Question

Two springs of spring constants $k_{1}$ and $k_{2}$ are joined in series and a mass $m$ is attached to them as shown in figure. The time-period of oscillations of the springs is

• A. $T=\pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{k_{1} k_{2}}}$
• B. $T=2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{k_{1} k_{2}}}$
• C. $T=2 \pi \sqrt{\frac{m}{k_{1}+k_{2}}}$
• D. $T=2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{2 k_{1} k_{2}}}$

Answer 1

Answer: (B)

$T=2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{k_{1} k_{2}}}$

Answer 2

In series combination spring constants of combination
$\frac{1}{k_{s}}=\frac{1}{k_{1}}+\frac{1}{k_{2}}$
$\Rightarrow k_{s}=\frac{k_{1} k_{2}}{k_{1}+k_{2}}$ Time period of combination
$T=2 \pi \sqrt{\frac{m}{k_{s}}}=2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{k_{1} k_{2}}}$