Question

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

• A. $\frac{k_{1} A}{k_{2}}$
• B. $\frac{k_{2} A}{k_{2} - k_{1}}$
• C. $\frac{k_{2} A}{k_{1} + k_{2}}$
• D. $\frac{k_{2} A}{k_{1}}$

Answer 1

Answer: (C)

If a force $F$ is applied to $M$ , say to the right, let A be the distance moved by $M$ . If the system is released, it executes simple harmonic motion of amplitude $A$ . If $A_{1}$ and $A_{2}$ ae the extensions in springs $k_{1}$ and $k_{2}$ then $A=\left(\right.A_{1}+A_{2}\left.\right)$ and
$F=k_{1}A_{1}=k_{2}A_{2}$
$\Rightarrow A_{1}=\frac{F}{k_{1}}$ and $A_{2}=\frac{F}{k_{2}}$
$\therefore A=A_{1}+A_{2}=F\left(\frac{1}{k_{1}} + \frac{1}{k_{2}}\right)$
$=\frac{F \left(k_{1} + k_{2}\right)}{k_{1} k_{2}}$
$\Rightarrow F=\frac{k_{1} k_{2} A}{\left(k_{1} + k_{2}\right)}$
The amplitude of point $P =$ amplitude of oscillation of spring $k_1$ which is
$A_{1}=\frac{F}{k_{1}}=\frac{k_{2} A}{\left(k_{1} + k_{2}\right)}$