A mass $0.9\,kg$, attached to a horizontal spring, executes $SHM$ with an amplitude $A _{1}$. When this mass passes through its mean position, then a smaller mass of $124\,g$ is placed over it and both masses move together with amplitude $A _{2}$. If the ratio $\frac{ A _{1}}{ A _{2}}$ is $\frac{\alpha}{\alpha-1}$, then the value of $\alpha$ will be$......$
JEE MAIN 2022, Diffcult
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$\frac{1}{2} kA ^{2}=\frac{ p ^{2}}{2\,m }$
$\Rightarrow\left(\frac{ A _{1}}{ A _{2}}\right)^{2}=\frac{ m _{2}}{ m _{1}}=\frac{1024}{900}$
$\Rightarrow \frac{ A _{1}}{ A _{2}}=\frac{32}{30}=\frac{16}{15}=\frac{16}{16-1}$
$\therefore \alpha=16$
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