Question

A simple pendulum oscillates with an angular amplitude \(θ\). If the maximum tension in the string is twice the minimum tension then \(θ\) is

• A. \(cos^{-1}(0.75)\)
• B. \(cos^{-1}(0.5)\)
• C. \(sin^{-1}(0.5)\)
• D. \(sin^{-1}(0.75)\)

Hint:

Find tension at the lowest point using energy conservation and centripetal force, and at the extreme point where speed is zero.

Answer 1

The Correct Option is A

Step 1: Work with heights.
Let the pendulum rise through height $h = L(1 - \cos\theta)$ above the lowest point.

Step 2: Tension at the bottom.
$v^2 = 2gh$, so $T_{max} = mg + \dfrac{mv^2}{L} = mg + \dfrac{2mgh}{L} = mg\left(1 + 2(1 - \cos\theta)\right)$.

Step 3: Tension at the end.
$T_{min} = mg\cos\theta$ since the bob is momentarily at rest.

Step 4: Equate.
$3 - 2\cos\theta = 2\cos\theta$ gives $\cos\theta = 3/4$.

Final Answer:
Option (A). \[ \boxed{\theta = \cos^{-1}(0.75)} \]