First slide

Acceleration due to gravity and its variation

Question

A tunnel is dug along the diameter of the earth. An object is held in the tunnel at a distance x from the centre of the earth. The magnitude of the gravitational force on the object is proportional to

Moderate

A

$\frac{1}{x}$
B

$\frac{1}{x^{2}}$
C

$x$
D

$x^{2}$

Solution

Let M and R respectively be the mass and radius of the earth and let m be the mass of the object (Figure) Gravitational force on the object is
$\begin{matrix} F = {mg}_{d} \\ = mg (1 - \frac{d}{R}) \\ = m \times \frac{MG}{R^{2}} \times (\frac{R - d)}{R}) \\ = \frac{GmMx}{R^{3}} \end{matrix}$
So the correct choice is (c).

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