Tunnel through Earth

To understand why the ball released from one end of a tunnel through the Earth will reach the other end in a specific time, we need to consider a few key concepts. Let's break it down step by step:

1. Assumptions:
- Neglecting the Earth's rotation means we are assuming that the Earth is not rotating while the ball is in motion through the tunnel.
- We are assuming a uniform density for the Earth, which means the density remains the same as we move from the surface to the center.

2. Gravitational Force:
- The force of gravity acts towards the center of the Earth, causing objects to be pulled towards it.
- As the ball moves through the tunnel, it will experience a gravitational force directed towards the center of the Earth.

3. Gravitational Field:
- Inside a uniform spherical shell, the gravitational force experienced by an object is zero. This is known as the "shell theorem."
- However, inside the Earth, the gravitational force gradually decreases as the distance from the center increases. This is because the mass below the object contributes less to the overall gravitational force.

4. Acceleration due to Gravity:
- The acceleration due to gravity near the Earth's surface is approximately 9.8 m/s^2.
- As the ball moves towards the center of the Earth, the acceleration due to gravity gradually decreases.

5. Ball's Motion:
- Initially, when the ball is released from one end of the tunnel, it will accelerate towards the center of the Earth due to gravity.
- As the ball moves towards the center, the acceleration due to gravity decreases, reducing its speed.
- At the center of the Earth, the gravitational force is zero, and the ball will stop momentarily.
- After reaching the center, the ball will start moving in the opposite direction, gradually slowing down as it moves towards the other end.
- It will eventually come to a stop at the other end of the tunnel, completing the motion.

6. Symmetry:
- The motion of the ball is symmetrical, meaning the time taken to reach the center from one end is the same as the time taken to reach the other end from the center.
- Therefore, the total time taken by the ball to travel from one end to the other is twice the time taken to reach the center.

7. Calculating the Time:
- The time taken to reach the center can be calculated using the formula for time of flight in free fall:
- t = sqrt(2d/g), where t is the time, d is the distance, and g is the acceleration due to gravity.
- Assuming the distance from one end to the center is equal to the radius of the Earth (R), the time taken to reach the center is:
- t = sqrt(2R/g)
- The total time taken by the ball to travel from one end to the other is twice this time:
- Total time = 2 * sqrt(2R/g)

8. Answer:
- The correct answer is option 'B', 42.3 minutes, as it is the closest value to the calculated total time taken by the ball to travel from one end to the other.

Total time period will be 84.6 min when ball released from one end and it come backs to same point as in oscillation.
When ball is released from one end then time taken to reach other end will be half of total time period. that will be 42.3 min.