Question

A small object is tied to a string and whirled in a vertical circle of radius L. What should be the minimum speed at the topmost point of the circle so that the string just remains taut?

• A. \( \sqrt{2gL} \)
• B. \( \sqrt{gL} \)
• C. \( \sqrt{3gL} \)
• D. Zero

Hint:

In circular motion, the minimum speed required to keep the string taut at the topmost point is determined by balancing the gravitational force and the centripetal force.

Answer 1

The Correct Option is B

At the topmost point of the vertical circular motion, the only forces acting on the object are the tension in the string and the gravitational force. To ensure the string remains taut, the centripetal force must be at least equal to the weight of the object. The centripetal force is provided by the tension in the string and gravity. Let \( T \) be the tension and \( mg \) the weight of the object. For the string to remain taut, the condition at the topmost point is: \[ T + mg = \frac{mv^2}{L} \] At the minimum speed, \( T = 0 \), so the equation becomes: \[ mg = \frac{mv^2}{L} \] Solving for \( v \), we get: \[ v = \sqrt{gL} \] Thus, the minimum speed required at the topmost point is \( \sqrt{gL} \).