Question

A particle of mass m is attached to three springs A, B and C of equal force constants k as shown in figure. if the particle is pushed slightly against the spring C and released, find the time period of oscillation.

• A. $T=2\pi \sqrt{\frac{m}{k}}$
• B. $T=2\pi \sqrt{\frac{2m}{k}}$
• C. $T=2\pi \sqrt{\frac{m}{2k}}$
• D. None of these

Answer 1

The correct option is C

$T=2\pi \sqrt{\frac{m}{2k}}$

When the particle is displaced towards C by a small $Delta$x the extension in spring A and B is $\frac{\Delta x}{\sqrt{2}}$

Net force on particle=$k\Delta x+\sqrt{(\frac{k\Delta x}{\sqrt{2}}{)}^2+(\frac{k\Delta x}{\sqrt{2}}{)}^2}$

$=k.\Delta x+k.\Delta x=2k\Delta x$

For SHM we know, $a=\frac{F}{m}={\omega }^2\Delta x$

$\Rightarrow \frac{2k\Delta x}{m}=\omega \Delta x$

$\Rightarrow \omega =\sqrt{\frac{2k}{m}}$ $\Rightarrow T=2\pi \sqrt{\frac{m}{2k}}$