A small sphere is suspended by a string from the ceiling of a car. If the car beings to move with a constant acceleration a, the inclination of the string to the vertical is:

Options

A. `tan^(-1)(a//g)` in the direction of motion
B. `tan^(-1)(a//g)` opposite to the direction of motion
C. `tan^(-1)(g//a)` in the direction of motion
D. `tan^(-1)(g//a)` opposite to the direction of motion

Explanation

To solve the problem of the inclination of the string to the vertical when a small sphere is suspended in a car that is accelerating with a constant acceleration \( a \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify Forces Acting on the Sphere**:
- The weight of the sphere acts vertically downward and is given by \( W = mg \), where \( m \) is the mass of the sphere and \( g \) is the acceleration due to gravity.
- When the car accelerates with acceleration \( a \), a pseudo force \( F_{pseudo} = ma \) acts on the sphere in the opposite direction of the car's acceleration (i.e., horizontally backward).

2. **Draw a Free Body Diagram**:
- Draw a diagram showing the sphere, the string, and the forces acting on the sphere.
- The string makes an angle \( \theta \) with the vertical.
- The forces can be represented as:
- The weight \( mg \) acting downward.
- The pseudo force \( ma \) acting horizontally backward.

3. **Resolve Forces**:
- The tension \( T \) in the string can be resolved into two components:
- A vertical component \( T \cos \theta \) which balances the weight of the sphere.
- A horizontal component \( T \sin \theta \) which balances the pseudo force.

4. **Set Up Equations**:
- From the vertical forces:
\[ T \cos \theta = mg \quad \text{(1)} \]
- From the horizontal forces:
\[ T \sin \theta = ma \quad \text{(2)} \]

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

6. **Find the Angle \( \theta \)**:
- To find \( \theta \), take the inverse tangent:
\[ \theta = \tan^{-1} \left( \frac{a}{g} \right) \]

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