Question

When a mass $m$ is connected individually to two springs $S_1$ and $S_2$, the oscillation frequencies are $\upsilon_{1}$ and $\upsilon_{2}$. If the same mass is attached to the two springs as shown in figure,

the oscillation frequency would be

• A. $\upsilon_{1}+\upsilon_{2}$
• B. $\sqrt{\upsilon_{1}^{2}+\upsilon^{2}_{2}}$
• C. $\left(\frac{1}{\upsilon_{1}}+\frac{1}{\upsilon_{2}}\right)^{-1}$
• D. $\sqrt{\upsilon_{1}^{2}-\upsilon^{2}_{2}}$

Answer 1

Answer: (B)

Let $k_1$ and $k_2$ be the spring constants of springs $S_1$ and $S_2$ respectively. Then

$\upsilon_{1} = \frac{1}{2\pi}\sqrt{\frac{k_{1}}{m}}\quad...\left(i\right) $
and $\upsilon_{2} = \frac{1}{2\pi} \sqrt{\frac{k_{2}}{m}} \quad...\left(ii\right)$
If $k$ is effective spring constant of two springs $S_1$ and $S_2$
Then, $k = k_1 + k_2$ ($\because$ springs are connected in parallel)
If $\upsilon$ is the effective frequency of oscillation when the mass $m$ is attached to the springs $S_1$and $S_2$ as shown in figure. Then
$\upsilon = \frac{1}{2\pi} \sqrt{\frac{k}{m}} = \frac{1}{2\pi} \sqrt{\frac{k_{1}+k_{2}}{m} } $
$= \frac{1}{2\pi} \sqrt{\frac{k_{1}}{m} +\frac{k_{2}}{m}} $
$ \upsilon = \frac{1}{2\pi}\sqrt{4\pi^{2}\upsilon_{1}^{2}+4\pi^{2}\upsilon_{2}^{2} }$ (Using $(i)$ and $(ii)$
$ = \sqrt{\upsilon_{1}^{2} + \upsilon_{2}^{2}}$