Question

A mass M, attached to a horizontal spring, executes S.H.M. with amplitude A1. When the mass M passes through its mean position then a smaller m is placed it and both of them move together with amplitude A2. The ratio of (A1/A2) is :-

Answer 1

The ratio of amplitudes A1/A2 when a mass is added to a mass-spring system in SHM can be found by using energy conservation, leading to A1/A2 = √((M+m)/M).

Step-by-step explanation:

To find the ratio of amplitudes A1/A2, we can use the principle of conservation of energy or the concept that the maximum speed of the mass-spring system in simple harmonic motion (SHM) is the same just before and after the smaller mass m is added. At the mean position, the entire energy is kinetic, and it can be represented as ½ kA1² for the initial mass M and ½ kA2² for the combined mass M+m after the small mass is added. Since the spring constant k is the same and the kinetic energy is conserved just as the small mass is placed on the bigger mass without changing the velocity, we equate the kinetic energies and solve for the amplitude ratio: A1/A2 = √((M+m)/M).