A simple pendulum is set up on a trolley which slides down a frictionless inclined plane making an angle `theta` with horizontal. The time period of pendulum is

Options

A. `2pisqrt((l)/(g))`
B. `2pisqrt((lcosthta)/(g))`
C. `2pisqrt((l)/(gcostheta))`
D. `2pisqrt((l)/(g(1-costheta)))`

Explanation

To find the time period of a simple pendulum set up on a trolley sliding down a frictionless inclined plane at an angle \( \theta \) with the horizontal, we can follow these steps:

### Step 1: Understand the Effective Length and Gravity
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L_{\text{effective}}}{g_{\text{effective}}}} \]
Where:
- \( L_{\text{effective}} \) is the effective length of the pendulum.
- \( g_{\text{effective}} \) is the effective acceleration due to gravity acting on the pendulum.

### Step 2: Determine the Effective Length
For a simple pendulum, the effective length \( L_{\text{effective}} \) is simply the length \( L \) of the pendulum itself, as it is the distance from the pivot point (hinge) to the center of mass of the pendulum bob.

### Step 3: Determine the Effective Gravity
When the trolley is sliding down the inclined plane, the pendulum experiences a pseudo force due to the acceleration of the trolley. The acceleration of the trolley is given by:
\[ a = g \sin \theta \]
This means that the effective gravity acting on the pendulum bob can be considered as:
\[ g_{\text{effective}} = g \cos \theta \]
This is because the pendulum will oscillate about a new equilibrium position where the gravitational force component acting along the direction of the string is \( g \cos \theta \).

### Step 4: Substitute Values into the Time Period Formula
Now we can substitute \( L_{\text{effective}} \) and \( g_{\text{effective}} \) into the time period formula:
\[ T = 2\pi \sqrt{\frac{L}{g \cos \theta}} \]

### Final Expression for the Time Period
Thus, the time period of the pendulum on the trolley sliding down the inclined plane is:
\[ T = 2\pi \sqrt{\frac{L}{g \cos \theta}} \]