A pendulum is located inside lift.If initially time period is pendulum is T. Find its time period if lift accelerate upwards with acceleration g/2

Answer

To solve the problem of finding the time period of a pendulum located inside a lift that is accelerating upwards with an acceleration of \( \frac{g}{2} \), we can follow these steps:

### Step 1: Understand the Initial Condition
The time period of a simple pendulum at rest (when the lift is not accelerating) is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where:
- \( T \) is the time period,
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

### Step 2: Analyze the Situation When the Lift Accelerates
When the lift accelerates upwards with an acceleration of \( \frac{g}{2} \), we need to determine the effective acceleration due to gravity that acts on the pendulum bob.

In a non-inertial frame (the accelerating lift), a pseudo force acts on the bob in the opposite direction of the lift's acceleration. Thus, the effective acceleration \( g_{\text{effective}} \) can be calculated as:
\[ g_{\text{effective}} = g + \text{(pseudo force)} \]
The pseudo force acting on the bob due to the lift's upward acceleration is \( \frac{g}{2} \). Therefore:
\[ g_{\text{effective}} = g + \frac{g}{2} = \frac{3g}{2} \]

### Step 3: Calculate the New Time Period
Now, we can use the effective acceleration to find the new time period \( T' \) of the pendulum when the lift is accelerating. The formula for the time period with the new effective gravity is:
\[ T' = 2\pi \sqrt{\frac{L}{g_{\text{effective}}}} = 2\pi \sqrt{\frac{L}{\frac{3g}{2}}} \]
This simplifies to:
\[ T' = 2\pi \sqrt{\frac{2L}{3g}} \]

### Step 4: Relate the New Time Period to the Initial Time Period
We can express the new time period \( T' \) in terms of the initial time period \( T \):
\[ T' = T \sqrt{\frac{2}{3}} \]
where \( T = 2\pi \sqrt{\frac{L}{g}} \).

### Conclusion
Thus, the time period of the pendulum when the lift accelerates upwards with acceleration \( \frac{g}{2} \) is:
\[ T' = T \sqrt{\frac{2}{3}} \]