A solid sphere and a hollow cylinder of the same mass and radius have a rolling

Question

A solid sphere and a hollow cylinder of the same mass and radius have a rolling race down an incline as in Example 13.9. They start at rest on an incline at a height h above a horizontal plane. The race then continues along the horizontal plane. The coefficient of rolling friction between each rolling object and the surface is the same. Both have mass M Both have radius R Write an expression for the distance S that each object will roll after reaching the bottom of the slope. (Assume that loss of energy due to rolling friction is negligible on the slope. Use the following as necessary: M, h, 9, R, P, and ur for the coefficient of rolling friction. Do not substitute numerical values; use variables only.) Ssphere Scylinder

Explanation / Answer

by using conservation of energy,

P.E=K.E+R.E

m*g*h=1/2*m*v^2+1/2*I*w^2

a)

in case of sold sphere,

m*g*h=1/2*m*v^2+1/2*(2/5*m*R^2)*(v/R)^2

m*g*h=7/10*(m*v^2)

===> h=(7/10g)*v^2

from the daigram, s=h*cot(beta)

s=(7/10g)*v^2*cot(beta)

b)

in case of hollow sphere,

m*g*h=1/2*m*v^2+1/2*I*w^2

m*g*h=1/2*m*v^2+1/2*m*R^2)*(v/R)^2

m*g*h=m*v^2

===> h=v^2/g

from the daigram, s=h*cot(beta)

s=(v^2/g)*cot(beta)

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