Question

The frequency of oscillation of system shown in the figure will be

• A. $\frac{1}{2 \pi} \sqrt{\frac{k}{M}}$
• B. $\frac{1}{2 \pi} \sqrt{\frac{2 k}{5 M}}$
• C. $\frac{1}{2 \pi} \sqrt{\frac{k}{5 M}}$
• D. $\frac{1}{2 \pi} \sqrt{\frac{2 k}{M}}$

Answer 1

Answer: (B)

Upper two springs are in series.
Therefore, their resultant spring constant $\frac{1}{k}$,
$=\frac{1}{k}+\frac{1}{k}=\frac{2}{k}$
$\therefore k'=\frac{k}{2}$ Lower two springs are in parallel.
Therefore their resultant spring constant,
$k''=k+k=2 k$
Now, springs of constant $k'$ and $k''$ are in series.
Therefore, their resultant spring constant
$\frac{1}{k_{1}}=\frac{2}{k}+\frac{1}{2 k}=\frac{4+1}{2 k} $
$k_{1}=\frac{2 k}{5}$
$\therefore $ Frequency of oscillation
$f=\frac{1}{T}=\frac{1}{2 \pi} \sqrt{\frac{k_{1}}{M}}$
$=\frac{1}{2 \pi} \sqrt{\frac{2 k}{5 M}}$