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A tunnel is dug along a chord of the ear...

A tunnel is dug along a chord of the earth at a perpendicular distance R/2 from the earth's centre The wall of the tunnel may be assumed to be frictionless. A particle is released from one end of the tunnel. The presion force by the particle on the wall, and the acceleration of the particle varies with x (distance of the particle from the centre) according to

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The correct Answer is:
B
Let the distance from center of earth to mass m then

d^2=x^2=(R^2/4)=(4x^2+R^2)/3`
`rarr d=(1/2)sqrt(4x^2+R^2)`
M be mass of the earth `m^1`, the mass of the sphere of radius `d/2`
`then M(4/3) piR^3rho`
`M^1=(4/3)pid^3rho`
`or M^1/M=d^3/R^3`

`:.` GravitatioN/Al force in m,
`F=(GM^1m)/d^2`
`=(Gd^2Mm)/(R^3d^2)=(Gmm)/R^3d`
NOrmal force exerted by the wall
`=Fcostheta`
` =(GMmd)/R^3xx R/(2d)=(GMmd)/(2R^2)`
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