Question

Suppose a vertical tunnel is dug along the diameter of earth, assumed to be a sphere to be of uniform mass density $\rho$. If a body of mass $m$ is thrown in this tunnel, then its acceleration at a distance $y$ from the centre is given by

• A. $\frac{4\pi }{3}G\rho ym$
• B. $\frac{1}{4}\pi \rho Gy$
• C. $\frac{4}{3}\pi \rho y$
• D. $\frac{4}{3}\pi G\rho y$

Answer 1

The correct option is D $\frac{4}{3}\pi G\rho y$

Net force $F_{net}$ acting at the point $\text{P}$ on mass $m$ due to the gravitational field $E$ is given by,
$$F_{net}=mE$$ If $a$ is the acceleration of mass $m$ due to the gravitational force, then we have

$F_{net}=ma=mE$

$\Rightarrow a=E$

If $M$ is the mass of Earth and $R$ is its radius, then the gravitation field at point $\text{P}$ will be,

$E=\frac{GMy}{R^3}$

Now substituting, $M=\frac{4}{3}\pi R^3\rho$

$\Rightarrow E=\frac{G\times \frac{4}{3}\pi R^3\rho \times y}{R^3}$

$\therefore E=\frac{4}{3}G\pi \rho y$

Hence, the acceleration of the particle is $a=\frac{4}{3}G\pi \rho y$

Hence, option (d) is the correct answer.