Question

Three springs of each force constant $k$ are connected as shown figure. Point mass $m$ is slightly displaced to compress $A$ and released. Time period of oscillation

• A. $2\pi \sqrt{\frac{m}{2k}}$
• B. $2\pi \sqrt{\frac{m}{3k}}$
• C. $2\pi \sqrt{\frac{m}{k}}$
• D. $2\pi \sqrt{\frac{m}{k+\sqrt{2}(k+1)}}$

Answer 1

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

Restoring force due to each spring for some extension $F_r=-kx$, Where, $x$=displacement of spring,

$k$=spring constant of each spring

Now, if Spring $3$ is extended by an amount $x$ a spring $2$ and $1$ will compressed by amount $x\cos {45}^0$. And restoring for each of this spring $F_r^{\prime }=-kx\cos {45}^0$.

The components of restoring force alone $OA$ and $OB$ are equal and opposite. So they will cancel each other. The component along $OC$ and will add up to give the resultant force.

$=F_r+F_r^{\prime }\cos {45}^0+F_r^{\prime }\cos {45}^0$

$=-kx(1+2\cos {}^2{45}^0)$

$=-kx(1+1)$

$=-2kx$

So, equation of motion mass $m$ is, $m\frac{d^{2}x}{{dt}^2}=-2kx$

$\Rightarrow m\frac{d^{2}x}{{dt}^2}+\left(\frac{2k}{m}\right)x=0$

Angular frequency $\omega =\sqrt{\frac{2k}{m}}$

So, the time period, $T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{2k}}$