Q. A body has a time period $T_{1}^{\ }$ under the action of one force and $T_{2}^{\ }$ under the action of another force, the square of the time period when both the forces are acting in the same direction is
A) $T_{1}^{2}T_{2}^{2}$
B) $T_{1}^{2}/T_{2}^{2}$
C) $T_{1}^{2} + T_{2}^{2}$
D) $T_{1}^{2}T_{2}^{2}/(T_{1}^{2} + T_{2}^{2}$)
Explanation
(d) $F_{1} = \frac{m4\pi^{2}a}{\pi^{2}}$ and $F_{2} = \frac{m4\pi^{2}a}{T_{2}^{2}}$
$F = F_{1} + F_{2} = \frac{4\pi^{2}ma}{T_{1}^{2}} + \frac{4\pi^{2}ma}{T_{2}^{2}}$
$= 4\pi^{2}ma\ \left( \frac{1}{T_{1}^{2}} + \frac{1}{T_{2}^{2}} \right)$
Or $\frac{4\pi^{2}ma}{T^{2}} = 4\pi^{2}ma\left( \frac{1}{T_{1}^{2}} + \frac{1}{T_{2}^{2}} \right)$
Or $\frac{1}{T^{2}} = \frac{1}{T_{1}^{2}} + \frac{1}{T_{2}^{2}}\ $
Or $\frac{1}{T^{2}} = \frac{T_{1}^{2} + T_{2}^{2}}{T_{1}^{2}T_{2}^{2}}\ $ or $T^{2} = \frac{T_{1}^{2}T_{2}^{2}}{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.