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, the time period of oscillation is

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

Answer 1

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

$F_{net}=F_A+F_B\cos {45}^0+F_C\cos {45}^0$

$=ky+2k{y}^{\prime }\cos {45}^0=ky+2k\left(y\cos {45}^0\right)\cos {45}^0=2ky$

$F_{net}=k^{\prime }y\Rightarrow k^{\prime }y=2ky\Rightarrow k^{\prime }=2k$

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

Hence,

option $\left(B\right)$ is correct answer.