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:

In pendulum motion, the tension in the string is affected by the displacement. The ratio of maximum to minimum tension can be used to calculate the amplitude of oscillation.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
In a simple pendulum, tension is maximum at the lowest point (mean position) and minimum at the highest point (extreme position). We are given the ratio of these tensions and need to find the angular amplitude.
Step 2: Key Formula or Approach:
1. Tension at extreme point (min): \( T_{min} = mg \cos \theta \).
2. Tension at mean point (max): \( T_{max} = mg(3 - 2 \cos \theta) \).
Step 3: Detailed Explanation:
Given \( T_{max} = 4 T_{min} \).
Substitute the formulas:
\[ mg(3 - 2 \cos \theta) = 4(mg \cos \theta) \]
Divide both sides by \( mg \):
\[ 3 - 2 \cos \theta = 4 \cos \theta \]
\[ 3 = 6 \cos \theta \]
\[ \cos \theta = \frac{3}{6} = 0.5 \]
\[ \theta = \cos^{-1}(0.5) \]
Step 4: Final Answer:
The value of \( \theta \) is \( \cos^{-1}(0.5) \), which corresponds to \( 60^\circ \).