A simple pendulum with a bob of mass m swings with an angular amplitude of `40^@`. When its angular displacement is `20^@`, the tension in the string is greater than `mg cos 20^@`

Explanation

To solve the problem, we need to analyze the forces acting on the bob of the pendulum when it is at an angular displacement of \(20^\circ\).

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Bob**:
- The forces acting on the bob of mass \(m\) are:
- The gravitational force \(mg\) acting downward.
- The tension \(T\) in the string acting along the string towards the pivot.

2. **Resolve the Gravitational Force**:
- The gravitational force can be resolved into two components:
- A component along the direction of the string: \(mg \cos(20^\circ)\)
- A component perpendicular to the string: \(mg \sin(20^\circ)\)

3. **Centripetal Force Requirement**:
- When the pendulum swings, it moves in a circular path, which requires a centripetal force. The net force acting towards the center of the circular path must provide this centripetal force.
- The net force towards the center is given by:
\[ F_{\text{net}} = T - mg \cos(20^\circ) \]
- This net force must equal the centripetal force required for circular motion:
\[ F_{\text{net}} = \frac{mv^2}{L} \]
where \(L\) is the length of the string and \(v\) is the tangential velocity of the bob at that position.

4. **Setting Up the Equation**:
- From the above, we can set up the equation:
\[ T - mg \cos(20^\circ) = \frac{mv^2}{L} \]
- Rearranging gives:
\[ T = mg \cos(20^\circ) + \frac{mv^2}{L} \]

5. **Analyzing the Velocity**:
- At \(20^\circ\), the pendulum is not at its maximum height (which is at \(40^\circ\)), hence the bob has some velocity \(v\) at this position.
- Since \(v\) is greater than zero, the term \(\frac{mv^2}{L}\) is also greater than zero.

6. **Conclusion**:
- Therefore, we can conclude that:
\[ T > mg \cos(20^\circ) \]
- This confirms that the tension in the string is indeed greater than \(mg \cos(20^\circ)\) when the pendulum is at \(20^\circ\).

### Final Answer:
The tension in the string when the angular displacement is \(20^\circ\) is greater than \(mg \cos(20^\circ)\).

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