A simple pendulum hangs from the roof of a train whose string is inclined towards the rear of the train. Then the train is

Options

A. at rest
B. moving with uniform velocity
C. moving with uniform acceleration
D. moving with uniform retardation.

Answer

To solve the problem, we need to analyze the situation of a simple pendulum hanging from the roof of a train that is inclined towards the rear of the train. We have to determine the state of motion of the train based on the inclination of the pendulum.

### Step-by-Step Solution:

1. **Understanding the Pendulum's Position**:
- A simple pendulum consists of a mass (bob) attached to a string. When the pendulum is at rest, the string hangs vertically downwards. If the string is inclined towards the rear of the train, it indicates that the pendulum is not in its equilibrium position.

2. **Analyzing the Forces**:
- The pendulum experiences two forces: the tension in the string and the gravitational force acting downwards. The inclination of the string suggests that there is a net force acting on the pendulum that is not balanced.

3. **Identifying the Motion of the Train**:
- If the train is at rest, the pendulum would hang straight down without any inclination. Therefore, the train cannot be at rest.
- If the train is moving with a uniform velocity, the pendulum would eventually reach a steady state where it hangs straight down again, as there would be no net force acting on it. Thus, this option is also not correct.

4. **Considering Uniform Acceleration**:
- If the train is accelerating forward, the pendulum will lag behind due to inertia. This causes the pendulum to swing back and create an angle with the vertical, inclined towards the rear of the train. This matches the given condition of the pendulum being inclined towards the rear.

5. **Considering Uniform Retardation**:
- If the train is decelerating (retardation), the pendulum would swing forward, creating an inclination towards the front of the train. This does not match the condition of the pendulum being inclined towards the rear.

6. **Conclusion**:
- Based on the analysis, the only scenario where the pendulum is inclined towards the rear of the train is when the train is moving with uniform acceleration. Therefore, the correct answer is that the train is moving with uniform acceleration.

### Final Answer:
The train is moving with uniform acceleration (Option C).