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

${\upsilon }_1=\frac{1}{2\pi }\sqrt{\frac{k_1}{m}}...\left(i\right)$
and ${\upsilon }_2=\frac{1}{2\pi }\sqrt{\frac{k_2}{m}}...\left(ii\right)$
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 $(ii)$
$=\sqrt{{\upsilon }_1^2+{\upsilon }_2^2}$