Question

A thin spherical shell of total mass M and radius R is held fixed. There is a small hole in the shell. A mass
m is released from rest at distance R from the hole. This mass subsequently moves under the gravitational force
of the shell. How long does the mass take to travel from the hole to the point diametrically opposite ?
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Answer 1

Answer:

Lets divide the scenario into two steps:

1) Movement till the hole

2) Movement inside the spherical shell through hole

For step 1:

As energy is conserved, change in kinetic energy + change in potential energy =0

⇒(Kf−Ki)+(Pf−Pi)=0

⇒2mv2=RGMm−2RGMm

Solving this we get v=(RGM)0.5

Now, for step 2:

Inside the shell, potential is constant and net force is zero. 

So, it moves with constant velocity of v=(RGM)0.5

⇒ Time taken =v2R=2(GMR3)0.5

Hence, x=2

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