Question

Assume that a tunnel is dug through earth from North pole to south pole and that the earth is a non-rotating, uniform sphere of density $\rho$. The gravitational force on a particle of mass m dropped into the tunnel when it reaches a distanc $r$ from the centre of earth is

• A. $\left(\frac{3}{4\pi }mGp\right)r$
• B. $\left(\frac{4\pi }{3}mGp\right)r$
• C. $\left(\frac{4\pi }{3}mGp\right)r^2$
• D. $\left(\frac{4\pi }{3}{m}^{2}Gp\right)r$

Answer 1

Answer: (B)

$\left(\frac{4\pi }{3}mGp\right)r$

The gravitational field intensity inside the earth at a distance $r(<R)$ from the centre is calculated as

$g_r=\frac{GM}{R^3}r$

where,

$M$ is the mass of the earth

$R$ is the radius of the earth

$G$ is the universal gravitational constant

Assuming earth to be a uniform sphere of density $\rho$, we can write its mass $M$ as

$Mass=density\times volume$

$\therefore M=\rho \times \frac{4\pi }{3}{R}^3$

Substituting in $g_r$, we get

$\therefore g_r=\frac{G\frac{4\pi \rho }{3}{R}^3}{R^3}r$

$\therefore g_r=\frac{4\pi }{3}G\rho r$

Thus, the force acting on a particle of mass $m$ at a distance $r(<R)$ from the centre of the earth is

$F=m{g}_r$

$\therefore F=\left(\frac{4\pi }{3}mG\rho \right)r$