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(C) The spring constant $k$ is inversely proportional to the length $\ell$ of the spring,i.e.,$k \propto 1/\ell$.Let the length of the spring in figur

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$3$

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(C) The spring constant $k$ is inversely proportional to the length $\ell$ of the spring,i.e.,$k \propto 1/\ell$.Let the length of the spring in figure $B$ be $\ell$. Then the length of the spring in figure $A$ is $3\ell$.Let the spring constant of the spring in figure $B$ be $k_B = k$. Then the spring constant of the spring in figure $A$ is $k_A = k/3$.In figure $A$,the equivalent spring constant is $K_A = k/3$.In figure $B$,there are three identical springs in parallel,each with spring constant $k$. Thus,the equivalent spring constant is $K_B = k + k + k = 3k$.The time period of a spring-mass system is given by $T = 2\pi \sqrt{M/K}$.Therefore,the ratio of the time periods is:$\frac{T_A}{T_B} = \sqrt{\frac{K_B}{K_A}} = \sqrt{\frac{3k}{k/3}} = \sqrt{9} = 3$.

The springs in the figure are identical,but the length of the spring in $A$ is three times that in $B$. The ratio of the time periods $T_A/T_B$ is:

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