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 assumedd to be frictionless. A particle is released from one end of the tunnel. The pressing 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

Options

A.
B.
C.
D.

Answer

B, C

Explanation

Net force towards the centre of the earth
`=mg'=(mGMx)/(R^(3))=(mgR^(2)x)/(R^(3))`
`=(mgx)//R`
Normal force, `N=mg'sintheta`
Thus, pressing force
`N=(mgx)/R R/(2x)`
`=N=(mg)/2` is constant and independent of `x`.
Hence, option b is correct.
Tangential force `F=ma=mg'costheta`
`a=g'costheta=(gx)/Rsqrt(((R^(2))/4-x^2))/x`
`impliesa=(gx)/Rsqrt(R^(2)-4x^(2))`
Curve is parabolic and at `x=R/2,a=0`
Hence option c is correct.