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 forces on the pendulum.
In a simple pendulum, the tension in the string varies as the pendulum oscillates. The tension is maximum at the lowest point of the swing and minimum at the maximum displacement. The tension in the string at any point is given by the formula:
\[ T = mg \cos \theta + \frac{mL\omega^2}{2}. \]

Step 2: Maximum and minimum tension.
At the lowest point, the tension is maximum:
\[ T_{\text{max}} = mg + \frac{mL\omega^2}{2}. \]
At the maximum displacement, the tension is minimum:
\[ T_{\text{min}} = mg \cos \theta. \]

Step 3: Relating the tensions.
We are told that the maximum tension is 4 times the minimum tension, so:
\[ T_{\text{max}} = 4T_{\text{min}}. \]
Substituting the expressions for \( T_{\text{max}} \) and \( T_{\text{min}} \):
\[ mg + \frac{mL\omega^2}{2} = 4mg \cos \theta. \]

Step 4: Simplifying the equation.
Simplifying the equation, we get:
\[ 1 + \frac{L\omega^2}{2g} = 4 \cos \theta. \]

Step 5: Final Answer.
Using the given values and solving for \( \theta \), we find that:
\[ \boxed{\cos^{-1} (0.75)}. \]