Question

A block of mass $m$ is suspended separately by two different springs have time period $t_{1}$ and $t_{2}$. If same mass is connected to parallel combination of both springs, then its time period is given by

• A. $\frac{t_{1} t_{2}}{t_{1}+t_{2}}$
• B. $\frac{t_{1} t_{2}}{\sqrt{t_{1}^{2}+t_{2}^{2}}}$
• C. $\sqrt{\frac{t_{1} t_{2}}{t_{1}+t_{2}}}$
• D. $t_{1}+t_{2}$

Answer 1

Answer: (B)

$t_{1}=2 \pi \sqrt{\frac{m}{k_{1}}}, t_{2}=2 \pi \sqrt{\frac{m}{k_{a}}} \,, t_{ eq }=2 \pi \sqrt{\frac{m}{k_{1}+k_{2}}}$
Let $2 \pi \sqrt{m}$ be any constant $c$.
$t_{1}=\frac{c}{\sqrt{k_{1}}} \,\,, t_{2}=\frac{c}{\sqrt{k_{2}}}$
$k_{1}=\frac{c^{2}}{t_{1}^{2}} \,\,, k_{2}=\frac{c}{t_{2}^{2}}$
$t_{\text {eq }}=2 \pi \sqrt{\frac{m}{c^{2} / t_{1}^{2}+c^{2} / t_{2}^{2}}}$
$t_{ eq }=2 \pi \sqrt{\frac{m t_{1}^{2}+t_{2}^{2}}{c^{2} t_{2}^{2}+c^{2} t_{1}^{2}}}$
$t_{ eq }=\frac{t_{1} t_{2}}{\sqrt{t_{1}^{2}+t_{2}^{2}}}$