A simple pendulum of length 5 m is suspended from the ceiling of a cart. Cart is sliding down on a frictionless surface having angle of inclination 60°. The time period of the pendulum is

Options

A. `2 pi s`
B. `pi s`
C. `4pi s`
D. `pi/2 s`

Answer

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

### Step 1: Understand the Forces Acting on the Pendulum
The pendulum experiences two forces due to gravity:
- The component of gravitational force acting parallel to the incline: \( mg \sin \theta \)
- The component of gravitational force acting perpendicular to the incline: \( mg \cos \theta \)

### Step 2: Identify the Effective Gravitational Acceleration
In this scenario, the effective gravitational acceleration (\( g_{\text{effective}} \)) that affects the pendulum's motion is the component of gravity acting perpendicular to the incline. This is given by:
\[ g_{\text{effective}} = g \cos \theta \]
where \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)) and \( \theta \) is the angle of inclination (given as \( 60^\circ \)).

### Step 3: Calculate \( \cos 60^\circ \)
The cosine of \( 60^\circ \) is:
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 4: Substitute Values into the Effective Gravitational Acceleration
Now substituting the values into the equation for \( g_{\text{effective}} \):
\[ g_{\text{effective}} = g \cos 60^\circ = 10 \times \frac{1}{2} = 5 \, \text{m/s}^2 \]

### Step 5: Use the Formula for the Time Period of a Pendulum
The time period \( T \) of a simple pendulum is given by the formula:
\[ T = 2\pi \sqrt{\frac{L}{g_{\text{effective}}}} \]
where \( L \) is the length of the pendulum (given as \( 5 \, \text{m} \)).

### Step 6: Substitute Values into the Time Period Formula
Substituting the values into the formula:
\[ T = 2\pi \sqrt{\frac{5}{5}} = 2\pi \sqrt{1} = 2\pi \]

### Step 7: Final Result
Thus, the time period of the pendulum is:
\[ T = 2\pi \, \text{seconds} \]

### Conclusion
The time period of the pendulum is \( 2\pi \) seconds.

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