Question

A particle of mass $m$ is attached to three identical massless springs of spring constant $k$ as shown in the figure. The time period of vertical oscillation of the particle is

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

Answer 1

Answer: (B)

When the particle of mass $m$ at $O$ is pushed by $y$ in the direction of $A$, the spring $A$ will be compressed by $y$ while spring $B$ and $C$ will be stretched by $y^{{}^{\prime }}=y\cos 45^{\circ }.$ So, that the total restoring force on the mass $m$ is along $OA$

$F_{\text{net }}=F_A+F_B\cos 45^{\circ }+F_C\cos 45^{\circ }$
$=ky+2k{y}^{{}^{\prime }}\cos 45^{\circ }$
$=ky+2k\left(y\cos 45^{\circ }\right)\cos 45^{\circ }$
$=2ky$
Also, $F_{\text{net }}=k^{{}^{\prime }}y$
$\Rightarrow k^{{}^{\prime }}y=2ky$
$\Rightarrow k^{{}^{\prime }}=2k$
$T=2\pi \sqrt{\frac{m}{k^{\prime }}}=2\pi \sqrt{\frac{m}{2k}}$