The total energy of a stretched spring is given by, E = (1/2)kx2
So, in this question the energy of the spring when its amplitude of the spring is = E = (1/2)k(A1)2
This ‘E’ is also the kinetic energy of the mass ‘M’ at its mean position.
Since, E is independent of mass, so, the energy remains same. This means, even after ‘m’ is added, the amplitude remains same.
Therefore,
A1 = A2
So,
A1 : A2 = 1 : 1
Hope it helps..

You can follow below given answer dude...

When a mass M attached to a horizontal spring executes Simple Harmonic Motion (SHM) with amplitude A1, and a smaller mass m is placed over it, both masses move together with amplitude A2. The ratio of A1 /A2 can be calculated using the following steps.



SHM is a type of periodic motion in which the restoring force is directly proportional to the displacement from the equilibrium position and is always directed towards it. In other words, the force acting on the object is proportional to the distance it is displaced from its equilibrium position, and its direction is towards the equilibrium position.



The equation for SHM is given by:

x = A cos(ωt + φ)

where x is the displacement of the object from its equilibrium position, A is the amplitude of the motion, ω is the angular frequency (ω = 2πf, where f is the frequency of the motion), t is the time, and φ is the phase angle.



When the mass M executes SHM with amplitude A1, the spring exerts a force on it given by:

F = -kx

where k is the spring constant. When the smaller mass m is placed over it, the force acting on the combined system is given by:

F = -(k/M+m)x

where M+m is the total mass of the system. The amplitude of the motion of the combined system can be calculated as follows:

A2 = (M/(M+m))A1



The ratio of A1 /A2 can be calculated as follows:

A1/A2 = (M+m)/M

Therefore, the ratio of amplitudes of SHM is (M+m)/M.



In conclusion, the ratio of amplitudes of SHM can be calculated using the equation (M+m)/M, where M is the mass attached to the spring and m is the smaller mass placed over it. Understanding the concept of SHM and deriving the equation for it is crucial in calculating the amplitude of the combined system and the ratio of amplitudes.