Simple hormonic motion

Question

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

Difficult

Solution

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

$(M+m)\omega \text{'}A\text{'}=M\omega A\phantom{\rule{0ex}{0ex}}\Rightarrow \sqrt{M+m}A\text{'}=\sqrt{M}A\phantom{\rule{0ex}{0ex}}\Rightarrow A\text{'}=\sqrt{\frac{M}{M+m}}A\phantom{\rule{0ex}{0ex}}$

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