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

Question

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 A 1 A 2 is α α - 1 , then the value of α will be _____ .

TestHub · Accepted Answer

At mean position, the velocity of the particle in S.H.M. is maximum, so kinetic energy is $K E = \frac{1}{2} k A^{2}$ , where, $k$ is spring constant.
We also know that, kinetic energy is related to momentum as $K E = \frac{p^{2}}{2 m}$ .
Now, from both equations, we have
$\frac{1}{2} k A^{2} = \frac{p^{2}}{2 m}$ (Here, momentum is conserved)
$\Rightarrow {(\frac{A_{1}}{A_{2}})}^{2} = \frac{m_{2}}{m_{1}} = \frac{1024}{900}$
$\Rightarrow \frac{A_{1}}{A_{2}} = \frac{32}{30} = \frac{16}{15} = \frac{16}{16 - 1}$
$∴ α = 16$

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