Assume that there is a tunnel in the shape of a circular arc through the earth. Wall of the tunnel is smooth. A ball of mass ‘m’ is projected into the tunnel at A with speed ‘v’. The ball comes out of the tunnel at B and escapes out of the gravity of earth. Mass and radius of earth are M and R respectively and radius of circular shaped tunnel is also R.
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a
The minimum possible value of ‘v’ is v0=2GMR
b
The minimum possible value of ‘v’ is v0=GM2R
c
If v=v0 , the normal force applied by the tunnel when it is closest to the centre of earth is 27GMm4R2
d
If v=v0 , the normal force applied by the tunnel when it is closest to the centre of earth is 13GMm4R2
answer is A.
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Detailed Solution
Escape velocity, vo=2GMRApplying energy conservation principle−GMmR+12mv2=−GMm2R33R2−R24+12mv12⇒12mv12=GMm2R311R24⇒mv12=11GMm4RN=mg'+mv12R=mg1−R/2R+11GMm4R2=mg2+11GMm4R2=13GMm4R2