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  • A massless rod is fitted with a sphere of mass ' M ' on its top end such that the centre of the sphere lies at distance ' <img alt="l" src="https://entrancecorner.oncodecogs.c

A massless rod is fitted with a sphere of mass ' M ' on its top end such that the centre of the sphere lies at distance ' l ' from the other end of the rod. If the rod is kept vertically and then allowed to fall, find the angular velocity of the rod when it makes an angle of \frac{\pi}{3} from the vertical. [Given, length of the rod =l=10 \mathrm{~cm}] , g=10 \mathrm{~m} / \mathrm{s}^2

Assume that the rod does not slip on the ground.

Option: 1

5 \mathrm{rad} / \mathrm{s}


Option: 2

5 \sqrt{3} \mathrm{rad} / \mathrm{s}


Option: 3

10 \mathrm{rad} / \mathrm{s}


Option: 4

\frac{5 \sqrt{3}}{2} \mathrm{rad} / \mathrm{s}


Answers (1)

best_answer

Applying conservation of energy:-

    \begin{aligned} & M g l=M g l \cos \alpha+\frac{1}{2}(Ml^2)\omega^2 \\ & 2 g=2 g \cos \alpha+l \omega^2 \\ & 2 g(1-\cos \alpha)=l \omega^2 \\ & 2 g\left(2 \sin ^2 \frac{\alpha}{2}\right)=l \omega^2 \\ & \omega^2=4\left(\frac{g}{l}\right) \sin ^2 \frac{\alpha}{2} \\ & \omega=2 \sin \left(\frac{\alpha}{2}\right) \sqrt{\frac{g}{l}} \\ & \omega=2 \sin \left(\frac{\pi}{6}\right) \sqrt{\frac{10}{0.1}} \\ & \omega=2 \times \frac{1}{2} \times 10=10 \mathrm{rad} / \mathrm{s} \end{aligned}

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SANGALDEEP SINGH

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