Question

A simple pendulum oscillates with an angular amplitude $\theta$. If the maximum tension in the string is twice the minimum tension, then $\theta$ is

• A. $\cos^{-1}(0.25)$
• B. $\cos^{-1}(0.75)$
• C. $\cos^{-1}(0.10)$
• D. $\cos^{-1}(0.50)$

Hint:

In pendulum problems, maximum tension occurs at mean position and minimum at extreme position.

Answer 1

The Correct Option is B

Step 1: Write expressions for tension.
For a simple pendulum:
Maximum tension (at mean position): \[ T_{\max} = mg + \dfrac{mv^2}{l} \] Minimum tension (at extreme position): \[ T_{\min} = mg\cos\theta \] Step 2: Use energy relation for velocity.
At mean position: \[ \dfrac{1}{2}mv^2 = mg l (1 - \cos\theta) \] \[ v^2 = 2gl(1 - \cos\theta) \] Step 3: Substitute into $T_{\max$. \[ T_{\max} = mg + 2mg(1 - \cos\theta) = mg(3 - 2\cos\theta) \] Step 4: Apply given condition.
\[ T_{\max} = 2T_{\min} \] \[ mg(3 - 2\cos\theta) = 2mg\cos\theta \] Step 5: Solve for $\theta$.
\[ 3 = 4\cos\theta \] \[ \cos\theta = 0.75 \] \[ \theta = \cos^{-1}(0.75) \]