Question

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

• A. $2\pi \sqrt{\frac{m}{2k}}$
• B. $2\pi \sqrt{\frac{m}{3k}}$
• C. $2\pi \sqrt{\frac{m}{5k}}$
• D. $2\pi \sqrt{\frac{m}{7k}}$

Answer 1

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

Let the particle is displaced slightly by distance 'x' against spring C.

$\text{Elongation of spring,}B=\frac{x}{\sqrt{2}}$

$\text{Elongation of spring,}C=\frac{x}{\sqrt{2}}$

Force on the mass along x direction

$=-kx-\frac{kx}{2}\cos 45\times 2=-2kx$

$F_{net}=-2kx$
$m\frac{d^{2}x}{d{t}^2}=-2kx$

$\frac{d^{2}x}{d{t}^2}=-\frac{2k}{m}x$

${\omega }_n=\sqrt{\frac{2k}{m}}$

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