question_answer

The mass \[M\]shown in the figure oscillates in simple harmonic motion with amplitude \[A\]. The amplitude of the point \[P\] is

A) \[\frac{{{k}_{1}}A}{{{k}_{2}}}\]

B) \[\frac{{{k}_{2}}A}{{{k}_{1}}}\]

C) \[\frac{{{k}_{1}}A}{{{k}_{1}}+{{k}_{2}}}\]

D) \[\frac{{{k}_{2}}A}{{{k}_{1}}+{{k}_{2}}}\]

Correct Answer: D

Solution :

In case (ii), the springs are shown in the maximum compressed position. If the spring of spring constant \[{{k}_{1}}\] is compressed by \[{{x}_{1}}\] and that of spring constant \[{{k}_{2}}\] is compressed by \[{{x}_{2}}\] then
\[{{x}_{1}}+{{x}_{2}}=A\]	? (i)
\[{{k}_{1}}{{x}_{1}}={{k}_{2}}{{x}_{2}}\Rightarrow {{x}_{2}}=\frac{{{k}_{1}}{{x}_{1}}}{{{k}_{2}}}\]	? (ii)
From (i) and (ii)
\[{{x}_{1}}+\frac{{{k}_{1}}{{x}_{1}}}{{{k}_{2}}}=A\Rightarrow {{x}_{1}}=\frac{{{k}_{2}}A}{{{k}_{2}}+{{k}_{1}}}\]