A hollow uniform sphere starts moving from rest at the top of a ramp. At the bot

Question

A hollow uniform sphere starts moving from rest at the top of a ramp. At the bottom the ramp curves upward, and when it is a distance H below the starting point the sphere leaves the ramp and travels vertically straight upward. (see diagram)

a) If there is no friction (so basically the sphere just slides like a block), what is the speed of the sphere when it leaves the ramp?

b) If there is no friction, how high will the sphere go (relative to the point it leaves the ramp)?

c) If there is enough friction for the sphere to roll without slipping, what is the speed of the sphere when it leaves the ramp?

d) If there is enough friction for the sphere to roll without slipping, how high will the sphere go (relative to the point it leaves the ramp)?

*hint: The only given quantity here is H, so that should be the only variable in your answers.

Explanation / Answer

a) If there is no friction ,the speed of the sphere when it leaves the ramp, Vi=0 m/s

b) If there is no friction, how high will the sphere go (relative to the point it leaves the ramp)?

Pe= KE

mgH= 1/2mvf2

H= mvf2/ 2mg

c) If there is enough friction for the sphere to roll without slipping, what is the speed of the sphere when it leaves the ramp

here Ke of the sphere= 1/2mvi2+1/2Iw2

vi=0 m/s; I of hollow sphere is 2/3mr2

Ke of the sphere=1/3mv2

Pe= KE

mgH= 1/3mv2

v= [3gH]1/2

d) If there is enough friction for the sphere to roll without slipping, how high will the sphere go (relative to the point it leaves the ramp)?

Ke of the sphere= 1/2mvi2+1/2Iw2

Pe= KE

mgH=1/2mv2+1/3mv2

mgH=5/6mv2

v= [1.2gH]1/2

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