Question

A body of mass $m$ hangs from a smooth fixed pulley $P_1$ by an inextensible string fitted with the springs of stiffness constants $k_1$ and $k_3$ The string passes over a smooth light pulley $P_2$ , which is connected with another ideal spring of stiffness constant $k_2$. Find the period of small oscillations of the body.

• A. $T=2\pi {\left(m(\frac{2}{k_1}+\frac{4}{k_2}+\frac{2}{k_3})\right)}^{\frac{1}{2}}$
• B. $T=2\pi {\left(m(\frac{1}{k_1}+\frac{4}{k_2}+\frac{1}{k_3})\right)}^{\frac{1}{2}}$
• C. $T=2\pi {\left(m(\frac{2}{k_1}+\frac{2}{k_2}+\frac{2}{k_3})\right)}^{\frac{1}{2}}$
• D. $T=2\pi {\left(m(\frac{1}{k_1}+\frac{1}{k_2}+\frac{1}{k_3})\right)}^{\frac{1}{2}}$

Answer 1

The correct option is B $T=2\pi {\left(m(\frac{1}{k_1}+\frac{4}{k_2}+\frac{1}{k_3})\right)}^{\frac{1}{2}}$

Let $x_1,x_2$ and $x_3$ be the elongations of the springs $k_1,k_2$ and $k_3$ respectively.


Using constraint relations we find the total displacement of $m$ as
$x=x_1+2x_2+x_3........\left(1\right)$

Spring$\left(3\right)$ experiences tension force due to mass $m$
From this, we get ,
$k_3{x}_3=mg\Rightarrow x_3=\frac{mg}{k_3}$
Since, a string is connected between the two springs, the tension force remains same . So, spring $\left(1\right)$ experiences same tension force due to mass $m$
$\therefore x_1=\frac{mg}{k_1}$
Tension experienced by spring $\left(2\right)$ is given by $2mg=k_2{x}_2\Rightarrow x_2=\frac{2mg}{k_2}$
Using these values in equation $\left(1\right)$ we obtain
$x=mg[\frac{1}{k_1}+\frac{4}{k_2}+\frac{1}{k_3}].......\left(2\right)$
Time period of oscillation $T=2\pi \sqrt{\frac{\text{displacement}}{\text{acceleration}}}=2\pi \sqrt{\frac{x}{g}}$
Using $\left(2\right)$ in the above equation we get,
$T=2\pi {\left(m(\frac{1}{k_1}+\frac{4}{k_2}+\frac{1}{k_3})\right)}^{\frac{1}{2}}$
Hence, option (b) is the correct answer.