Two 0.85Mtrs -long pendulums hang side by side. The masses of the pendulum bobs are 50g and 70g . The lighter bob is pulled aside until its string is horizontal and is then released from rest. It swings down and collides elastically with the other bob at the bottom of its arc.

Question

Two 0.85Mtrs -long pendulums hang side by side. The masses of the pendulum bobs are 50g and 70g . The lighter bob is pulled aside until its string is horizontal and is then released from rest. It swings down and collides elastically with the other bob at the bottom of its arc.

Part A
To what height does the lighter bob rebound?

Answer 1

To determine the height to which the lighter bob rebounds after colliding elastically with the other bob, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system remains constant as long as no external forces are acting on it.

In this case, when the lighter bob reaches its maximum height, it possesses potential energy due to its height above the equilibrium position. At the same time, it has zero kinetic energy since it is momentarily at rest. When it collides elastically with the other bob, its potential energy is converted into kinetic energy. Finally, when it reaches its maximum height again after the collision, it once again possesses potential energy.

The equation for conservation of mechanical energy in this situation is:

m1gh1 + 0.5m1v1^2 = m1gh2 + 0.5m1v2^2

where m1 is the mass of the lighter bob, g is the acceleration due to gravity, h1 is the initial height of the lighter bob, v1 is the initial velocity of the lighter bob, m2 is the mass of the heavier bob, and h2 is the final height of the lighter bob after the collision.

Since the lighter bob starts from rest and its string is horizontal, its initial velocity (v1) is equal to zero. Also, after the collision, the heavier bob comes to rest momentarily, so its final velocity (v2) is also equal to zero. Thus, the equation simplifies to:

m1gh1 = m1gh2

To find the final height to which the lighter bob rebounds (h2), we can rearrange the equation as follows:

h2 = (m1h1) / m1

Substituting the values given in the question, where m1 = 50g = 0.05kg and h1 = 0.85m, we can calculate h2:

h2 = (0.05kg * 0.85m) / 0.05kg
h2 = 0.85m

Therefore, the lighter bob rebounds to a height of 0.85 meters.