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Oscillations

A system of s...

Question

A system of springs with their springs constants are as shown in figure. The frequency of oscillation of the mass m will be (assuming the springs to be massless)

A

\frac{1}{2 π} \sqrt{\frac{k_{1} k_{2} (k_{3} + k_{4})}{[(k_{1} + k_{2}) + (k_{3} + k_{4}) + k_{1} k_{4}] m}}

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B

\frac{I}{2 π} \sqrt{\frac{k_{1} k_{2} (k_{3} + k_{4})}{[(k_{1} + k_{2}) + (k_{3} + k_{4}) + k_{1} k_{2}] m}}

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C

\frac{1}{2 π} \sqrt{\frac{k_{1} k_{2} (k_{3} + k_{4})}{[(k_{1} + k_{2}) + (k_{3} + k_{4}) + k_{1} k_{2}] m}}

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D

\frac{1}{2 π} \sqrt{\frac{(k_{1} + k_{2}) + (k_{3} + k_{4}) {+ k}_{1} k_{2}}{[(k_{1} + k_{2}) + (k_{3} + k_{4}) + k_{1} k_{2}] m}}

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Solution

The correct option is C $\frac{1}{2 π} \sqrt{\frac{k_{1} k_{2} (k_{3} + k_{4})}{[(k_{1} + k_{2}) + (k_{3} + k_{4}) + k_{1} k_{2}] m}}$
Equivalent spring constant, $k_{e q} = \frac{\frac{k_{1} k_{2}}{k_{1} + k_{2}} \times (k_{3} + k_{4})}{\frac{k_{1} k_{2}}{k_{1} + k_{2}} + (k_{3} + k_{4})} = \frac{k_{1} k_{2} (k_{3} + k_{4})}{(k_{1} + k_{2}) + (k_{3} + k_{4}) + k_{1} k_{2}}$

$∴$ , The frequency of oscillation of the mass m, $f = \frac{1}{2 π} \sqrt{\frac{k_{e q}}{m}} = \frac{1}{2 π} \sqrt{\frac{k_{1} k_{2} (k_{3} + k_{4})}{m [(k_{1} + k_{2}) + (k_{3} + k_{4}) + k_{1} k_{2}]}}$

Hence, option $(C)$ is correct.

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