\(2\,m\) લંબાઈ અને \(A\) જેટલું આડછેદનું ક્ષેત્રફળ અને \(d\) ઘનતા ધરાવતો એક પાટલો નિયમિત સળીયો,તેની લંબાઈ ને લંબરુપે કેન્દ્ર માંથી પસાર થતી અક્ષને અનુલક્ષીને \(\omega\) જેટલા કોણીય વેગથી ભ્રમણ કરે છે. ચાક્ગતિ ઊર્જાના સ્વરૂપમાં \(\omega\) નું મૂલ્ય \(\sqrt{\frac{\alpha E }{ Ad }}\) છે. \(\alpha\) નું મૂલ્ય \(...............\) હશે.

A \(2\)
B \(1\)
C \(4\)
D \(3\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(3\)

Step-by-step Solution

Detailed explanation

\(( KE )_{\text {Rotational }}=\frac{1}{2} I \omega^2= E\) \(E =\frac{1}{2} \frac{ m \ell^2}{12} \omega^2\) \(E =\frac{1}{2} \frac{ dA \ell^3}{12} \omega^2\) \(E =\frac{ dA (2)^3}{24} \omega^2\) \(\sqrt{\frac{3 E }{ dA }}=\omega\) \(\alpha=3 \text { Ans. }\)

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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