Physics - SHM Question with Solution | TestHub

Let the restoring forces be $F_1=-k_{1}x$ and $F_2=-k_{2}x$.

The time periods are $T_1=2\pi \sqrt{\frac{m}{k_1}}$ and $T_2=2\pi \sqrt{\frac{m}{k_2}}$.

From these, we get $k_1=\frac{4{\pi }^{2}m}{T_1^2}$ and $k_2=\frac{4{\pi }^{2}m}{T_2^2}$.

When both forces act in the same direction, the net restoring force is $F=F_1+F_2=-(k_1+k_2)x$.

The effective force constant is $k_{eff}=k_1+k_2$.

The new time period $T$ is given by $T=2\pi \sqrt{\frac{m}{k_{eff}}}$, which implies $k_{eff}=\frac{4{\pi }^{2}m}{T^2}$.

Substituting the values of $k_1$, $k_2$, and $k_{eff}$:

$\frac{4{\pi }^{2}m}{T^2}=\frac{4{\pi }^{2}m}{T_1^2}+\frac{4{\pi }^{2}m}{T_2^2}$

$\frac{1}{T^2}=\frac{1}{T_1^2}+\frac{1}{T_2^2}=\frac{T_1^2+T_2^2}{T_1^2{T}_2^2}$

$T^2=\frac{T_1^2{T}_2^2}{T_1^2+T_2^2}$

Answer: $\frac{T_1^2{T}_2^2}{(T_1^2+T_2^2)}$