Question

Suppose a smooth tunnel is dug along a straight line joining two points on the surface of the earth and a particle is dropped from rest at its one end. Assume that mass of earth is uniformly distributed over its Volume.then.

• A. the particle will emerge from the other end with velocity $\sqrt{\frac{G{M}_e}{2R_e}}$ where $M_e$ and $R_e$ are earth mass and radius respectively,
• B. the particle will come to rest at centre of the tunnel because at this position, particle is closest to earth centre
• C. potential energy of the particle will be equal to zero at centre of tunnel if it is along a diameter.
• D. acceleration of the particle will be proportional its distance from midpoint of the tunnel.

Answer 1

The correct option is D acceleration of the particle will be proportional its distance from midpoint of the tunnel.

The particle will emerge from the other end with velocity as only conservative acts on the system its speed will be same at other end of the tunnel to that of the speed at initial point. Hence it will arrive at zero velocity. Hence statement in correct. The particle will have non zero velocity as potential energy decreases as it moves towards center of the earth, the loss in potential energy is equal to gain in kinetic energy hence statement is incorrect.

Potential energy of the particle will be minimum at center its value being.

$PE=\frac{{3GM}_m}{2R}$ where $M$ and $m$ are mass of earth and particle hence statement is incorrect Gravitational field in a solid sphere is given as $\frac{GM}{R^2}r$ where $r$ is proportional to distance from center of earth so statement is correct.

So, $\left(A\right)$ $\left(B\right)$ $\left(C\right)$ incorrect $\left(D\right)$ correct.