Question

A simple pendulum of length \( \ell \) has a bob of mass \( m \). It executes S.H.M. of small amplitude \( A \). The maximum tension in the string is

• A. \( 2mg \)
• B. \( mg \)
• C. \( mg \left( \frac{A}{\ell} + 1 \right) \)
• D. \( mg \left( \frac{A^2}{\ell^2} + 1 \right) \)

Hint:

In pendulum motion, the maximum tension occurs at the lowest point where both gravitational and centripetal forces are maximum.

Answer 1

The Correct Option is D

Step 1: Understanding the forces involved.
The maximum tension in the string occurs at the lowest point of the pendulum's swing, when both the gravitational force and the centripetal force act together.
Step 2: Maximum tension expression.
At the lowest point, the maximum tension is: \[ T = mg + \frac{m v^2}{\ell} \] where \( v \) is the maximum velocity. For small amplitude oscillations, \( v = \sqrt{gA} \), so: \[ T = mg + m \left( \frac{g A}{\ell} \right) = mg \left( 1 + \frac{A^2}{\ell^2} \right) \]
Step 3: Conclusion.
The maximum tension is \( mg \left( \frac{A^2}{\ell^2} + 1 \right) \), so the correct answer is (D).