A simple pendulum is suspended from the roof of a train. If the train is moving with an acceleration `49 cm//s^(2)`. Then the angle of inclination of the string about the vertical will be

Options

A. `20^(@)`
B. zero
C. `30^(@)`
D. `3^(@)`

Explanation

To find the angle of inclination of the pendulum string about the vertical when the train is accelerating, we can follow these steps:

### Step 1: Understand the Forces Acting on the Pendulum
When the train accelerates, the pendulum experiences a pseudo force in the opposite direction of the train's acceleration. The forces acting on the pendulum bob are:
- The tension (T) in the string, acting along the string.
- The weight (mg) of the pendulum bob, acting vertically downwards.
- The pseudo force (ma) acting horizontally in the opposite direction of the train's acceleration.

### Step 2: Set Up the Free Body Diagram
In the frame of reference of the train, we can analyze the forces:
- The tension can be resolved into two components:
- \( T \cos \theta \) acting vertically (balancing the weight).
- \( T \sin \theta \) acting horizontally (balancing the pseudo force).

### Step 3: Write the Equations of Motion
From the free body diagram, we can write the following equations based on the balance of forces:
1. Vertically (upward tension balances weight):
\[ T \cos \theta = mg \]
2. Horizontally (tension's horizontal component balances the pseudo force):
\[ T \sin \theta = ma \]

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

### Step 5: Substitute the Values
Given:
- \( a = 49 \, \text{cm/s}^2 = 0.49 \, \text{m/s}^2 \)
- \( g = 10 \, \text{m/s}^2 \)

Substituting these values into the equation:
\[ \tan \theta = \frac{0.49}{10} = 0.049 \]

### Step 6: Calculate the Angle
To find the angle \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(0.049) \]

Using a calculator:
\[ \theta \approx 2.80^\circ \]

### Step 7: Round to the Nearest Degree
Rounding \( 2.80^\circ \) gives approximately \( 3^\circ \).

### Conclusion
The angle of inclination of the pendulum string about the vertical is approximately \( 3^\circ \).

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