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Rigid-body rotation about a moving axis & Moment of inertiaLab Date: 3/16/2016Report Date: 3/23/2016Stephanie IoriPHY 2048 – 013Casey Curley
Purpose:The purpose of the first part of this lab experiment was to study the rotation of a rigid-body about a moving axis. In other words, we simply want to see how a round object would rotate about its axis while moving down a ramp.Theory underlying the experiment:The moment of inertia of a round rigid-body can be expressed as: I=cM R2, where c= 2/5 for a solid sphere, ½ for a solid cylinder and disk, 2/3 for a thin-wall hallow sphere, and 1 for a thin-wall hallow cylinder. The concept of “rigid-body rotation about a moving axis” states that every possible motion of a rigid body can be represented as a combination of translational motion of the center of massand rotation about an axis through the center of mass. The kinetic energy of a rigid body would then be:K=12M vcm2+12Icmω2,where Mis the mass, vcmis the speed of the center of the mass, Icmis the moment of inertia, andωis theangular speed of the rigid body. Ω(ω)will equal vcmRif the rigid body rolls along a plan without slipping. The total mechanical energy would be conserved:K1+U1=K2+U2,where the subscripts represent the top and the bottom of an inclined plane, respectively. This is becauseno work is done by kinetic friction if the rigid body rolls along a plan without slipping, and the effect of rolling friction can be ignored provided that the body and the surface on which it rolls are perfectly rigid. With that being said, only the conservative force of gravity does work on the rigid body. If the height of the inclined plane is represented by h, then the previous equation becomes:0+Mgh=12M vcm2+12Icmω2=12M vcm2+12cM R2(vcmR)2=12(1+c)M vcm2So, the speed of the rigid body at the bottom of the incline would be:vcm=√2gh1+cThe speed doesn’t depend on either the mass Mof the rigid body or its radius R. All solid spheres have the same speed at the bottom because they have the same c, even though their masses and radii are different. All uniform solid cylinders have the same speed, etc. The smaller the value of c, the faster the body will move at any point on its way down. So, small-c bodies will always beat large-cbodies because they have less of their kinetic energy tied up in rotation and have more available for translation.