A simple pendulum of length `l` is suspended from the celing of a cart which is sliding without friction on as inclined plane of inclination theta . What will be the time period of the pendulum?

Explanation

To find the time period of a simple pendulum suspended from the ceiling of a cart sliding down an inclined plane, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Pendulum:**
- The pendulum experiences gravitational force acting downwards, which can be resolved into two components due to the incline:
- \( G \sin \theta \) (parallel to the incline)
- \( G \cos \theta \) (perpendicular to the incline)

2. **Consider the Pseudo Force:**
- Since the cart is accelerating down the incline, we also need to consider the pseudo force acting on the pendulum bob. This pseudo force is equal to \( G \sin \theta \) acting in the opposite direction to the acceleration of the cart.

3. **Net Effective Force:**
- The effective force acting on the pendulum bob in the direction of the pendulum's motion can be expressed as:
- The gravitational force component acting down the incline is \( G \cos \theta \).
- The pseudo force acting up the incline is \( G \sin \theta \).
- Thus, the net effective force acting on the pendulum bob is:
\[ G_{\text{effective}} = G \cos \theta \]

4. **Using the Formula for Time Period:**
- The formula for the time period \( T \) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g_{\text{effective}}}} \]
- Substituting \( g_{\text{effective}} \) with \( G \cos \theta \), we have:
\[ T = 2\pi \sqrt{\frac{L}{G \cos \theta}} \]

5. **Final Expression for Time Period:**
- Therefore, the time period of the pendulum suspended from the ceiling of the cart sliding down the incline is:
\[ T = 2\pi \sqrt{\frac{L}{g \cos \theta}} \]

### Final Answer:
The time period of the pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g \cos \theta}} \]