A horizontal spring block system of mass M executes simple harmonic motion. When the block is passing through its equilibrium position, an object of mass m is put on it and the two move together. Find the new amplitude

Suppose the velocities in the equilibrium position before and after placing the object m and v and v’, and the amplitudes are A and A’ respectively. By conservation of linear momentum we have

$$\begin{gathered} (M+m)\omega \text{'}A\text{'}=M\omega A \\ \Rightarrow \sqrt{M+m}A\text{'}=\sqrt{M}A \\ \Rightarrow A\text{'} =\sqrt{\frac{M}{M+m}}A \end{gathered}$$