Question

Two springs having force constants $k$ each are arranged in parallel and in series. A mass $M$ is attached to two arrangements separately. If time period in first case is $T_1$ and in second case is $T_2$, then ratio $\frac{T_1}{T_2}$ is :

• A. 1.5
• B. 3.2
• C. 0.5
• D. 2.1

Answer 1

Answer: (C)

In first set up, the springs are joined in parallel and in second, the springs are joined in series.
When springs are connected in parallel, the effective spring constant is

$k_1=k+k=2k$
Hence, time period
$T_1=2\pi \sqrt{\frac{M}{k_1}}=2\pi \sqrt{\frac{M}{2k}}$ ...(i)
When springs are connected in series, the effective spring constant is

$\frac{1}{k_2}=\frac{1}{k}+\frac{1}{k}=\frac{2}{k}$
$\Rightarrow k_2=\frac{k}{2}$
Therefore, time period,
$T_2=2\pi \sqrt{\frac{M}{k_2}}$
$=2\pi \sqrt{\frac{M}{k/2}}$
$T_2=2\pi \sqrt{\frac{2M}{k}}$ ...(ii)
Dividing Eq. (i) by Eq. (ii), we have
$\frac{T_1}{T_2}=\sqrt{\frac{M/2k}{2M/k}}=\sqrt{\frac{1}{4}}=\frac{1}{2}=0.5$