ࡱ> 1305@ &bjbj22 ,NXX6666R::::t$R#.#######$$R'C#+C#66CX#"""<6lL#"#"""," _9:= "#n#0#"'!'"D&,6666'"p"C#C#RR 6"RR6Jills Angular Velocity Notes 1. First draw a unit circle (leave room for a larger circle of radius 3 for later) with an arrow (vector) from the origin to the point A(1,0). (Draw it bigger than the figure shown here!)  Time Out! Whats a radian? A radian is a radius! Look at the bigger circle above, do you see how much a radian is? (Its the length of that B-ray!) So if you go ccw around the B-circle 1 radian youll still be in the first quadrant? If you go 3 radians ccw, youll end up in Quadrant II just a little before the 180o line, yes? Question: How many of these radians (radiuss, I know, radii) will go all the way around? About 6.28 radians(?) Well, circumference equals  EMBED Equation.DSMT4  radiuss,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Also: 360o =  EMBED Equation.DSMT4 Rad which is where the conversion formula: EMBED Equation.DSMT4  comes from. Question: Can you close your eyes and see the B-ray rotate at 3 radians/sec? (Good!) Now look at point A and point B together as they rotate ccw. Assume the B-arrow is right on top of the A-arrow. Question: Which has the greater angular velocity? Answer: The same.  EMBED Equation.DSMT4  Question: Which point moves with the greater linear speed? Answer: B (3 times as fast) Using our geometry of similar figures, since the ratio of the radii is 1 to 3, the ratio of the arc lengths  EMBED Equation.DSMT4  is also 1 to 3. So B travels 3 times the distance in the same amount of time. Linear Speed is distance traveled divided by time. So vB = 3vA On the unit circle the distance traveled numerically equals the angular distance traveled when measured in radians, but in general were not on the unit circle so another crucial formula is:  EMBED Equation.DSMT4  *using rad/sec for omega ex/ vB = 3(ft/radius)  EMBED Equation.DSMT4   EMBED Equation.DSMT4 =  EMBED Equation.DSMT4  Angular Velocity Notes p.2 2. Period vs Frequency  EMBED Equation.DSMT4  Question: If omega equals  EMBED Equation.DSMT4 , how long does it take to cover 360o? Answer: 360/30 = 12 sec. This is the period (fundamental period). We write T = 12 sec/revolution or just T = 12 sec. (The capital T tells us its special! ( ) Question: If it takes 12 seconds per revolution, then how many revolutions are turned in 1 second? Answer:  EMBED Equation.DSMT4  Notice the units of T and f are reciprocals. Period is the reciprocal of frequency. Most math books leave out the concept of frequency. (Youll see angular frequency.) In physics, youll learn that the SI units of frequency used to be cps (cycles per second). Now, the units are call hertz. So when the frequency of the voltage is 60Hz that means the voltage oscillates at 60 cycles per second. RPM (revolutions per minute) is a very common frequency unit. Question: If T = 12 sec, what is omega,  EMBED Equation.DSMT4  Answer:  EMBED Equation.DSMT4  A big, big (really big!) formula to memorize is:  EMBED Equation.DSMT4  *using radians Ex/ What is the angular velocity of a pulley that turns 20 revolutions every 60 seconds? Looking at the units (rev/sec), we see that were give frequency: f =  EMBED Equation.DSMT4 rev/sec. So using our formula (that we just memorized!)  EMBED Equation.DSMT4  Angular Velocity Notes p. 3 3. Jills Problems The tractor, pulley, or gear-ratio problem. In these problems (unlike pts A & B), the omegas (angular velocities) are not equal but the linear speeds are equal. Also in gear-ratio problems, linear speed is equal,  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4 . The Gear-Ratio Theorem In words, if the ratio of 2 gears is 3 to 1, then their angular velocities and frequencies (since  EMBED Equation.DSMT4 ) are in a ratio of 1 to 3 (a reciprocal relationship).  Solution: First of all, note were give fD = 20 rpm =  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4  Secondly, since  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  (a)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  vD = 2( EMBED Equation.DSMT4 ) =  EMBED Equation.DSMT4  (b) vC = vD since the belt rides on both wheels (like a pulley problem). (c) fc = ? From our gear-ratio theorem:  EMBED Equation.DSMT4  so  EMBED Equation.DSMT4   EMBED Equation.DSMT4  fc =  EMBED Equation.DSMT4  Oh yeah, they wanted the answer in rev/min!!! We coulda/shoulda left fC in rev/min, so fC = 20 rev/min &  EMBED Equation.DSMT4  EMBED Equation.DSMT4  fC =  EMBED Equation.DSMT4 rpm Im betting the books answers for parts (a) and (b) were in ft/min??? The teeter-totter problem! 37 degree angular displacement in 0.7 sec. Stan is 8 ft from the fulcrum. Ben is 5 ft from the fulcrum. Find the angular velocity for each.  EMBED Equation.DSMT4  (1 degree with using  EMBED Equation.DSMT4 ) = .923 rad/sec Angular Velocity Notes p. 4 3. Jills Problems (continued) The train wheel with the extra 1 inch flange problem! The wheel that rides on the rail has a radius of 15 inches. Its linear speed (with respect to its center) matches the linear speed of the train (and the center of the wheel) as it moves down the railroad track. The flange is sort of a second wheel which extends 1 inch further to keep the 15 inch wheel on the rails. Let RA = 15 inches and RB = 16 inches. If vTrain = 60 mph = 88.2 ft/s, What is the speed on the outer edge of the flange, vB = ? Solution: In this problem,  EMBED Equation.DSMT4  and vA = vTrain = 88.2 ft/s First find omega first using  EMBED Equation.DSMT4 . Watch out, well need RA = 15in = 1.25 ft (changing inches to feet) so 88.2 ft/s =  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . Now use  EMBED Equation.DSMT4  again but with vB =  EMBED Equation.DSMT4 = 1130 in/s = 94.1 ft/s > 88.2 ft/s. 4. Concept Questions/Problems What is the angular velocity of a person on earth with respect to the earths axis. Using  EMBED Equation.DSMT4  have the student spend time trying to figure out what the time period, T, is here. Answer: T = 1 day or 24 hours or hence,  EMBED Equation.DSMT4  = 15 deg/hr. What is the angular velocity of the earth about the sun? (Assume a circular orbit.) Using  EMBED Equation.DSMT4  have the student spend time trying to figure out what the time period, T, is here. Answer: T = 1 year or 365 days or hence,  EMBED Equation.DSMT4 < 1 deg/day. What is the speed of a person standing on the equator with respect to the earths axis? Answer: Use  EMBED Equation.DSMT4  and an earth radius of 6.38 x 106 m (meters) v = 1.67 x 106 m/hr or 1.67 x 103 km/hr or 1040 mph What is the speed of the earth moving around the sun? (Assume the sun is sitting still.) Answer: Use  EMBED Equation.DSMT4  and an earth-sun mean radius of 1.50 x 1011 m (meters) v =  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  A(1,0) B(3,0) So what is omega  EMBED Equation.DSMT4  or angular velocity? Imagine the A-ray rotating counterclockwise (ccw) at 30 degrees per second. Can you see the arrow going around as you count 1 hippopotamus, 2 hippopotamus, 3 hippo etc? Then you understand angular velocity! Well, of course theres more Rather than degrees per second, we tend to use radians per second, so:  EMBED Equation.DSMT4 . The rear wheel of the tractor has a 2ft radius, RD = 2 ft and RC = 0.9 ft. The rear wheel rotates at 20 rev/min. (a) What is the linear speed of wheel D? (on its edge) (b) What is the linear speed of wheel C? (on its edge) (c) What is the angular frequency of wheel C?  ( ) 4 5     ! 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