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  4. Two springs of force constants K1 and K2 , respectively, are connected to a mass m , as shown. The frequency of oscillation of the mass is f . If both K1 and K2 are made four times their original values, the frequency of oscillation becomes <img class=img-fluid question-image alt=Question src=https://cdn.tardigrade.in/q/nta/p-flytdi9hgmq6.png />

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Q. Two springs of force constants $K_{1}$ and $K_{2}$ , respectively, are connected to a mass $m$ , as shown. The frequency of oscillation of the mass is $f$ . If both $K_{1}$ and $K_{2}$ are made four times their original values, the frequency of oscillation becomes

Question

NTA AbhyasNTA Abhyas 2020Oscillations

Solution:

In the given figure, two springs are connected in parallel. Therefore, the effective spring constant is given by
$K_{\text{eff}}=K_{1}+ \, K_{2}$
Solution
Frequency of oscillation,
$\textit{f}=\frac{1}{2 \pi }\sqrt{\frac{K_{\text{eff}}}{\textit{m}}}=\frac{1}{2 \pi }\sqrt{\frac{K_{1} + K_{2}}{\textit{m}}}$ ...........(i)
As $K_{1}$ and $K_{2}$ are increased by four times
New frequency,
$f^{\prime}=\frac{1}{2 \pi} \sqrt{\frac{4\left(K_1+K_2\right)}{m}}=2 f \quad$ ( using (i))