Question

Two identical springs of spring constant k are connected 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. 2:1
• B. 1.5
• C. 2.1
• D. 2.5

Answer 1

Answer: (B)

In first set up, the springs are joined in series and in second, the springs are joined in parallel. When springs are connected in series the effective spring constant is
$\frac{1}{k}=\frac{1}{k}+\frac{1}{k}=\frac{2}{k}$
$\Rightarrow$ $k=\frac{k}{2}$
Hence, frequency $n=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$
$=\frac{1}{2\pi }\sqrt{\left(\frac{k}{2m}\right)}$ ..(i)
When springs are connected in parallel, the effective force constant is
$k=k+k=2k$
Therefore, frequency is $n=\frac{1}{2\pi }\sqrt{\frac{2k}{m}}$ ..(ii)
Dividing Eq. (i) by Eq. (ii), we get
$\frac{n}{n}=\frac{\sqrt{(k/2m)}}{\sqrt{2k/m}}=\frac{1}{2}$