To solve the problem of the pendulum bob, we'll use the principles of conservation of energy and conservation of momentum.
Step 1: Calculate the velocity of the left bob before collision.
When the left pendulum bob is released from a height , it converts potential energy into kinetic energy at the lowest point (just before collision). The potential energy at height is given by , where is the mass and is the acceleration due to gravity. At the lowest point, kinetic energy is equal to potential energy:
Simplifying, we get:
Step 2: Use conservation of momentum for the collision.
The left bob (mass ) collides with the right bob (of same mass ) which is initially at rest. The initial momentum before collision is:
After the collision, they stick together, so the combined mass is and let the velocity be . By conservation of momentum:
Solving for , we get:
Step 3: Calculate the height the two bobs rise to after collision.
After collision, the kinetic energy of the combined mass is converted to potential energy at the highest point. Thus, we have:
, where is the maximum height reached after collision.
Substituting for :
Thus, the two bobs rise to a height of after collision, which matches the correct option.
