Question

Two identical springs of spring constant $k$ each are connecetd in series and parallel as shown in figure. A mass $M$ is suspended from them. The ratio of their frequencies of vertical oscillation will be

• A. $1:2$
• B. $2:1$
• C. $4:1$
• D. $1:4$

Answer 1

The correct option is A $1:2$

Case I : Two springs of equal spring constant are in series arrangement.
$\therefore$ effective spring constant $\left(K_S\right)=\frac{K\times K}{2K}=\frac{K}{2}$
frequency of oscillation $\left(n_s\right)=\frac{1}{2\pi }\sqrt{\frac{K_S}{m}}=\frac{1}{2\pi }\sqrt{\frac{K}{2m}}...\left(1\right)$
Case II : Two springs of equal spring constant are in parallel arrangement.
$\therefore$ effective spring constant $\left(K_P\right)=2K$
Frequency of oscillation $\left(n_P\right)=\frac{1}{2\pi }\sqrt{\frac{K_P}{m}}$
$=\frac{1}{2\pi }\sqrt{\frac{2K}{m}}...\left(2\right)$
From (1) & (2) we get
$\frac{n_S}{n_P}=\frac{\frac{1}{2\pi }\sqrt{\frac{K_S}{m}}}{\frac{1}{2\pi }\sqrt{\frac{K_P}{m}}}=\frac{1}{2}$