Question 189 of 280

A thin uniform rod of length $\ell$ and mass $m$ is swinging freely about a horizontal axis passing through its end. Its maximum angular speed is $\omega$. Its centre of mass rises to a maximum height of

A$\frac{1}{3} \frac{\ell^2 \omega^2}{g}$B$\frac{1}{6} \frac{\ell \omega}{\mathrm{g}}$C$\frac{1}{2} \frac{\ell^2 \omega^2}{g}$D$\frac{1}{6} \frac{\ell^2 \omega^2}{\mathrm{~g}}$

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Answer: (D) $\frac{1}{6} \frac{\ell^2 \omega^2}{\mathrm{~g}}$

$$\begin{aligned} & \text { } \\ & \text { T.E }=T . E_{\mathrm{f}} \\ & \frac{1}{2} I \omega^2=m g h \\ & \frac{1}{2} \times \frac{1}{3} m \ell^2 \omega^2=m g h \Rightarrow h=\frac{1}{6} \frac{\ell^2 \omega^2}{\mathrm{~g}} \end{aligned}$$