Question

The bob ${{A}}$ of a pendulum released from $30^\circ$ to the vertical hits another bob ${{B}}$ of the same mass at rest on a table as shown in fig. How high does the bob ${{A}}$ rise after the collision? Neglect the size of the bobs and assume the collision to be elastic.

Answer 1

Hint: According to the conservation of momentum, the momentum of a system is constant when no external forces are acting on the system. Here momentum will be transferred from bob ${{A}}$ to bob ${{B}}$ which satisfies the conservation of momentum. When momentum is being transferred then bob ${{A}}$ will come to rest and it will not rise anymore after the collision.

Complete step by step solution:
Given: Masses of bob ${{A}}$ and bob ${{B}}$ are equal
Bob ${{A}}$ of a pendulum released from ${{3}}{{{0}}^{{0}}}$ to the vertical
Bob ${{A}}$ hits bob ${{B}}$ where bob ${{B}}$ is at rest
Let us assume that the collision is elastic. In an elastic collision, two equal masses where one is stationary and the other is moving with some velocity, the stationary mass acquires the same velocity. However, the moving mass comes to rest immediately after the collision.
Here, in this case a complete transfer of momentum takes place from a moving mass to a stationary mass.
Thus, bob ${{A}}$ of mass ${{m}}$ after colliding with bob ${{B}}$of same mass i.e. ${{m}}$ , will come to rest while the other bob ${{B}}$ will start moving with the same velocity of bob ${{A}}$ at the instant of collision.

Therefore, bob ${{A}}$ will not rise at all.

Note: Even though the momentum of each particle changes, the momentum of the system remains constant as long as there is no external force acting on it. Momentum is always conserved in any collision (i.e. elastic collision or non-elastic collision) but kinetic energy is not conserved in a non-elastic collision, the kinetic energy is converted into other forms of energy.