Question

A body of mass 'm' tied to one end of a string is whirled in a vertical circle of radius 'R' with zero tension in the string at its highest point. The angle made by the string with the vertical when the kinetic energy of the body becomes half of its maximum kinetic energy is:

• A. \( \theta = \cos^{-1} \left( \frac{1}{4} \right) \)
• B. \( \theta = \sin^{-1} \left( \frac{1}{4} \right) \)
• C. \( \theta = \tan^{-1} \left( \frac{1}{4} \right) \)
• D. \( \theta = \cos^{-1} \left( \frac{1}{2} \right) \)

Hint:

In vertical circular motion, energy conservation principles help determine velocity, height, and tension at different points.

Answer 1

The Correct Option is A

Step 1: Understanding Energy Conservation in a Vertical Circle 
- The total mechanical energy at any position is given by: \[ E = KE + PE \] - At the highest point, velocity is minimum and tension is zero. 

Step 2: Finding Kinetic Energy at a Given Angle 
- The maximum kinetic energy occurs at the lowest point. - At an angle \( \theta \), kinetic energy becomes half of this maximum value. - Using energy conservation: \[ \frac{1}{2} m v^2 + mgR(1 - \cos \theta) = \text{constant} \] Solving for \( \theta \), we get: \[ \theta = \cos^{-1} \left( \frac{1}{4} \right) \] 

Step 3: Conclusion 
Since \( \theta = \cos^{-1} \left( \frac{1}{4} \right) \) satisfies the condition, Option (1) is correct.