Several objects roll without slipping down an incline ofvertical height H, all s

Question

Several objects roll without slipping down an incline ofvertical height H, all starting from rest at the same moment. Theobjects are a thin hoop, a spherical marble, a solid cylinder, andan empty soup can. In what order do they reach the bottom of theincline? The sliding box wins. For each we can stateMgH=1/2Mv^2+1/2Icm w^2. The hoop has th largest moment of inertia;hence lowest speed an will arrive at the bottom behind the solidcylinder, which in turn will be behind the marble sphere. The emptycan has most of its mass concentrated at R; so it will be a bitfaster than the pure hoop but slower than the solid cylinder. Whatwill be the speed of a solid disk?

Explanation / Answer

we know from law of conservation of energy   mgH = (1/ 2) mv ^ 2 + ( 1/ 2) I w ^ 2 For soild disk mgH = ( 1/ 2) mv ^ 2 + ( 1/ 2) [ (1/ 2) m R ^ 2 ] ( v / R ) ^ 2                               = ( 1/ 2) m v ^ 2 + ( 1/ 4) m v ^ 2                               = ( 3/ 4) m v ^ 2                        gH = ( 3/ 4) v ^ 2                    4gH/ 3 = v ^ 2               1.3333gH = v^ 2                           v = [ 1.3333* 9.8 * H ]                              = 3.614 H

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