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)?

Explanation / Answer

Here,

a)

at the bottom of ramp ,

v = sqrt(2 * g* H)

b)

as there is no friction , the energy loss will be zero

the final height is H

c) for the sphere to roll down the slope

v = r * w

Using conservation of energy

0.5 * m * v^2 + 0.5 * (0.4 * m * r^2) * (v^2/r^2) = m *g * H

0.7 * v^2 = g * H

v = sqrt(1.43*g*H)

the speed of the sphere is sqrt(1.43*g*H)

d)

for the height of sphere ,

m * g * h = 0.5 * m * (1.43 * g * H)

h= 0.714 * H

the sphere will reach is 0.714*H

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