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(B) The time period of a spring-mass system is given by $t = 2\pi \sqrt{\frac{m}{k}}$.For the two springs,we have $t_1 = 2\pi \sqrt{\frac{m}{k_1}}$ an

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$T^2 = t_1^2 + t_2^2$

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(B) The time period of a spring-mass system is given by $t = 2\pi \sqrt{\frac{m}{k}}$.For the two springs,we have $t_1 = 2\pi \sqrt{\frac{m}{k_1}}$ and $t_2 = 2\pi \sqrt{\frac{m}{k_2}}$.Squaring these,we get $t_1^2 = 4\pi^2 \frac{m}{k_1}$ and $t_2^2 = 4\pi^2 \frac{m}{k_2}$.When two springs are connected in series,the effective spring constant $k$ is given by $\frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2}$,which implies $k = \frac{k_1 k_2}{k_1 + k_2}$.The time period $T$ for the series combination is $T = 2\pi \sqrt{\frac{m}{k}}$.Squaring this,$T^2 = 4\pi^2 \frac{m}{k} = 4\pi^2 m \left( \frac{1}{k} \right) = 4\pi^2 m \left( \frac{1}{k_1} + \frac{1}{k_2} \right)$.Substituting the expressions for $t_1^2$ and $t_2^2$,we get $T^2 = 4\pi^2 \frac{m}{k_1} + 4\pi^2 \frac{m}{k_2} = t_1^2 + t_2^2$.

$A$ particle at the end of a spring executes simple harmonic motion with a period $t_1$,while the corresponding period for another spring is $t_2$. If the period of oscillation with the two springs in series is $T$,then

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