The period of oscillation of a simple pendulum of length (L) suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination (prop), is given by.

Options

A. `2 pi sqrt((L)/(g cos prop))`
B. `2 pi sqrt((L)/(g cos prop))`
C. `2 pi sqrt((L)/(g))`
D. `2 pi sqrt((L)/(g tan prop))`

Explanation

To find the period of oscillation of a simple pendulum of length \( L \) suspended from the roof of a vehicle moving down an inclined plane with inclination \( \alpha \), we can follow these steps:

### Step 1: Analyze the Forces Acting on the Pendulum
When the pendulum is suspended from the roof of the vehicle, two main forces act on it: the gravitational force and the pseudo force due to the acceleration of the vehicle.

- The gravitational force can be broken down into two components:
- \( mg \cos \alpha \) (perpendicular to the direction of the pendulum's motion)
- \( mg \sin \alpha \) (parallel to the direction of the pendulum's motion)

### Step 2: Consider the Effective Acceleration
Since the vehicle is moving down the incline, we need to consider the effective acceleration acting on the pendulum. The pseudo force due to the vehicle's acceleration down the incline is equal to \( mg \sin \alpha \).

### Step 3: Determine the Effective Gravitational Acceleration
In the non-inertial frame of the vehicle, the effective gravitational acceleration \( g_{\text{effective}} \) acting on the pendulum can be calculated as:
\[ g_{\text{effective}} = g \cos \alpha \]
This is because the component of gravitational force acting along the direction of the pendulum's motion is balanced by the pseudo force.

### Step 4: Use the Formula for the Period of a Pendulum
The formula for the period \( T \) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g_{\text{effective}}}} \]
Substituting \( g_{\text{effective}} = g \cos \alpha \) into the formula, we get:
\[ T = 2\pi \sqrt{\frac{L}{g \cos \alpha}} \]

### Step 5: Final Expression for the Period
Thus, the period of oscillation of the pendulum is:
\[ T = 2\pi \sqrt{\frac{L}{g \cos \alpha}} \]