TS EAMCET · Physics · Oscillations

A body of mass \(0.3 \mathrm{~kg}\) hangs by a spring with a force constant of \(50 \mathrm{~N} / \mathrm{m}\). The amplitude of oscillations is damped and reaches \(\frac{1}{e}\) of its original value in about 100 oscillations. If \(\omega\) and \(\omega^{\prime}\) are the angular frequencies of undamped and damped oscillations respectively, then percentage of \(\left(\frac{\omega-\omega}{\omega}\right)\) is

A \(\left(\frac{1}{800 \pi}\right)\)
B \(\left(\frac{\pi^2}{600}\right)\)
C \(\left(\frac{1}{800 \pi^2}\right)\)
D \(\left(\frac{\pi}{400}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\left(\frac{1}{800 \pi^2}\right)\)

Step-by-step Solution

Detailed explanation

Hint The displacement of a spring mass oscillator, \(X=A e^{-b t / 2 m} \cos \left(\omega^{\prime} t+\phi\right)\) where, \(\omega^{\prime}=\) angular frequency of damped oscillation and \(\omega^{\prime}=\omega_0 \sqrt{1-\frac{b^2}{4 m^2 \omega_0^2}}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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