Question

A ball of mass \( m \) is attached to the free end of an inextensible string of length \( \ell \). Let \( T \) be the tension in the string. The ball is moving in a horizontal circular path about the vertical axis. The angular velocity of the ball at any particular instant will be

• A. \( \sqrt{\frac{T}{m \ell}} \)
• B. \( \sqrt{\frac{T \ell}{m}} \)
• C. \( \sqrt{\frac{m \ell}{T}} \)
• D. \( \sqrt{\frac{T m}{\ell}} \)

Hint:

For circular motion, the centripetal force is provided by the tension in the string, and the angular velocity can be calculated using \( \omega = \sqrt{\frac{T}{m \ell}} \).

Answer 1

The Correct Option is A

Step 1: Understanding the problem.
The ball is moving in a horizontal circular path, and the forces acting on it are tension in the string and the centripetal force required for circular motion.

Step 2: Using the centripetal force formula.
The centripetal force required to maintain circular motion is given by \( F_c = m \omega^2 \ell \), where \( \omega \) is the angular velocity and \( \ell \) is the length of the string. The tension \( T \) provides the centripetal force. Thus, \( T = m \omega^2 \ell \).

Step 3: Solving for angular velocity.
Solving for \( \omega \), we get: \[ \omega = \sqrt{\frac{T}{m \ell}} \] Thus, the angular velocity of the ball is \( \sqrt{\frac{T}{m \ell}} \).

Step 4: Conclusion.
The correct answer is (A).