First slide

Acceleration due to gravity and its variation

Question

A tunnel is dug along a diameter of the earth. Find the force on a particle of mass M placed in the tunnel at a distance x from the center.

Moderate

A

$\frac{GMm}{R^{3}}$
B

$\frac{GMmx}{R^{2}}$
C

$\frac{GMmx}{R^{3}}$
D

$\frac{{GMmx}^{2}}{R^{3}}$

Solution

Mass of earth ${M=Vρ=(4/3)πR}^{3} ρ$
$Mass of imaginary sphere of radius x is M' \Rightarrow {M=(4/3)πx}^{3} {ρ (or) M∝R}^{3} ⇒ \frac{M^{'}}{M} = \frac{R'^{3}}{R^{3}} ⇒ \frac{M^{'}}{M} = \frac{x^{3}}{R^{3}} ⇒ M^{'} = \frac{M x^{3}}{R^{3}} - - - (1) g' = \frac{G M'}{x^{2}} = \frac{G \frac{M x^{3}}{R^{3}}}{x^{2}} f r o m (1) g' = \frac{G M x}{R^{3}} - - - (2) \Rightarrow F = m g' = m \frac{G M x}{R^{3}} f r o m e q u a t i o n (2)$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App