आंतरिक त्रिज्या \(10 \,cm\), बाह्य त्रिज्या \(20 \,cm\) तथा लम्बाई \(30 \,cm\) के एक खोखले बेलन का जड़त्व आघूर्ण, उसकी अक्ष के परितः I है। उसी द्रव्यमान के एक ऐसे खोखले एवं पतले बेलन की त्रिज्या, जिसका अपने अक्ष के परितः जड़त्व आघूर्ण \(I\) ही है, होगी। (\(cm\) में)

A \(12\)
B \(16\)
C \(14\)
D \(18\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(16\)

Step-by-step Solution

Detailed explanation

\(r = 10 cm\) mass per unit \(R = 20 cm\) area \(Mass = m\) \(\sigma = \frac{m}{{\pi \left( {{R^2} - {r^2}} \right)}}\) Consider an element of radius \(x\) and thickness \(dx\) Mass of element \(dm =\) \(\sigma \,2\pi \times x\left( {dx} \right)\) Moment of inertia of element,…

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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