A ball of radius 'r' is made to oscillate in a bowl of radius 'R'. The time period of its oscillation will be (R r) :

Question

A ball of radius 'r' is made to oscillate in a bowl of radius 'R'. The time period of its oscillation will be (R > r) :

Accepted Answer

The problem involves a ball of radius 'r' oscillating in a bowl of radius 'R', where R > r. We are tasked with determining the time period of its oscillation. The correct answer is given as Option 3: $2\pi\sqrt{\frac{R-r}{g}}$. Let's explore why this is the correct choice and why the other options are incorrect.

### Explanation for Option 3:

1. **Physical Context**: When a ball oscillates in a bowl, it behaves similarly to a simple pendulum. The effective length of this pendulum is determined by the geometry of the system, specifically the difference in radii between the bowl and the ball.

2. **Effective Length**: The effective length of the pendulum in this scenario is $R - r$. This is because the center of mass of the ball, which dictates the pendulum's behavior, is effectively at a distance $R - r$ from the bottom of the bowl.

3. **Time Period Formula**: For a simple pendulum, the time period $T$ is given by:
   $$
   T = 2\pi\sqrt{\frac{L}{g}}
   $$
   where $L$ is the effective length and $g$ is the acceleration due to gravity. Substituting $L = R - r$, we get:
   $$
   T = 2\pi\sqrt{\frac{R-r}{g}}
   $$
   This matches Option 3, confirming it as the correct answer.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $2\pi\sqrt{\frac{r}{g}}$**
   - This option incorrectly assumes that the time period depends solely on the radius of the ball. However, the oscillation is influenced by the geometry of the entire system, not just the ball. The radius 'r' alone does not account for the effective length of the pendulum.

2. **Option 2: $2\pi\sqrt{\frac{R}{g}}$**
   - This option assumes that the time period depends only on the radius of the bowl. While the bowl's radius is a factor, it is the difference between the bowl's radius and the ball's radius that determines the effective length of the pendulum.

3. **Option 4: $2\pi\sqrt{\frac{R+r}{g}}$**
   - This option suggests that the effective length is the sum of the radii, which does not align with the physical setup. The oscillation is determined by the distance from the center of the ball to the bottom of the bowl, which is $R - r$, not $R + r$.

In conclusion, Option 3 correctly accounts for the effective length of the pendulum formed by the ball oscillating in the bowl, making it the correct answer. The other options fail to accurately represent the physical dynamics of the system.