Question

In the arrangement shown in figure, the pulleys are small and light and the springs are ideal. $K_1=25{\pi }^2\text{N/m},K_2=2K_1,K_3=4K_1$ and $K_4=4K_1$ are the force constants of the springs. Calculate the period of small vertical oscillations of the block of mass $m=2\text{kg}$.

• A. $\frac{1}{2}\text{s}$
• B. $\frac{4}{5}\text{s}$
• C. $\frac{8}{5}\text{s}$
• D. $\frac{5}{8}\text{s}$

Answer 1

The correct option is C $\frac{8}{5}\text{s}$

In static equilibrium of the block, tension in the string is exactly equal to its weight.

Let a vertically downward force $F$ be applied on the block to pull it downwards. Equilibrium is again restored when tension in the string is increased by the same amount $F$. Hence, total tension in the string becomes equal to $(mg+F)$.

Strings are further elongated due to extra tension $F$ in the strings; tension in each spring increases by $2F$. Hence, increase in elongation of the springs is $\frac{2F}{K_1},\frac{2F}{K_2},\frac{2F}{K_3}$ and $\frac{2F}{K_4}$, respectively.

According to the geometry of the arrangement, downward displacement of the block from its equilibrium positon is

$y=2(\frac{2F}{K_1}+\frac{2F}{K_2}+\frac{2F}{K_3}+\frac{2F}{K_4})...\left(1\right)$

If the block is released now, it starts to accelerate upwards due to extra tension $F$ in the string. It means, the restoring force on the block is equal to $F$. From equation $\left(1\right)$, acceleration

$a=\frac{F}{m}=\frac{y}{4m(\frac{1}{K_1}+\frac{1}{K_2}+\frac{1}{K_3}+\frac{1}{K_4})}$

Since the acceleration of the block is restoring and is directly proportional to displacement $y$, the block performs SHM.

Time period of oscillations, $T=\sqrt{\frac{y}{a}}$

$\Rightarrow T=2\pi \sqrt{4m(\frac{1}{K_1}+\frac{1}{K_2}+\frac{1}{K_3}+\frac{1}{K_4})}$

$\Rightarrow T=4\pi \sqrt{m(\frac{1}{K_1}+\frac{1}{K_2}+\frac{1}{K_3}+\frac{1}{K_4})}$

Substituing the values, we get

$\Rightarrow T=4\pi \sqrt{2(\frac{1}{25{\pi }^2}+\frac{1}{2\times 25{\pi }^2}+\frac{1}{4\times 25{\pi }^2}+\frac{1}{4\times 25{\pi }^2})}$

$\therefore T=\frac{8}{5}\text{s}$

Hence, option (c) is the correct answer.