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  4. A thin uniform heavy rod of length l hangs from a horizontal axis passing through one end. The initial angular velocity ω that must be imparted to it to rotate it through 90° is :

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Q. A thin uniform heavy rod of length $l$ hangs from a horizontal axis passing through one end. The initial angular velocity $\omega$ that must be imparted to it to rotate it through $90^{\circ}$ is :Physics Question Image

System of Particles and Rotational Motion

Solution:

By conservation of energy we use
$\frac{1}{2} I \omega^{2}=m g \frac{l}{2}$
image
$\frac{m l^{2}}{3} \omega^{2}=m g l$
$\Rightarrow \omega^{2}=\frac{3 g}{l}$
$\Rightarrow \omega=\sqrt{\frac{3 g}{l}}$