Question

A particle of mass m is attached with three springs A, B and C of equal force constants k as shown in figure. The particle is pushed slightly against the spring C and released, the time period of oscillation will be:

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

Answer 1

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

It we displace the particle by ${}^{\prime }{y}^{\prime }$ towards $A$, The spring $(B,C)$ will Pull the block and spring $A$ push it, The net force on the block will be in vertical direction as the horizontal force balance the other.

elongation of in spring $A$ is $y$ and componention in $B,C$ will be $y\cos {45}^0$ [component of $4$ in $B,C$]

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

$=ky+2k{y}^{\prime }\cos {45}^0$

$=ky+2k\left(4\cos {45}^0\right)=2ky$

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

$\Rightarrow k^{\prime }=2k$

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

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