A simple pendulum of length l is suspended through the ceiling of an elevator. Find the time period of small oscillations if the elevator a. is going up with an acceleration `a_0`. b. is going down with an acceleration `a_0` and c. is moving with a uniform velocity.

Explanation

To solve the problem of finding the time period of small oscillations of a simple pendulum suspended in an elevator under different conditions, we can break it down into three parts based on the elevator's motion:

### Part (a): Elevator going up with an acceleration \( a_0 \)

1. **Identify the effective acceleration**: When the elevator is accelerating upwards, the effective acceleration acting on the pendulum is the sum of gravitational acceleration \( g \) and the elevator's acceleration \( a_0 \). Thus, the effective acceleration becomes:
\[ g_{\text{eff}} = g + a_0 \]

2. **Use the formula for the time period**: The time period \( T \) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g_{\text{eff}}}} \]
Substituting the effective acceleration:
\[ T = 2\pi \sqrt{\frac{L}{g + a_0}} \]

### Part (b): Elevator going down with an acceleration \( a_0 \)

1. **Identify the effective acceleration**: When the elevator is accelerating downwards, the effective acceleration acting on the pendulum is the difference between gravitational acceleration \( g \) and the elevator's acceleration \( a_0 \). Thus, the effective acceleration becomes:
\[ g_{\text{eff}} = g - a_0 \]

2. **Use the formula for the time period**: The time period \( T \) of the pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g_{\text{eff}}}} \]
Substituting the effective acceleration:
\[ T = 2\pi \sqrt{\frac{L}{g - a_0}} \]

### Part (c): Elevator moving with uniform velocity

1. **Identify the effective acceleration**: When the elevator is moving with uniform velocity, there is no acceleration acting on the pendulum. Therefore, the effective acceleration is simply \( g \):
\[ g_{\text{eff}} = g \]

2. **Use the formula for the time period**: The time period \( T \) of the pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g_{\text{eff}}}} \]
Substituting the effective acceleration:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]

### Summary of Results:
- **For elevator going up with acceleration \( a_0 \)**:
\[ T = 2\pi \sqrt{\frac{L}{g + a_0}} \]
- **For elevator going down with acceleration \( a_0 \)**:
\[ T = 2\pi \sqrt{\frac{L}{g - a_0}} \]
- **For elevator moving with uniform velocity**:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]