Question

In the figure, pendulum bob on the left side is pulled aside to a height $ h $ from its initial position. After it is released, it collides with the right pendulum bob at rest, which is of the same mass. After the collision, the two bobs stick together and rise to a height

• A. \( \frac{3h}{4} \)
• B. \( \frac{2h}{3} \)
• C. \( \frac{h}{2} \)
• D. \( \frac{h}{4} \)

Hint:

For perfectly inelastic collisions where two objects stick together, the total kinetic energy after the collision is less than the total energy before the collision. The new height reached by the bobs can be calculated using the principle of conservation of mechanical energy.

Answer 1

The Correct Option is D

In this problem, when the pendulum bob on the left is pulled to a height \( h \) and released, it has potential energy equal to \( mgh \). As it collides with the other bob at rest, the system undergoes an inelastic collision (since the bobs stick together). 
Using the principle of conservation of mechanical energy and the law of conservation of momentum, we know that: \[ mgh = \frac{1}{2} (m + m) v^2 \] where \( m \) is the mass of each bob and \( v \) is the velocity of the two bobs after collision. 
Since the bobs stick together, the energy after collision is shared between the combined mass of the bobs. 
After the collision, the maximum height reached by the bobs is related to the initial energy and the combined mass: \[ \text{New height} = \frac{h}{4} \] 
Thus, the new height is \( \frac{h}{4} \).