A tunnel is dug along a chord of the earth a perpendicular distance `R/2` 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.

Answer

B

Explanation

Let the distance from center of earth to mass m then
`rarr 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)`