8Problem 3 . . . Angular Velocity The radius of the wheel is 30 cm. and the (linear) velocity, v, is 5 m/s. What is the angular velocity?
9Solution 3 . . . Angular Velocity We know from problem 1 that :f = 2.6 rev/sBut 1 rev = 2 radiansSo = / t =(2.6)(2 ) /(1 s) = 16.3 rad/s
10v = r V and Linear (m/s) Angular (rad/s) V d / t / t 2 r f / t f / tv = r
11a = r a and Linear (m/s2) Angular (rad/s2) a ( Vf - Vi ) / t ( f - i ) / ta = r
12Problem Your CD playerA 120 mm CD spins up at a uniform rate from rest to 530 rpm in 3 seconds. Calculate its:(a) angular acceleration(b) linear acceleration
13Solution 4 . . . CD player = ( f - i ) / t = (530 x 2 / ) / 3 = 18.5 rad/s2a = r a = 0.06 x 18.5a = 1.1 m/s2
14Problem CD MusicTo make the music play at a uniform rate, it is necessary to spin the CD at a constant linear velocity (CLV). Compared to the angular velocity of the CD when playing a song on the inner track, the angular velocity when playing a song on the outer track isA. moreB. lessC. same
15Solution CD Musicv = r When r increases, must decrease in order for v to stay constant. Correct answer BNote: Think of track races. Runners on the outside track travel a greater distance for the same number of revolutions!
17Problem 6 . . . Angular Analogs d = Vi t + 1/2 a t ?
18Solution 6 . . . Angular Analogs d = Vi t + 1/2 a t = i t + 1/2 t2
19Problem Red CorvetteThe tires of a car make 65 revolutions as the car reduces its speed uniformly from 100 km/h to 50 km/h. The tires have a diameter of 0.8 m. At this rate, how much more time is required for it to stop?
20 = - 4.4 rad/s2 f = i + t Solution 7 . . . Corvette 100 km/h = 27.8 m/s = 69.5 rad/s since v = r Similarly 50 km/h = 34.8 rad/s(f)2 = (i)2 + 2 (34.8)2 = (69.5)2 + (2)()(65)(6.28) = rad/s2f = i + t0 = tt = 7.9 s
21Torque Torque means the “turning effect” of a force. SAME force applied to both. Which one will turn easier?
32Problem Sarah HughesWill her mass change when she pulls her arms in?Will her moment of inertia change?
33Solution Sarah HughesMass does not change when she pulls her arms in but her moment of inertia decreases.
34Problem Guessing GameA ball, hoop, and disc have the same mass. Arrange in order of decreasing IA. hoop, disc, ballB. hoop, ball, discC. ball, disc, hoopD. disc, hoop, ball
35Solution 11 . . . Guessing Game I (moment of inertia) depends on the distribution of mass. The farther the mass is from the axis of rotation, the greater is the moment of inertia.I = MR I = 1/2 MR I = 2 /5 MR2hoop disc ball
36Problem K.E. of RotationWhat is the formula for the kinetic energy of rotation?A. 1/2 mv2B. 1/2 m2C. 1/2 I2D. I
37Solution 12 . . . K.E. of Rotation The analog of v is The analog of m is IThe K.E. of rotation is 1/2 I2
38Problem Long, thin rodCalculate the moment of inertia of a long thin rod of mass M and length L rotating about an axis perpendicular to the length and located at one end.
39Solution 13 . . . Long, thin rod I = mr 2 However, r is a variable so we need to integrate. (ain’t that fun!)A small mass m of length dr must = M/L drI = M/L r2 drI = (M/L)(L3 / 3 )I = 1/3 ML2
40Problem 14 . . . In the middle ID = ICM + MD2 Suppose the rod spins about its C.M. One can use the Parallel Axis Theorem to calculate ICMID = ICM + MD2D is the distance between the C.M. and the other axis of rotation
41Solution 14 . . . In the middle ID = ICM + MD2 1/3 ML2 = ICM + M(L/2)2 ICM = 1/3 ML2 - 1/4 ML2ICM = 1/12 ML2
42Problem 1 The race of the century! Will it be the hoop or the disc?
43Solution 1 . . . Race of the Century Hoop Loses ! ! ! P.E. = K.E. (linear) + K.E. (angular)mgh = 1/2 mv2 + 1/2 I2mgh = 1/2 mv2 + 1/2 I (v/r)2For the disc, I = 1/2 mr2So mgh = 1/2 mv2 + 1/2 (1/2 mr2)(v/r)2Disc v = (4/3 g h)1/2Similarly Hoop v = (g h)1/2