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.
Let d be the distance from center of earth to mass 'm' then,
d2=x2+(R24)=4x2+R24
ād=(12)ā4x2+R2
M be mass of the earth Mā², the mass of the sphere of radius d
M=(43)ĻR3Ļ
Mā²=(43)Ļd3Ļ
Mā²M=d3R3
ā“ Gravitational force on mass m due to the sphere of mass Mā² is,
F=GMā²md2
=Gd3MmR3d2=GMmR3d
So, Normal force exerted by the wall
=FcosĪø=GMmdR3ĆR2d=GMmd2R2