Question

A block of mass $M$ is attached to a horizontal spring and is executing SHM, with amplitude $A_1$. When the mass $M$ passes through its mean position, a smaller mass $m$ is placed over it gently and both of them move together with amplitude $A_2$. The ratio of $\left(\frac{A_1}{A_2}\right)$ is

• A. $\frac{M+m}{M}$
• B. $\sqrt{\frac{M}{M+m}}$
• C. $\sqrt{\frac{M+m}{M}}$
• D. $\frac{M}{M+m}$

Answer 1

The correct option is C $\sqrt{\frac{M+m}{M}}$

Mean position is the equilibrium point.
i.e $F_{\text{net}}=0$
So, linear momentum is conserved.
$P_i=P_f\Rightarrow M{V}_1=(M+m)V_2$
where, $V_1=$ velocity of block $M$ at mean position.
$V_2=$ velocity of combination of blocks at mean position.
We know,
$V_1=A_1{\omega }_1:V_2=A_2{\omega }_2$
where ${\omega }_1=\sqrt{\frac{K}{M}},{\omega }_2=\sqrt{\frac{K}{M+m}}$
$\Rightarrow M{A}_1{\omega }_1=(M+m)A_2{\omega }_2$
$\Rightarrow M{A}_1\sqrt{\frac{K}{M}}=(M+m)A_2\sqrt{\frac{K}{M+m}}$
$\therefore \frac{A_1}{A_2}=\frac{\left(\sqrt{\frac{K}{M+m}}\right)(M+m)}{\sqrt{\frac{K}{M}}\times M}=\sqrt{\frac{M+m}{M}}$