Question

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. $T_{2}^{2} T_{1}^{2}$
• C. $T_{1}^{2}+T_{2}^{2}$
• D. $T_{1}^{2} T_{2}^{2} /\left(T_{1}^{2}+T_{2}^{2}\right)$

Answer 1

Answer: (D)

$As , F_{1}=\frac{m 4 \pi^{2} a }{\pi^{2}}$ and $F_{2}=\frac{m 4 \pi^{2} a }{T_{2}^{2}}$
Net force, $F =F_{1}+F_{2}=\frac{4 \pi^{2} m a}{T_{1}^{2}}+\frac{4 \pi^{2} m a}{T_{2}^{2}} $
$=4 \pi^{2} m a\left(\frac{1}{T_{1}^{2}}+\frac{1}{T_{2}^{2}}\right) $
or $\frac{4 \pi^{2} m a}{T^{2}} =4 \pi^{2} m a\left(\frac{1}{T_{1}^{2}}+\frac{1}{T_{2}^{2}}\right)$
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 $T^{2}=\frac{T_{1}^{2} T_{2}^{2}}{T_{1}^{2}+T_{2}^{2}}$