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 $c$.
$t_1=\frac{c}{\sqrt{k_1}},t_2=\frac{c}{\sqrt{k_2}}$
$k_1=\frac{c^2}{t_1^2},k_2=\frac{c}{t_2^2}$
$t_{\text{eq }}=2\pi \sqrt{\frac{m}{c^2/t_1^2+c^2/t_2^2}}$
$t_{eq}=2\pi \sqrt{\frac{m{t}_1^2+t_2^2}{c^2{t}_2^2+c^2{t}_1^2}}$
$t_{eq}=\frac{t_1{t}_2}{\sqrt{t_1^2+t_2^2}}$