Question 170 of 194

In the given figure, a mass $M$ is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is $k.$ The mass oscillates on a frictionless surface with time period $T$ and amplitude $A.$ When the mass is in equilibrium position, as shown in the figure, another mass $m$ is gently fixed upon it. The new amplitude of oscillation will be:

A$A\sqrt{\dfrac{M+m}{M}}$B$A\sqrt{\dfrac{M}{M-m}}$C$A\sqrt{\dfrac{M-m}{M}}$D$A\sqrt{\dfrac{M}{M+m}}$

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Answer: (D) $A\sqrt{\dfrac{M}{M+m}}$

Momentum of system remains conserved.
${p}_{i}={p}_{f}$
$MA\omega =\left(m+M\right){A}^{'}\omega$
$MA\sqrt{\dfrac{k}{M}}=\left(m+M\right){A}^{'}\sqrt{\dfrac{k}{m+M}}$
${A}^{'}=A\sqrt{\dfrac{M}{M+m}}$