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A tunnel is dug inside the earth across ...

A tunnel is dug inside the earth across one of its diameters. Radius of earth is R and its mass is M. A particle is projected inside the tunnel with velocity `sqrt((2GM)/(R))` from one of its ends then maximum velocity attained by the particle in the subsequent motion is `sqrt((nGM)/(R))` (assuming tunnel to be frictionless). Find n

A

`V_(0)=sqrt((GM)/(2R))`

B

`V_(0)=sqrt((GM)/R)`

C

`R=(pi)/2sqrt((R^(3))/(GM))`

D

`T=pisqrt((R^(3))/(GM))`

Text Solution

Verified by Experts

The correct Answer is:
B, C
`(-GMm)/(2R)=(-GMm)/R+1/2mv_(0)^(2)`
`impliesv_(0) =sqrt((GM)/R)`
while in the tunnel,
`(d^(2)x)/(dt^(2))=-((GM)/(R^(3)))x`
`v_(0)^(2)=(GM)/R=omega^(2)(a^(2)-R^(2))impliesa=sqrt(2)R`
`implies T_(AtoB)=1/4xx2pisqrt((R^(3))/(GM))`
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