Question

A hollow cylinder, a solid cylinder, and a billiard ball are all released at the top of a ramp and roll to the bottom without slipping

Part A

Rank them according to the fraction of the kinetic energy that is rotational as they roll.

Rank from greatest to least. To rank items as equivalent, overlap them.

Part B

What are the ratios of speeds of a hollow and a solid cylinders when they reach the bottom of the ramp?

Express your answer using two significant digits.

Part C

What are the ratios of speeds of a hollow cylinder and a billiard ball when they reach the bottom of the ramp?

Express your answer using two significant digits.

Part D

What are the ratios of speeds of a solid cylinder and a billiard ball when they reach the bottom of the ramp?

Express your answer using two significant digits.

Answer 1

fraction = Krot/Kroll = (1/2)*Ic*w^2 / (1/2)*(Ic + m*R^2)*w^2

fraction = Ic/(Ic+m*R^2)

for hollow cylinder Ic = m*R^2

fraction = 1/2

for solid Ic = (1/2)*m*R^2

fraction = 1/3

for ball Ic = (2/5)*m*R^2

fraction = 2/7

hollow > solid > ball

+++++++++++++++++++++++++++++

part B

initial potential energy at top = final KE at the bottom

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

w = sqrt( 2*m*g*h/(Ic+mR^2) )

linear speed v = R*w

vhollow / vsolid = R*m*g*h/sqrt(mR^2+mR^2) / Rm*g*h/sqrt(mR^2/2 +mR^2)

vhollow / vsolid = 0.71 / 0.82 = 0.87

======================

part(C)

vhollow / vball = R*m*g*h/sqrt(mR^2+mR^2) / Rm*g*h/sqrt(2mR^2/5 + mR^2)

vhollow / vball = 0.71 / 0.845 = 0.84

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part(D)

vsolid / vball = Rm*g*h/sqrt(mR^2/2 +mR^2) / Rm*g*h/sqrt(2mR^2/5 + mR^2)

vsolid / vball = 0.82 / 0.845 = 0.97