Question

A block of mass $M_1$ is constrained to move along with a moveable pulley of mass $M_2$ which is connected to a spring of force constant $k$ , as shown in the figure. If the mass of the fixed pulley is negligible and friction is absent everywhere, then the period of small oscillations of the system is

• A. $2\pi \sqrt{\frac{4{\text{M}}_1+{\text{M}}_2}{k}}$
• B. $2\pi \sqrt{\frac{4{\text{M}}_2+{\text{M}}_1}{\text{k}}}$
• C. $4\pi \sqrt{\frac{2{\text{M}}_1+{\text{M}}_2}{k}}$
• D. $\pi \sqrt{\frac{4{\text{M}}_1-{\text{M}}_2}{k}}$

Answer 1

Answer: (A)

If the mass $M_1$ is displaced by $x$ and has speed $v$ , spring will extend by $x/2$ and $M_2$ will have speed $v/2$ . Energy of this system can be written as
$-M_{1}gx+\frac{1}{2}{M}_1{v}^2+M_{2}g\frac{X}{2}+\frac{1}{2}{M}_2{\left(\frac{v}{2}\right)}^2+\frac{1}{2}k{\left(\frac{x}{2}\right)}^2=$ const.
$-{\text{M}}_1\text{gx}+\frac{1}{2}{\text{M}}_1{\text{v}}^2+\frac{{\text{M}}_2}{2}\text{gx}+\frac{1}{8}{\text{M}}_2{\text{v}}^2+\frac{1}{8}k{\text{x}}^2=\text{const.}$
Differentiate wrt time
$-{\text{M}}_1\text{gv}+{\text{M}}_1\text{v}\frac{\text{dv}}{\text{dt}}+\frac{{\text{M}}_2\text{g}}{2}\text{v}+\frac{1}{4}{\text{M}}_2\text{v}\frac{\text{dv}}{\text{dt}}+\frac{1}{8}k\text{x.v}=0$
$\Rightarrow -{\text{M}}_1\text{g}+{\text{M}}_1\frac{\text{dv}}{\text{dt}}+\frac{{\text{M}}_2\text{g}}{2}+\frac{1}{4}{\text{M}}_2\frac{\text{dv}}{\text{dt}}+\frac{1}{8}k\text{x}=0$

$\left(M_1+\frac{M_2}{4}\right)\frac{dv}{dt}=-\frac{k}{4}x+M_{1}g-\frac{M_{2}g}{2}$
$a=\frac{dv}{dt}=-\left(\frac{k}{4M_1+M_2}\right)x+\frac{2\left(2M_1-M_2\right)g}{4M_1+M_2}$
Here, ${\omega }^2=\frac{k}{4M_1+M_2}\Rightarrow T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{4M_1+M_2}{k}}$