Q. Let $T_{1}and\ T_{2}$ be the time period of spring $A\ and\ B$ when mass M is suspended from one end of each spring. If both springs are taken in series and the same mass M is suspended from the series combination, the time period is T, then
A) $T = T_{1} + T_{2}$
B) $\frac{1}{T} = \frac{1}{T_{1}} + \frac{1}{T^{2}}$
C) $T^{2} = T_{1}^{2} + T_{2}^{2}$
D) $\frac{1}{T^{2}} = \frac{1}{T_{1}^{2}} + \frac{1}{T_{2}^{2}}$
Explanation
(c) $T_{1} = 2\pi\sqrt{\frac{M}{k_{1}}}\ \ \ or\ \ \ k_{1} = \frac{4\pi^{2}M}{T_{1}^{2}}$
And $k_{2} = \frac{4\pi^{2}M}{T_{2}^{2}}$
In series combination,
$k_{eff} = \frac{k_{1}k_{2}}{k_{1} + k_{2}} = \frac{4\pi^{2}M}{T_{1}^{2} + T_{2}^{2}}$
$\therefore\ \ \ \ T = 2\pi\sqrt{\frac{M}{k_{eff}}} = \sqrt{T_{1}^{2} + T_{2}^{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.