# A particle at the end of a spring executes simple harmonic motion with a period t1 , while the corresponding period for another spring is t2 . If the period of oscillation with the two springs in series is T, then Option 1) Option 2) Option 3) Option 4)

As we learnt in

Series combination of spring -

- wherein

$\frac{1}{K_{eq}}= \frac{1}{K_{1}}+ \frac{1}{K_{2}}$

$K_{1}and\ K_{2}$ are spring constants of spring 1 & 2 respectively.

When springs are in series, $\dpi{100} k=\frac{k_{1}k_{2}}{k_{1}+k_{2}}$

For first spring,  $\dpi{100} t_{1}=2\pi \sqrt{\frac{m}{k_{1}}}$

For second spring  $\dpi{100} t_{2}=2\pi \sqrt{\frac{m}{k_{2}}}$

$\dpi{100} \therefore \; \; \; t_{1}^{2}+t_{2}^{2}=\frac{4\pi ^{2}m}{k_{1}}+\frac{4\pi ^{2}m}{k_{2}}=4\pi ^{2}m\left ( \frac{k_{1}+k_{2}}{k_{1}k_{2}} \right )$

$\dpi{100} or \; \; \; t_{1}^{2}+t_{2}^{2}=\left [ 2\pi \sqrt{\frac{m(k_{1}+k_{2})}{k_{1}k_{2}}} \right ]^{2}$

$\dpi{100} or \; \; \; t_{1}^{2}+t_{2}^{2}=T^{2}$

Correct option is 2.

Option 1)

This is an incorrect option.

Option 2)

This is the correct option.

Option 3)

This is an incorrect option.

Option 4)

This is an incorrect option.

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