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Question

A tunnel is dug along a chord of the earth at a perpendicular distance R2 From the earth's centre. The wall of the tunnel may be assumed to be frictionless. Find the force exerted by the wall on a particle of mass m when it is at a distance x from the centre of the tunnel.

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Solution

Let d be the distance from center of earth to mass 'm' then,
d2=x2+(R24)=4x2+R24
⇒d=(12)√4x2+R2
M be mass of the earth M′, the mass of the sphere of radius d
M=(43)Ļ€R3ρ
M′=(43)Ļ€d3ρ
M′M=d3R3

∓ Gravitational force on mass m due to the sphere of mass M′ is,
F=GM′md2
=Gd3MmR3d2=GMmR3d
So, Normal force exerted by the wall
=FcosĪø=GMmdR3ƗR2d=GMmd2R2



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