Question

The time period of oscillations of a block attached to a spring is $t_{1}$ . When the spring is replaced by another spring, the time period of the block is $t_{2}$ . If both the springs are connected in series and the block is made to oscillate using the combination, then the time period of the block is

• A. $\textit{T}=t_{1}+t_{2}$
• B. $\textit{T}^{2}=t_{1}^{2}+t_{2}^{2}$
• C. $T^{- 1}=t_{1}^{- 1}+t_{2}^{- 1}$
• D. $T^{- 2}=t_{1}^{- 2}+t_{2}^{- 2}$

Answer 1

Answer: (B)

When springs are in series, $k=\frac{k_{1} k_{2}}{k_{1} + k_{2}}$
For first spring, $t_{1}=2\pi \sqrt{\frac{\textit{m}}{k_{1}}}$
For Second spring, $t_{2}=2\pi \sqrt{\frac{\textit{m}}{k_{2}}}$
$\therefore $ $t_{1}^{2}+t_{2}^{2}=\frac{4 \left(\pi \right)^{2} \textit{m}}{k_{1}}+\frac{4 \left(\pi \right)^{2} \textit{m}}{k_{2}}=4\left(\pi \right)^{2}\textit{m}\left(\frac{k_{1} + k_{2}}{k_{1} k_{2}}\right)$
or $t_{1}^{2}+t_{2}^{2}=\left(\left[2 \pi \sqrt{\frac{\textit{m} \left(k_{1} + k_{2}\right)}{k_{1} k_{2}}}\right]\right)^{2}$
or $t_{1}^{2}+t_{2}^{2}=\textit{T}^{2}$ .