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} $