a mass $M$, which is attached to a spring.

Key Equations:

Time Period for Mass $(M + m)$:

$ T_2 = 2 \pi \sqrt{\frac{m + M}{k}} $

Time Period for Mass $M$ Only:

$ T_1 = 2 \pi \sqrt{\frac{M}{k}} $

Velocity Considerations:

Assume $v_1$ is the velocity of the mass $M$ at the mean position and $v_2$ is the velocity of the combined mass $(m+M)$.

Using Conservation of Linear Momentum:

$ M v_1 = (m + M) v_2 $

Since $v_1 = A_1 \omega_1$ and $v_2 = A_2 \omega_2$:

$ M (A_1 \omega_1) = (m + M) (A_2 \omega_2) $

Rewriting for the amplitude ratio $\frac{A_1}{A_2}$:

$ \frac{A_1}{A_2} = \frac{(m + M)}{M} \times \frac{\omega_2}{\omega_1} $

Given $\omega_1 = \frac{2 \pi}{T_1}$ and $\omega_2 = \frac{2 \pi}{T_2}$, we have:

$ \frac{\omega_2}{\omega_1} = \frac{T_1}{T_2} $

Substituting Time Periods:

Substituting from earlier equations:

$ \frac{T_1}{T_2} = \sqrt{\frac{M}{m + M}} $

Therefore:

$ \frac{A_1}{A_2} = \frac{(m + M)}{M} \times \sqrt{\frac{M}{m + M}} $

$ \frac{A_1}{A_2} = \sqrt{\frac{m + M}{M}} $

This provides the required ratio of the amplitudes when a smaller mass is attached, leading to a new simple harmonic motion with the described properties.

Simple Harmonic Motion

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