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A tunnel is dug along a chord of the earth at a perpendicular distance R/2 from the earth's center. 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:

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$F g = \frac{G M m r}{R^{3}}$

Pressing force $= F_{g} \cos θ = \frac{G M m r \cos θ}{R^{3}}$

$= \frac{G M m}{2 R^{2}} = C o n s \tan t$

$a = \frac{F g \sin θ}{2 R^{2}} = \frac{G M r \sin θ}{R^{3}} a = \frac{G M y}{R^{3}}$

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