A small ball is suspended by a string from the ceiling of a car. As the car accelerates at a rate ‘a’ the string makes an angle 'a' with the vertical in equilibrium. Then the tension in the string is :

Options

A. `m sqrt(a^(2) + g^(2))`
B. `m a sqrt(a^(2) + g^(2))`
C. `m g sin theta`
D. `m g cos theta`

Explanation

To find the tension in the string when the ball is in equilibrium and the string makes an angle θ with the vertical as the car accelerates, we can follow these steps:

### Step 1: Identify the Forces Acting on the Ball
When the car accelerates, two forces act on the ball:
1. The gravitational force (weight) acting downward: \( F_g = mg \)
2. The pseudo force acting horizontally due to the car's acceleration: \( F_p = ma \)

### Step 2: Analyze the Forces in the Equilibrium Condition
In equilibrium, the ball is not accelerating, which means the net force acting on it is zero. The tension \( T \) in the string must balance these forces. The tension can be resolved into two components:
- A vertical component \( T \cos(\theta) \) that balances the weight \( mg \).
- A horizontal component \( T \sin(\theta) \) that balances the pseudo force \( ma \).

### Step 3: Set Up the Equations
From the vertical forces:
\[ T \cos(\theta) = mg \] (1)

From the horizontal forces:
\[ T \sin(\theta) = ma \] (2)

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

### Step 5: Solve for Tension \( T \)
Now, we can express \( T \) in terms of \( mg \) and \( \theta \):
From equation (1):
\[ T = \frac{mg}{\cos(\theta)} \tag{3} \]

### Step 6: Substitute \( \tan(\theta) \) into the Tension Equation
Using \( \tan(\theta) = \frac{a}{g} \), we can find \( \cos(\theta) \):
\[ \cos(\theta) = \frac{g}{\sqrt{g^2 + a^2}} \]
Substituting this back into equation (3):
\[ T = \frac{mg}{\frac{g}{\sqrt{g^2 + a^2}}} = mg \cdot \frac{\sqrt{g^2 + a^2}}{g} = m\sqrt{g^2 + a^2} \]

### Final Answer
Thus, the tension in the string is:
\[ T = m\sqrt{g^2 + a^2} \]