Question

A body of mass ${}^{\prime }{m}^{\prime }$ hangs from three springs, each of spring constant ${}^{\prime }{k}^{\prime }$ as shown in the figure. If the mass is slightly displaced and let go, the system will oscillate with time period

• A. $2\pi \sqrt{\frac{k}{m}}$
• B. $2\pi \sqrt{\frac{3m}{2k}}$
• C. $2\pi \sqrt{\frac{2m}{3k}}$
• D. $2\pi \sqrt{\frac{3k}{m}}$

Answer 1

The correct option is B $2\pi \sqrt{\frac{3m}{2k}}$

By using the concept of combination of springs :
Since spring $1$ and $2$ are connected parallel to each other. Net spring constant can be calculated as :
$k_{\text{eq}}=k_1+k_2$
$k_{\text{eq}}=k+k=2k$
now we can replace both the springs by single spring having spring constant $\left(2k\right)$.

These both springs are connected in series. So we can calculate the spring constant.
$k_{\text{eq}}=\frac{k_1{k}_2}{k_1+k_2}=\frac{\left(k\right)\left(2k\right)}{k+2k}=\frac{2k^2}{3k}=\frac{2k}{3}$
Finally we replace both the springs with single spring having spring constant $\left(2k/3\right)$
Now, it has become a simple spring mass system. Now the time period can be calculated as general formula.
$T=2\pi \sqrt{\frac{m}{2k/3}}\Rightarrow T=2\pi \sqrt{\frac{3m}{2k}}$