the string are fastened, they cannot movethe ends must be nodes. Only waves for which the string length equals an integral number of half-wave lengths will persist in the string. Put another way, the string will vibrate at only certain discrete frequencies, corresponding to those allowed wavelengths. The allowed frequencies are called normal modes of the string. The normal modes are also known as the strings natural frequencies of vibration. Every physical system has natural vibration modes, based on its dimensions and elasticity. For a string, those are its length and its tension and its mass per unit length.   The normal modes frequencies for the string are  EMBED Equation.3 , where v is the wave speed in the string. c. Spectra An arbitrary wave form may be expressed as a superposition of sine waves having differing amplitudes. This exactly analogous to expressing a function as a Fourier Series.  The Fourier components are just the normal modes, fn. 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Entropy 3. Heat Engines     PAGE  PAGE 2  Lets take a closer look at our (+/-) signs. The force is in the r direction. The displacement is in the +r direction, and r is always (+). The anti-derivative gives a (-) sign. The W is negative change in U. Hence, U is (-).    EMBED Equation.3     Note: the units are a part of the measurement as important as the number. They must always be kept together. 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̶T28Lf-Hq9nU< y5 HwNgY-ǥ!2 !N 2wSC7R# s7T ;j3mL#OB3Uz1.42'LڜֈGGq{7'UB2`(8SoC Lڬ&ofj= &QpvOoEdN+:$ⱏ\9*4 FMicrosoft Equation 3.0 DS Equation Equation.39q <a x >=v x t=v x2 "v x1 t 2 "t 1CompObjafObjInfocEquation Native d_1271684746F023023Ole hCompObjifObjInfokEquation Native l: FMicrosoft Equation 3.0 DS Equation Equation.39q v x2 FMicrosoft Equation 3.0 DS Equation Equation.39q_12716847760F023023Ole mCompObjnfObjInfo p{ v x1 FMicrosoft Equation 3.0 DS Equation Equation.39qF v x2 =v x1 +<a x >t Equation Native q:_1271685550DI#F023023Ole rCompObj"$sfObjInfo%uEquation Native v_1271684982(F023023Ole y2 "t 1 () FMicrosoft Equation 3.0 DS Equation Equation.39q#Ё t 1 =0 FMicrosoft Equation 3.0 DS EqCompObj')zfObjInfo*|Equation Native }?_1271684999&?-F023023Ole ~CompObj,.fObjInfo/Equation Native Puation Equation.39q4H: v x1 =v o FMicrosoft Equation 3.0 DS Equation Equation.39q_1271685764+N2F023023Ole CompObj13fObjInfo4Equation Native T_12716857727F023023Ole CompObj68f8G <a x >=a x FMicrosoft Equation 3.0 DS Equation Equation.39q[+ v x (t)=v o +a x "tObjInfo9Equation Native w_1271685148<F023023Ole  FMicrosoft Equation 3.0 DS Equation Equation.39q@x  <v x >a"xt FMicrosoft Equation 3.0 DS EqCompObj;=fObjInfo>Equation Native \_1271685239:!AF023023Ole CompObj@BfObjInfoCEquation Native uation Equation.39qd, <v x >a"v x2 +v x1 2 FMicrosoft Equation 3.0 DS Equation Equation.39q_1271685311FF023 23Ole CompObjEGfObjInfoHEquation Native _1271685714KF 23 23Ole CompObjJLfp: xt=v x2 +v x1 2x 2 =x 1 +v x2 +v x1 2"t FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoMEquation Native ,_12716858205SPF 23 23Ole  x 2 =x 1 +v x1 +a x t+v x1 2"t=x 1 +v x1 t+12a x t 2 FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjOQfObjInfoREquation Native -_1271685971UF 23 23h t FMicrosoft Equation 3.0 DS Equation Equation.39qh t=v x2 "v x1 <a x >Ole CompObjTVfObjInfoWEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39q؍ x=v x2 +v x1 2v x2 "v x1 a x =_1271686037hZF 23 23Ole CompObjY[fObjInfo\Equation Native _1271753954_F 23 23Ole CompObj^`fv x22 "v x12 2a x FMicrosoft Equation 3.0 DS Equation Equation.39qx ` x=x o +v xo t+12a ObjInfoaEquation Native _1271754050]gdF 23 23Ole x t 2 FMicrosoft Equation 3.0 DS Equation Equation.39qP`< x 1 =x o +v xo t+12a x t 2 =0 m+CompObjcefObjInfofEquation Native l_1271754236iF 23 230 m/s"10 s+12"2 m/s 2 "10 s() 2 =100 m FMicrosoft Equation 3.0 DS Equation Equation.39q , v x1 =v     sYZ !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWX[\^]_`bacedfhgijklmonpqrtuwvyxz{|}~Ole CompObjhjfObjInfokEquation Native  xo +a x t=0 m/s+2 m/s 2 "10 s=20 m/s FMicrosoft Equation 3.0 DS Equation Equation.39q\ȿ|= x 2 =x_1271754353b{nF 23 23Ole CompObjmofObjInfopEquation Native x_1271755053sF 23 23Ole CompObjrtf 1 +v x1 t+12a x t 2 =100 m+20 m/s"30 s+12"0 m/s 2 "30 s() 2 =700 m FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfouEquation Native q_1271754825xF 23 23Ole UȊ\ v x2 =v x1 =20 m/s FMicrosoft Equation 3.0 DS Equation Equation.39qx ` tCompObjwyfObjInfozEquation Native -_1271754837vq}F 23 23Ole CompObj|~fObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qI| v x32 "v x22 =2"a x x 3 "x 2 ()x 3 =x 2 +v x32 "v x22 2"a x =700 m+0"20 m/s() 2 2""4 m/s 2 ()=750 m FMicrosoft Equation 3.0 DS Equation Equation.39q_1271765591lF 23 23Ole CompObjfObjInfoEquation Native _1271765786F 23 23Ole CompObjf  !$'()*+.123456789:;>CHMPSVY\]^adefgjmnqtux{|}~x ` y=y o +v yo "t+12a y "t 2 v y =v yo +a y "t FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native )_1271765820F 23 23Ole   x ` " FMicrosoft Equation 3.0 DS Equation Equation.39q x ` " FMicrosoft Equation 3.0 DS EqCompObj fObjInfo Equation Native  )_1271765923F 23 23Ole CompObjfObjInfoEquation Native 9uation Equation.39qx ` yE"0 m FMicrosoft Equation 3.0 DS Equation Equation.39qnx ` v y2 _1271766346F 23 23Ole CompObjfObjInfoEquation Native _1271766508F 23 23Ole "CompObj#f"v yo2 =2a y y"y o ()v y2 =v yo2 +2a y y"y 0 ()=5 m/s() 2 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ObjInfo%Equation Native &E_1271767513F 23 23Ole , y "v yo a y =0 m/s"20 m/s"9.8 m/s 2 =2.59 s FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj-fObjInfo/Equation Native 0_1336309418F 23 23O ` y=y o +v yo "t+12a y "t 2 0 m=20 m+5 m/s"t+12"9.8 m/s 2 ()"t 2 t=""5 m/s() 25 m 2 /s 2 "44.9 m/s 2 ()"20 m()  9.8 m/s  =520.429.8 s=2.59 s or -1.57 s FMicrosoft Equation 3.0 DS Equation Equation.39qOle <CompObj=fObjInfo?Equation Native @-"j 2r  FMicrosoft Equation 3.0 DS Equation Equation.39q"ܣ 2v  FMicrosoft Equation 3.0 DS Eq_1336309425 F 23 23Ole ACompObjBfObjInfoDEquation Native E-_1336309432F 23 23Ole FCompObjGfuation Equation.39q"pL 2a  FMicrosoft Equation 3.0 DS Equation Equation.39q;3W a x =0 ObjInfoIEquation Native J-_1271837619F 23 23Ole KCompObjLfObjInfoNEquation Native OO_1271837647F 23 23m/s FMicrosoft Equation 3.0 DS Equation Equation.39q;' a y ="gOle QCompObjRfObjInfoTEquation Native UC_1427899614F 23 23Ole WCompObjXfObjInfoZ FMicrosoft Equation 3.0 DS Equation Equation.39qؼ 2v =2v  o +2a "t=v xo ,v yo []+a x ,a y []"tEquation Native [_1271837881F 23 23Ole _CompObj`f FMicrosoft Equation 3.0 DS Equation Equation.39q; v x =v xo +a x t=v xo v y =v yo +a y t=v yo +("g)"tObjInfobEquation Native c_1271837996F 23 23Ole h FMicrosoft Equation 3.0 DS Equation Equation.39q;Xd v xo =v o cos o v yo =v o sin oCompObjifObjInfokEquation Native l_1336309371F 23 23Ole oCompObjpfObjInforEquation Native s FMicrosoft Equation 3.0 DS Equation Equation.39q"{Ԡ 2r =2r  o +2v  o t+122a "t 2 FMicrosoft Equation 3.0 DS Eq_1271838184F 23 23Ole vCompObjwfObjInfoyuation Equation.39q;& l x=x o +v xo ty=y o +v yo t+12a y t 2 =y o +v yo t+12("g)"t 2Equation Native zB_1271838446F 23 23Ole CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q;3W v y =0 m/s FMicrosoft Equation 3.0 DS EqObjInfoEquation Native O_1271838556F 23 23Ole CompObjfObjInfoEquation Native  _1271838844F 23 23uation Equation.39q;W v y2 "v yo2 =2a y y"y o ()y=y o +v y2 "v yo2 2a y =0 m+0 m 2 /s 2 "40 m/s"sin53 o () 2 2"9.8 m/s 2 ()=52.1 m FMicrosoft Equation 3.0 DS Equation Equation.39q;C`a v y =vOle CompObjfObjInfoEquation Native _ yo +a y tt=v y "v yo a y =0 m/s"40 m/s"sin53 o "9.8 m/s=3.26 s FMicrosoft Equation 3.0 DS Equation Equation.39q_1271839139F 23 23Ole CompObjfObjInfo; W y=y o +v yo t+12a y t 2 12a y t 2 +v yo t+y o "y()=0t 2 +2v yo a y t+2y o "y()a y Equation Native <_1271840390*F 23 23Ole CompObjf=0t 2 "6.52 s"t+5.10 s 2 =0 s 2 t="("6.52 s) (6.52 s) 2 "4"5.1 s 2 2=0.910 s and 5.61 s FMicrosoft Equation 3.0 DS Equation Equation.39q;yW v x =v xo +a x t=v o cos o =40 m/s"cos53 o =24.1 m/sv y =v yo +a y t=v o sin oH$%ObjInfoEquation Native _1271840976F 23 23Ole  +("g)"t=40 m/s"sin53 o +("9.8 m/s 2 )"t     =31.9 m/s"9.8 m/s 2 "t FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjfObjInfoEquation Native w_1271840989F 23 23;[W v x =24.1v y =23.0 FMicrosoft Equation 3.0 DS Equation Equation.39q;_X v x =2Ole CompObjfObjInfoEquation Native {4.1v y ="23.0 FMicrosoft Equation 3.0 DS Equation Equation.39q;CW v x =v xo +a x t=20 m/sv y =v yo +a?#" `[2LK<~T1ʻ `!LK<~T1, .@(G yxڥKhAgvf$Őڇijy4FTXA(lxib0A=Aܼ"VT*("A[Dj}8lԔ4ooK2!&P%Ò -(Efa~iƹv/!*P4 5ASǸ@gbHOv @ҵG?2 swgS\9;||}lf0 h;8gv&zczzWg- v,S 1Һpur%~(GEk!az{X5k!뽖ZN;[rҬ]ҮL1tkx]Z;2b}B~&ZuF1âǜNm}T&9Jf|þt*q|!o*>%FJ2I~MkB_Q)gL~n,!OC8qDZR ?ѠU։;ebD]tkpRΚܯS0y ,  rWyP/]H7igW\7}ˆ]4s_5q4Rs||>زiz:1~DF~ԃEN186Xa4l 2Dd \ ^ c $A[? ?#" `\22y4uUcl ^Q `!V2y4uUcl ĸ@!؊3(+$xڝ1l@=;uMBBTN@) L %@j#,OJL@T LLH@ H{wɎ[>ݻgA @|2X 8 ;M+]#ݬp,`q~íWDM}Mֺ]\LƮa_ɪr\:wν?B{~IkZ|Wmxt|À Jq{dӠTy8W&TH~]!&Y⻊ߖI۞ɯğ_-P\@7oZ`?C\>ܣ  q&{ ]2@Dd \ _ c $A\? ?#" `]2yDfů`!;l `!dyDfů`!; 1 Mx2xڥMlEggzۄmJ7iu9UqJ#1Ʒ8DJ ¥ AjPz*=TT\ 82p(PZi~ϛv@!Ӷ^DVx,F0FF Ť?r)/ ؝hjPgVx p-0Sy9oFۘE_$L i9t916l!v<*DI^BN򐮷o7c<~i|td\'^<ݺB<խ+Ţ/]]yWWfEKUc D9k_1+dczä9⛨qRK?jjv5 ><+n? 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