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)

$As,F_1=\frac{m4{\pi }^{2}a}{{\pi }^2}$ and $F_2=\frac{m4{\pi }^{2}a}{T_2^2}$
Net force, $F=F_1+F_2=\frac{4{\pi }^{2}ma}{T_1^2}+\frac{4{\pi }^{2}ma}{T_2^2}$
$=4{\pi }^{2}ma\left(\frac{1}{T_1^2}+\frac{1}{T_2^2}\right)$
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 $T^2=\frac{T_1^2{T}_2^2}{T_1^2+T_2^2}$