Let `theta` denote the angular displacement of a simple pendulum oscillating in a vertical plane. If the mass of the bob is m, the tension in the string is `mg cos theta`

Options

A. always
B. never
C. at the extreme position
D. at the mean position

Answer

To solve the problem regarding the tension in the string of a simple pendulum at different positions, we will analyze the forces acting on the pendulum bob.

### Step-by-Step Solution:

1. **Understanding the Forces**:
- The forces acting on the pendulum bob are the tension (T) in the string and the gravitational force (mg), where m is the mass of the bob and g is the acceleration due to gravity.
- The gravitational force can be resolved into two components: one acting along the direction of the string (mg cos θ) and the other acting perpendicular to the string (mg sin θ).

2. **Tension at an Angle θ**:
- When the pendulum is at an angle θ from the vertical, the tension in the string can be expressed as:
\[ T = mg \cos \theta + \frac{mv^2}{r} \]
- Here, \(v\) is the linear velocity of the bob, and \(r\) is the length of the string (radius of the circular motion).

3. **Extreme Positions**:
- At the extreme positions of the pendulum's swing (the maximum displacement), the velocity \(v\) of the bob is zero because it momentarily stops before reversing direction.
- Therefore, the term \(\frac{mv^2}{r}\) becomes zero.

4. **Calculating Tension at Extreme Positions**:
- Substituting \(v = 0\) into the tension equation gives:
\[ T = mg \cos \theta + 0 \]
- Thus, at the extreme positions, the tension in the string is:
\[ T = mg \cos \theta \]

5. **Mean Position**:
- At the mean position (the lowest point of the swing), the tension is at its maximum because the bob has maximum velocity and the centripetal force is at its peak. However, the question specifically asks about the tension at extreme positions.

6. **Conclusion**:
- Therefore, at extreme positions, the tension in the string of the pendulum is zero because the bob is momentarily at rest, leading to:
\[ T = 0 \]

### Final Answer:
At extreme positions, the tension in the string of the pendulum is zero.

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