Question

A simple pendulum oscillates with an angular amplitude \( \theta \). If the maximum tension in the string is 4 times the minimum tension then the value of \( \theta \) is

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

Hint:

Write the tension at the extreme position and at the lowest position in terms of the amplitude, then use the given ratio.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Tension is minimum at the extreme position, where the speed is zero, and maximum at the lowest point, where the speed is greatest.

Step 2: Minimum and Maximum Tension
At the extreme position, \(T_{min} = mg\cos\theta\). By energy conservation, \(v^2 = 2gl(1-\cos\theta)\) at the bottom, so \(T_{max} = mg + \dfrac{mv^2}{l} = mg(3 - 2\cos\theta)\).

Step 3: Calculation
Given \(T_{max} = 4T_{min}\): \(3 - 2\cos\theta = 4\cos\theta\), so \(6\cos\theta = 3\) and \(\cos\theta = 0.5\).

Step 4: Compare the options
Hence \(\theta = \cos^{-1}(0.5)\), which matches the second option.

Final Answer:
The angular amplitude is the inverse cosine of 0.5. \[ \boxed{\text{(B) }\cos^{-1}(0.5)} \]