Question

A bob of heavy mass \(m\) is suspended by a light string of length \(l\). The bob is given a horizontal velocity \(v_0\) as shown in figure. If the string gets slack at some point P making an angle \( \theta \) from the horizontal, the ratio of the speed \(v\) of the bob at point P to its initial speed \(v_0\) is : 

• A. \( \left( \frac{1}{2 + 3 \sin \theta} \right)^{1/2} \)
• B. \( \left( \frac{\cos \theta}{2 + 3 \sin \theta} \right)^{1/2} \)
• C. \( \left( \frac{\sin \theta}{2 + 3 \sin \theta} \right)^{1/2} \)
• D. \( (\sin \theta)^{1/2} \)

Hint:

Use conservation of energy between the initial point and point P. The condition for the string to get slack is that the tension becomes zero. Analyze the radial forces at point P to relate the velocity \(v\) to the angle \( \theta \). Combine these two equations to find the ratio \(v/v_0\). Remember to correctly resolve the gravitational force along the radial direction based on the given angle \( \theta \) with the horizontal.