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Let $T_1$ and $T_2$ be the time periods of springs $A$ and $B$ when mass $M$ is suspended from one end of each spring. If both springs are taken in series and the same mass $M$ is suspended from the series combination, the time period is $T$, then
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The correct answer is:
$T^2=T_1^2+T_2^2$
$T_1=2 \pi \sqrt{\frac{M}{k_1}}$ or $k_1=\frac{4 \pi^2 M}{T_1^2}$ and $k_2=\frac{4 \pi^2 M}{T_2^2}$
In series combination, $k_{\text {eff }}=\frac{k_1 k_2}{k_1+k_2}=\frac{4 \pi^2 M}{T_1^2+T_2^2}$
$$
\therefore T=2 \pi \sqrt{\frac{M}{k_{\mathrm{eff}}}}=\sqrt{T_1^2+T_2^2} \text {. }
$$
In series combination, $k_{\text {eff }}=\frac{k_1 k_2}{k_1+k_2}=\frac{4 \pi^2 M}{T_1^2+T_2^2}$
$$
\therefore T=2 \pi \sqrt{\frac{M}{k_{\mathrm{eff}}}}=\sqrt{T_1^2+T_2^2} \text {. }
$$
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