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The mass M shown in the figure oscillates in SHM with amplitude A

Question:- The mass M shown in the figure oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is

Solution :-

As the springs are connected in series, so restoring force is same

$\displaystyle F={{k}_{1}}{{x}_{1}}={{k}_{2}}{{x}_{2}}$

Amplitude A is given by,

$\displaystyle A={{x}_{1}}+{{x}_{2}}=\frac{F}{{{k}_{1}}}+\frac{F}{{{k}_{1}}}=F\left( \frac{1}{{{k}_{1}}}+\frac{1}{{{k}_{1}}} \right)$

$\displaystyle \Rightarrow F=\frac{{{k}_{1}}{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}A$

Now the amplitude of the point P is,

$\displaystyle {{x}_{1}}=\frac{F}{{{k}_{1}}}=\frac{{{k}_{2}}}{{{k}_{1}}+{{k}_{2}}}A$

Hence correct option is (D).