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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 pressing force by the particle on the wall, and the acceleration of the particle vary with $x$ (distance of the particle from the centre of earth ) according to

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answer is B, C.

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Detailed Solution

Net force towards centre of earth

$= m g^{ι} = (m g x) / R$

Normal force $N$ = $m g^{ι}$ $\sin θ$

Thus pressing force $N = \frac{m g x}{R} \frac{R}{2 x}$

$N = \frac{m g}{2}$ is constant and independent of $x$

Hence ( 2)

Tangential force $(F)$ = $m a$ $= m g^{ι} co s θ$

$Q = g^{ι} \cos θ = \frac{g x \sqrt{x^{2} - \frac{R^{2}}{4}}}{R x}$ $a = \frac{g x}{R} \sqrt{R^{2} - 4 X^{2}}$

Curve is parabolic and at $x = \frac{R}{2}$ , $a = 0$ Hence ( 3 )

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