एक खोखला आईसक्रीम शंकु को चित्र में दिखाया गया है। (इसका ऊपरी भाग खुला है।) यदि इसका द्रव्यमान \(M\), ऊपरी भाग की त्रिज्या \(R\), तथा ऊँचाई, \(H\) हो, तो इसकी अक्ष के सापेक्ष जड़त्व आघूर्ण है?

A \(\frac{ MR ^{2}}{2}\)
B \(\frac{ MH ^{2}}{3}\)
C \(\frac{ MR ^{2}}{3}\)
D \(\frac{ M \left( R ^{2}+ H ^{2}\right)}{4}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{ MR ^{2}}{2}\)

Step-by-step Solution

Detailed explanation

Area \(=\pi R \ell=\pi R \left(\sqrt{ H ^{2}+ R ^{2}}\right)\) Area of element \(d A =2 \pi rd \ell==2 \pi r \frac{ dh }{\cos \theta}\) mass of element \(dm =\frac{ M }{\pi R \sqrt{ H ^{2}+ R ^{2}}} \times \frac{2 \pi rdh }{\cos \theta}\)…

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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