Question

A hollow sphere with radius .5 M rolls down a 10 M long ramp angled at 30 degrees.

a) Find the speed of the center of the sphere at the bottom of the ramp.

b) Find the time it takes for the sphere to make 10 complete rotations once it's at the bottom of the ramp.

c) After it makes the 10 rotations, the radius of the sphere doubles due to internal forces within the sphere. Now how long does it take to make 10 complete rotations?

Answer 1

for rolling without slipping ,

v = rw

Now,

using conservation of energy, we have

mgh = 1/2mv² + 1/2Iw²

where, I = (2/3)mr²

so,

mgL sin 30 = 1/2mv² + 1/2 * (2/3) mr² * v² / r²

gL sin 30 = v² / 2 + v² / 3

gL sin 30 = 0.8333 v²

so,

v = 7.668 m/s

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time = distance / speed

here, distance = 2 * pi * r = 2 * pi * 10

so,

w = v / r = 7.668 / 0.5 = 15.336 rad/sec

so,

time = 4.1 sec

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(c)

when radius is doubles,

time = 8.19 sec

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