Question

The time period of system shown below is

• A. $2 \pi \sqrt{\frac{m}{3 k}}$
• B. $2 \pi \sqrt{\frac{2 m}{k}}$
• C. $2 \pi \sqrt{\frac{m}{4 k}}$
• D. $2 \pi \sqrt{\frac{m}{k}}$

Answer 1

Answer: (C)

Since, the three springs with spring constants $k, k$ and $2 k$ are attached to the mass $m$ in parallel. So, net spring constant $=k+k+2 k=4 k$

$\therefore$ The net restoring force on mass $m$,
$\left| F _{\text {net }}\right|=-4 k x$
As, force in SHM, $\left| F _{ SHM }\right|=-m \omega^{2} x$
On comparing Eqs. (i) and (ii), we get
$m \omega^{2} =4 k $
$\omega =\sqrt{\frac{4 k}{m}} $
$T =\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{m}{4 k}}$