द्रव्यमान \(M\) तथा त्रिज्या \(R\) की एक पतली डिस्क का प्रति इकाई क्षेत्रफल द्रव्यमान \(\sigma( r )= kr ^{2}\) है जहाँ \(r\) केन्द्र से दूरी है। डिस्क के केन्द्र से जाने वाली तथा इसके लम्बवत् अक्ष के परितः जड़त्व आघूर्ण होगा।

A \(\frac{{M{R^2}}}{2}\)
B \(\frac{{M{R^2}}}{3}\)
C \(\frac{{M{R^2}}}{6}\)
D \(\frac{{2M{R^2}}}{3}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{{2M{R^2}}}{3}\)

Step-by-step Solution

Detailed explanation

\({I_{Disc}} = \int\limits_0^R {\left( {dm} \right){r^2} \Rightarrow {I_{Disc}} = \int\limits_0^R {\left( {\sigma 2\pi rdr} \right){r^2}} } \) \({I_{Disc}} = \int\limits_0^R {\left( {k{r^2}2\pi rdr} \right)} {r^2}\,\,\,\,\,\,\,\,\,Mass\,of\,disc\)…

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.

Same subject

सूची-\(I\) का सूची-\(II\) के साथ मिलान कीजिए।

	List \(-I\)		List \(-II\)
\(A\).	श्यानता गुणांक	\(I\).	\([M L^2T^{–2}]\)
\(B\).	पृष्ठ तनाव	\(II\).	\([M L^2T^{–1}]\)
\(C\).	कोणीय संवेग	\(III\).	\([M L^{-1}T^{–1}]\)
\(D\).	घूर्णन गतिज ऊर्जा	\(IV\).	\([M L^0T^{–2}]\)

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