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?

Options

A. the particle will emerge from the other end will velocity `sqrt((GM_(e))/(2R_(e)))`, where `M_(e) and R_(e)` are earth's mass and radius respectively
B. the overlapping will come to rest at centre of the tunnel
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 to its distance from mid-point of the tunnel

Answer

To solve the problem, we need to analyze the statements provided regarding a particle dropped in a smooth tunnel dug through the Earth. We assume that the mass of the Earth is uniformly distributed. Let's evaluate each statement step by step.

### Step 1: Understanding the Motion of the Particle
When a particle is dropped into a tunnel that goes through the Earth, it will experience gravitational force. As it falls towards the center, the gravitational force acting on it will change due to the uniform distribution of mass.

### Step 2: Statement Analysis
1. **Statement A**: "The particle will emerge from the other end with a velocity of \(\sqrt{\frac{g M_e}{2 R_e}}\)."
- **Analysis**: This statement is correct. The particle will convert its potential energy at the starting point into kinetic energy as it falls, and it will reach the other end with the same speed it had at the starting point due to conservation of energy.

2. **Statement B**: "The particle will come to rest at the center of the tunnel."
- **Analysis**: This statement is incorrect. The particle will not come to rest at the center; instead, it will continue to oscillate back and forth through the tunnel due to the restoring force of gravity acting on it.

3. **Statement C**: "The potential energy of the particle will be equal to zero at the center of the tunnel if it is along a diameter."
- **Analysis**: This statement is incorrect. The potential energy is not zero at the center; it is at a minimum. The potential energy is defined relative to a reference point, and it reaches its minimum value at the center of the Earth.

4. **Statement D**: "The acceleration of the particle will be proportional to its distance from the midpoint of the tunnel."
- **Analysis**: This statement is correct. According to Hooke's Law, the gravitational force (and thus the acceleration) acting on the particle is proportional to its distance from the center of the Earth (or midpoint of the tunnel).

### Conclusion
From the analysis:
- **Correct Statements**: A and D
- **Incorrect Statements**: B and C

Thus, the statements that are not correct are B and C.

### Final Answer
The statements that are not correct are:
- Statement B: "The particle will come to rest at the center of the tunnel."
- Statement C: "The potential energy of the particle will be equal to zero at the center of the tunnel if it is along a diameter."