Question

$T_1$ is the time period of oscillation of a body suspended from a spring and $T_2$ is the time period of oscillation of the same body suspended from another spring. If the same body suspended from a series combination of the same two springs, then its time period of oscillation is:

• A. $T_1+T_2$
• B. $\frac{T_1{T}_2}{T_1+T_2}$
• C. $\frac{T_1{T}_2}{\sqrt{{T_1}^2+{T_2}^2}}$
• D. $\sqrt{{T_1}^2+{T_2}^2}$

Answer 1

The correct option is D $\sqrt{{T_1}^2+{T_2}^2}$

$T=2\pi \sqrt{\frac{m}{k}}$
for combination of springs $\frac{1}{k}=\frac{1}{k_1}+\frac{1}{k_2}$
$\therefore T_1=2\pi \sqrt{\frac{m}{k_1}}$
$T_2=2\pi \sqrt{\frac{m}{k_2}}$
$T=2\pi \sqrt{\frac{m}{k}}$
$T_1^2+T_2^2=\left(2\pi {)}^2\right(\frac{m}{k_1}+\frac{m}{k_2})=(2\pi {)}^2\frac{m}{k}=T^2$
Thus we get
$T=\sqrt{T_1^2+T_2^2}$