Question

When a mass m is connected individually to two springs $S_{1}and{S}_2$, the oscillation frequencies are $v_{1}and{v}_2$. If the same mass is attached to the two springs as shown in figure, the oscillation frequency would be:

• A. $v_1+v_2$
• B. $\sqrt{v_1^2+v_2^2}$
• C. $[\frac{1}{v_1}+\frac{1}{v_2}{]}^{-1}$
• D. $\sqrt{v_1^2-v_2^2}$

Answer 1

Answer: (B)

$\sqrt{v_1^2+v_2^2}$

Let $k_{1}and{k}_2$ be the spring constants of springs $S_1$ and $S_2$ respectively then.
$v_1=\frac{1}{2\pi }\sqrt{\frac{k_1}{m}}$
and $v_2=\frac{1}{2\pi }\sqrt{\frac{k_2}{m}}$
If k is effective spring constant of two springs $S_{1}and{S}_2$.
Then, $k=k_1+k_2$ ($\therefore$ springs are connected in parallel)
If v 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
$v=\frac{1}{2\pi }\sqrt{\frac{k}{m}}=\frac{1}{2\pi }\sqrt{\frac{k_1+k_2}{m}}=\frac{1}{2\pi }\sqrt{\frac{k_1}{m}+\frac{k_2}{m}}$

$v=\frac{1}{2\pi }\sqrt{4{\pi }^2{v}_1^2+4{\pi }^2{v}_2^2}$

$\sqrt{v_1^2+v_2^2}$