Question

Please answer this two part question

A solid sphere of mass m and radius a rolls without slipping down a ramp that has a height h and is inclined at angle 0. The sphere is initially motionless. It takes 10 s for the sphere to roll to the bottom of the ramp. 2. a) Would a hoop of the same mass and radius take the same time, or more, or less? Explain. MR The hoop has a maller coeficient from homunt of The hoop would take less time to reach the be b) Would a solid sphere of twice the mass and radius take the same time, or more, or less? Explain.

Answer 1

2)t = 10 s

a)the moment of inertia of hoop is:

Ih = 2/3 m a^2

Is = 2/5 m a^2

from conservation of energy

m g h = 1/2 m v^2 + 1/2 x 2/3 m a^2 x v^2/a^2 = 1/2 m v^2 + 1/3 m v^2 = 5/6 m v^2

v = sqrt (6 g h/5) [hoop]

similarly for speher we get

v' = sqrt (10 g h/7) [sphere]

v' is more than v, v = d/t => t = d/v

more the velocity less the time, so the hoop will take more time as its moment of inertia is more than sloid sphere.

b)I' = 2/5 (2m) (2a)^2 = 15/6 m a^2

I' > I(solid)

Since the moment of inertia is now bigger, the time taken will be higher.