A rubber ball with water, having a small hole in its bottom is used as the bob of a simple pendulum. The time-period of such a pendulum:a)Increases with timeb)First increases and then decreases finally having same value as at the beginningc)Decreases with timed)Is a constantCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

When rubber ball completely filled with water its centre of gravity will be at its centre, as water will fall through hole its COG will shift towards lower side leading to increase in length of pendulum and thus T, when very small amount of water will be left in rubber ball its COG will again shifts upward causing decrease in length and thus T, and finally when rubber ball becomes empty its COG will be at its centre and T will remains same as earlier.

Answer

When rubber ball completely filled with water its centre of gravity will be at its centre,as water will fall through hole its cog will shift towards lower side leading to increase in length of pendulum and thus T ,when very small amount of water will be left in rubber ball its c.o.g will again shifts upward causing decrease in length and thus T ,and finally when rubber ball becomes empty its c.o.g will be at its centre and T will remains same as earlier

Answer

Explanation:

When a rubber ball filled with water is used as the bob of a simple pendulum, the time-period of the pendulum initially increases and then decreases, finally reaching the same value as at the beginning. This can be explained by considering the following points:

1. Initial Increase in Time-period:
- When the rubber ball is filled with water, it increases the effective length of the pendulum.
- The effective length is the distance between the point of suspension and the center of mass of the bob.
- As the water fills the ball, the center of mass shifts downwards, increasing the effective length.
- The time-period of a simple pendulum is directly proportional to the square root of the effective length.
- Therefore, when the effective length increases, the time-period also increases.

2. Maximum Time-period:
- As the water continues to fill the ball, the effective length of the pendulum reaches a maximum value.
- At this point, the time-period is maximum, and the pendulum takes the longest time to complete one oscillation.

3. Decrease in Time-period:
- When the ball has a small hole at the bottom, water starts leaking out slowly.
- As water leaks out, the center of mass of the bob gradually shifts upwards.
- This reduces the effective length of the pendulum.
- As the effective length decreases, the time-period also decreases.

4. Same Time-period as at the Beginning:
- Eventually, the rubber ball loses all the water, and the center of mass returns to its initial position.
- This means that the effective length of the pendulum is the same as it was at the beginning.
- Consequently, the time-period of the pendulum also becomes the same as it was initially.

Therefore, the time-period of a simple pendulum with a rubber ball filled with water initially increases due to an increase in the effective length. It then decreases as water leaks out and the effective length decreases. Finally, the time-period reaches the same value as at the beginning when all the water is lost, and the center of mass returns to its initial position.

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