આકૃતિમાં દર્શાવ્યા અનુસાર \('a'\) ત્રિજ્યા ધરાવતી વર્તુળાકાર તક્તિમાંથી \(\left(\frac{ a }{2}\right)\) જેટલી ત્રિજ્યા ધરાવતું વર્તુળાકાર છિદ્ર કાપવામાં આવે છે. બિંદુ \('O'\) ની સાપેક્ષ બાકી રહેતા ભાગ માટે ગુરુત્વીય કેન્દ્ર .......... છે.

A \(\frac{1}{6} a\)
B \(\frac{10}{11} a\)
C \(\frac{5}{6} a\)
D \(\frac{2}{3} a\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{5}{6} a\)

Step-by-step Solution

Detailed explanation

Let \(\sigma\) be the uniform mass density of disc then \(x _{ COM }=\frac{\left(\sigma \pi a ^{2}\right) a -\sigma \pi\left(\frac{ a ^{2}}{4}\right) \times \frac{3 a }{2}}{\sigma \pi a ^{2}-\frac{\sigma \pi a ^{2}}{4}}\) \(=\frac{a-\frac{3 a}{8}}{1-\frac{1}{4}}=\frac{5 a}{6}\)

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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