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 the earth is uniformly distributed over its volume. Then, which of the following statements are not correct?

• A. The particle will emerge from the other end with velocity $\left(G{M}_e\right)/\left(2R_e\right)$, where $M_e$ and $R_e$ are earth's mass and radius, respectively
• B. The particle will come to rest at the centre of the tunnel because at this position, the particle is closest to the earth's centre
• C. Potential energy of the particle will be equal to zero at centre of the tunnel if it is along diameter
• D. Acceleration of the particle will be proportional to its distance from the mid-point of the tunnel

Answer 1

Answer: (A), (B), (C)

The correct options are
A The particle will emerge from the other end with velocity $\left(G{M}_e\right)/\left(2R_e\right)$, where $M_e$ and $R_e$ are earth's mass and radius, respectively
B The particle will come to rest at the centre of the tunnel because at this position, the particle is closest to the earth's centre
C Potential energy of the particle will be equal to zero at centre of the tunnel if it is along diameter

Particle start to execute simple harmonic motion when dropped in the tunnel at its one end with mean position as centre of tunnel. So, acceleration of the particle is directly proportional to its distance from centre of tunnel. So, $\left(d\right)$ is correct.

Particle has zero velocity at one end as it is dropped from rest at that end, So, ends of tunnel are its extreme positions. Hence, at other end, velocity will be zero. So, $\left(a\right)$ is wrong.

When executing SHM, velocity is maximum at its mean position. so, at centre (mean position)its velocity will be maximum possible.

Hence $\left(b\right)$ is wrong.

When partcle moves from extreme position to centre of tunnel, its velocity increases.

So, potential energy decreases and Kinetic energy increases. But initial potential energy of particle is negative. hence it becomes more negative & is least at mid point of tunnel because KE is maximum there =, it means gravitational PE can never be equal to zero. Hence $\left(c\right)$ is wrong.