Question

Some springs are combined in series and parallel arrangement as shown in the figure and a mass $m$ is suspended from them. The ratio of their frequencies will be

• A. $1:1$
• B. $2:1$
• C. $\sqrt{3}:2$
• D. $4:1$

Answer 1

The correct option is C $\sqrt{3}:2$

Case I :
Both springs are in series,
$\therefore$ effective spring constant $(K_s{)}_1=\frac{K_1{K}_2}{K_1+K_2}=\frac{K}{2}$
Frequency of oscillation $(f_s{)}_1=\frac{1}{2\pi }\sqrt{\frac{K_s}{m}}$
$=\frac{1}{2\pi }\sqrt{\frac{K}{2m}}...\left(1\right)$

Case II :
Two springs are in parallel combination, which is in series arrangement with other spring.
Effective spring constant $(K_s{)}_2=\frac{\left(2K\right)\left(K\right)}{3K}=\frac{2K}{3}$

Frequency of oscillation $(f_s{)}_2=\frac{1}{2\pi }\sqrt{\frac{(K_s{)}_2}{m}}$
$=\frac{1}{2\pi }\sqrt{\frac{2K}{3m}}...\left(2\right)$
From (1) & (2)
$\frac{(f_s{)}_1}{(f_s{)}_2}=\frac{\sqrt{3}}{2}$