Question

A tunnel is dug along a chord of the earth at a perpendicular distance $\frac{R}{2}$ 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 varies with X (distance of the particle from the centre) according to:

• A.
• B.
• C.
• D.

Answer 1

Answer: (B), (C)

Net force towards centre of earth
= mg' = $\frac{mgX}{R}$
Thus pressing force N = $\frac{mgX}{R}\frac{R}{2X}$
N = $\frac{mg}{2}$= constant and independent of X
tangential force F = ma = mg'
a = $=\frac{gX}{R}\frac{\sqrt{X^2-\frac{R^2}{4}}}{X}$
a = $\frac{g}{R}\sqrt{X^2-\frac{R^2}{4}}$
curve is parabolic and at X = $\frac{R}{2}$, a = 0