2Angular VelocityAngular velocity,w, is the rate of change in angular displacement. (radians per second.)w = Angular velocity in rad/s.qtAngular velocity can also be given as the frequency of revolution, f (rev/s or rpm):w = 2pf Angular frequency f (rev/s).
3Angular AccelerationAngular acceleration is the rate of change in angular velocity. (Radians per sec per sec.)The angular acceleration can also be found from the change in frequency, as follows:
4Force and Linear Acceleration When an object is subject to a net force, it undergoes an acceleration. (Newton’s 2nd)Torque and Angular AccelerationWhen a rigid object is subject to a net torque, it undergoes an angular acceleration.
5Inertia of Rotation t Force does for translation what Consider Newton’s second law for the inertia of rotationLinear Inertia, m = F/am = = 5 kg20 N4 m/s2F = 20 Na = 4 m/s2Rotational Inertia, II = = = 2.5 kg m2(20 N)(0.5 m) 2 rad/s2taF = 20 NR = 0.5 ma = 2 rad/s2Force does for translation whattorque does for rotation:
6Moment of InertiaThis mass analog is called the moment of inertia, I, of the objectis defined relative to rotation axisSI units are kg m2
7More About Moment of Inertia I depends on both the mass and its distribution.If an object’s mass is distributed further from the axis of rotation, the moment of inertia will be larger.
8Common Moments of Inertia Common moments of inertia are on page 251.
9Example 1 A circular hoop and a disk each have a mass of 3 kg and a radius of 20 cm. Compare their rotational inertias.RI = mR 2HoopI = kg m2RI = ½mR 2DiskI = kg m2
10Important Analogies t m I x f For many problems involving rotation, there is an analogy to be drawn from linear motion.R4 kgwtwo = 50 rad/s t = 40 N mmxfIA resultant torque t produces angular acceleration a of disk with rotational inertia I.A resultant force F produces negative acceleration a for a mass m.
11Example 2 Treat the spindle as a solid cylinder. a) What is the moment of Inertia of the spindle?b) If the tension in the rope is 10 N, what is the angular acceleration of the wheel?c) What is the acceleration of the bucket?d) What is the mass of the bucket?M
12Solution a) What is the moment of Inertia of the spindle? Given: M = 5 kg, R = 0.6 mM= 0.9 kgm2
13Solution b) If the tension in the rope is 10 N, what is a? Given: I = 0.9 kg m2, T = 10 N, r = 0.6 ma = (0.6m)(10 N)/(0.9 kg∙m2)a = 6.67 rad/s2c) What is the acceleration of the bucket?Given: r=0.6 m, a = 6.67 rad/sMa = (6.67 rad/s2)(0.6 m)a=4 m/s2
14Solution d) What is the mass of the bucket? Given: T = 10 N, a = 4 m/s2M = 1.72 kgM
16Combined Rotation and Translation vcmFirst consider a disk sliding without friction. The velocity of any part is equal to velocity vcm of the center of mass.vRPNow consider a ball rolling without slipping. The angular velocity about the point P is same as for disk, so that we write:Or
17Two Kinds of Kinetic Energy Kinetic Energy of Translation:K = ½mv2vRPKinetic Energy of Rotation:K = ½I2Total Kinetic Energy of a Rolling Object:KE due to rotationKE of center-of-mass motion
18Example 3What is the kinetic energy of the Earth due to the daily rotation?Given: Mearth=5.98 x1024 kg, Rearth = 6.63 x106 m.First, find w= 7.27 x10-5 rad/s= 2.78 x1029 J
20Analogous Formulas F = ma = I K = ½mv2 K = ½I2 Work = Fx Work = tq Linear MotionRotational MotionF = ma = IK = ½mv2K = ½I2Work = FxWork = tqPower = FvPower = IFx = ½mvf2 - ½mvo2 = ½If2 - ½Io2
21Example 4 A solid sphere rolls down a hill of height 40 m. What is the velocity of the ball when it reaches the bottom? (Note: We don’t know r or m!)v = 23.7 m/s
22Conserved if no net outside torques Angular MomentumRigid bodyPoint particleAnalogy between L and pAngular MomentumLinear momentumL = Iwp = mvt = DL/DtF = Dp/DtConserved if no net outside torquesConserved if no net outside forces
23Example 5A 65-kg student sprints at 8.0 m/s and leaps onto a 110-kg merry-go-round of radius 1.6 m. Treating the merry-go-round as a uniform cylinder, find the resulting angular velocity. Assume the student lands on the merry-go-round while moving tangentially.
24Solution Known: M, R, m, v0 Find: wF First, find L0 Next, find Itot Now, given Itot and L0, find w= rad/s
25= Summary of Formulas: Conservation: I = SmR2 mgho mghf Height? ½Iwo2 ½mvo2=mghf½Iwf2½mvf2Height?Rotation?velocity?