Class Notes on Rotational Motion - Galileo and Einstein

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12Varying Moment of InertiaNewton did not actually write his Second Law as F = ma = mdv/dt, but as F = dp/dt, where p = mv iscalled the momentum, and in fact this is the correct description of nature in cases where the mass of anobject varies, such as a rocket expelling fuel, or a falling raindrop growing from condensation.Similarly, the rotational Second Law = Icould be written more generally asddtto allow for the possibility of a varying moment of inertia, and this is much more common than a varyingmass. Think of a spinning dancer or skater throwing her arms out and bringing them in, or of a divercurling up and uncurling his rotating body. The external torques are pretty small under these conditions,the variation in spin speed is largely due to the changing moment of inertia and the constancy of thetotal angular momentum IThe observed variations in angular velocity confirm the validity of the moregeneral law.Rotational Kinetic EnergyA rotating object has kinetic energy even if its center of mass is at rest. Think of it as composed of manysmall masses m i at distances r i from the center of mass.Then if the angular velocity is m i has speed v i = r i , and kinetic energy ½ m i r i 2 2 , and summing these,If a net torque is acting, this energy will change:ddtI KE of rotation = ½ I 2 .d dt1 22I I Iso the rate of working, the power, of a torque equals the magnitude of the torque multiplied by theangular velocity, just as a force F moving something at speed v is working at a rate Fv.This equation can also be integrated:dd I dt dt ddt1 22 The total change in kinetic energy equals the integral of torque times angular distance, valid even if thetorque is varying.Rolling Down a RampFor example, we’ll take the case of a hoop of radius R rolling down a ramp without slipping. Suppose atsome instant the center of the hoop is moving with speed v. Of course, other points on the hoop are
13moving at different speeds. The point at the bottom in contact with the ramp isn’t moving at all at thatinstant. But in the frame of reference in which the center of mass is momentarily stationary, that pointis moving backwards at R. This means that the angular speed and the linear speed of the center ofmass are related by v = R.2vv = Rωvω0Velocities relative to the ramp for noslip:zero at point of contact, v atcenter of hoop, 2v at farthest point.Velocities of parts of hoop relative toits center: v = Rω perpendicular tothe radius.What is the total energy of the rolling hoop? Imagine as usual that it is made up of a large number ofsmall masses m i . If the small mass m i has velocity v i , the total energy is ½ mv i 2 . Since its both movingalong and rolling, let’s write the velocity of the small mass m i as v v uwhere u iis the velocity of the small mass m i relative to the center of mass.Then the total kinetic energy of all the small massesi 2CM CM1 2 1 1 2 1 22mivi 2mi v ui 2MvCM vCM mui i 2mui i.i i i iThe first term is just the ordinary kinetic energy of linear motion, the last term is the same as the kineticenergy of rotation for the center of mass at rest. The middle term is zero, because the sum is just the d linear momentum in the center of mass frame, which is zero. ( mui i miri0in the framedtin which the center of mass is at rest.)iii
Page 1 and 2: Rotational MotionMichael Fowler, UV
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Class Notes on Rotational Motion - Galileo and Einstein

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