Question

An ideal smooth massless inextensible string passes through a bead of mass m as shown whose ends are fixed at A and B. Initially the system is in equilibrium. Now the bead is given a velocity $\sqrt{2gl}$ in horizontal direction and in the plane of the string as shown. Let $T_0$ be the tension in the string and $r_0$ be the radius of curvature of path of bead just after the velocity is given. Then choose the correct option(s).

• A. $T_0 = \frac{31mg}{30}$
• B. $r_0 = \frac{35l}{3}$
• C. $T_0 = \frac{41mg}{40}$
• D. $r_0 = \frac{25l}{3}$

Answer 1

Answer: (A), (B)

Options (A) and (B) are correct.

Answer 2

The solution involves a careful "impulsive-constraint" analysis using energy-momentum principles.

Core Solution Summary:

1. Geometry Setup: Define the geometry with fixed points A and B, 8l apart, and string segments of length 5l each. The bead forms an isosceles triangle.

2. Equilibrium Analysis: In equilibrium, the vertical components of the tensions balance the weight mg. This leads to: $T_0 = \frac{31mg}{30}$

3. Post-Impulse Analysis: After the bead receives a horizontal velocity $\sqrt{2gl}$, the net force (vector sum of tensions) has a component perpendicular to the velocity, causing centripetal acceleration. This gives: $r_0 = \frac{35l}{3}$