A simple pendulum with a bob of mass m is suspended from the roof of a car moving with a horizontal accelertion a. The angle made by the string with verical is

Options

A. `tan^(-1)((a)/(g))`
B. `tan^(-1)(1-(a)/(g))`
C. `cos^(-1)((a)/(g))`
D. `sin^(-1)((a)/(g))`

Explanation

To solve the problem of finding the angle made by the string of a simple pendulum with the vertical when the pendulum is in a car that is accelerating horizontally, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Forces Acting on the Bob**:
- The forces acting on the bob of mass \( m \) are:
- The gravitational force \( mg \) acting downward.
- The tension \( T \) in the string acting along the string at an angle \( \theta \) from the vertical.
- A pseudo force \( F_{pseudo} = ma \) acting horizontally in the opposite direction of the car's acceleration due to the non-inertial frame of reference.

2. **Draw a Free Body Diagram**:
- Draw the bob and indicate the tension \( T \) at an angle \( \theta \) with the vertical. The vertical component of tension is \( T \cos \theta \) and the horizontal component is \( T \sin \theta \).

3. **Set Up the Equations**:
- In the horizontal direction (along the direction of acceleration), the pseudo force must be balanced by the horizontal component of the tension:
\[ T \sin \theta = ma \quad \text{(1)} \]
- In the vertical direction, the weight of the bob must be balanced by the vertical component of the tension:
\[ T \cos \theta = mg \quad \text{(2)} \]

4. **Divide the Two Equations**:
- To eliminate \( T \), divide equation (1) by equation (2):
\[ \frac{T \sin \theta}{T \cos \theta} = \frac{ma}{mg} \]
- This simplifies to:
\[ \tan \theta = \frac{a}{g} \]

5. **Solve for the Angle \( \theta \)**:
- To find \( \theta \), take the arctangent of both sides:
\[ \theta = \tan^{-1} \left( \frac{a}{g} \right) \]

### Final Answer:
The angle made by the string with the vertical is:
\[ \theta = \tan^{-1} \left( \frac{a}{g} \right) \]