Question

Find the time period of the spring mass system shown in the figure below.

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

Answer 1

The correct option is B $T=2\pi \sqrt{\frac{m}{K}}$

From the diagram we can deduce that, forces balance only when there is elongation $x_0$ in the spring.
At point of equilibrium, $F_{\text{net}}=0$
$\Rightarrow mg\sin \theta =K{x}_0\dots \dots \left(1\right)$


where $x_0$ is elongation in the spring at natural length.
Let us displace the spring further by $x$.
Then,
$F_{\text{net}}=mg\sin \theta -Kx-K{x}_0$


Using $\left(1\right)$ in the above equation, we get
$F_{\text{net}}=-Kx$
$\Rightarrow a=-\left(\frac{K}{m}\right)x$
Comapring this with $a=-{\omega }^{2}x$, we get
$\omega =\sqrt{\frac{K}{m}}$
$\Rightarrow T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{K}}$
Thus, option (b) is the correct answer.