Question

a uniform rod of mass m and length l rotates in a horizontal plane with an angular velocity w about a vertical axis passing through one end. the tension in the rod at a distance x from the axis is?

Answer 1

$$\begin{gathered} TheM.I.oftherodverticalaxispas\sin gthroughitsone \\ oftheendsisgivenby; \\ I=\frac{1}{4}m{l}^2+\frac{1}{12}m{l}^2=\frac{1}{3}m{l}^2 \\ Theenergyofthesystem=\frac{1}{2}I{\omega }^2=\frac{1}{2}\times \frac{1}{3}m{l}^2{\omega }^2=\frac{1}{6}m{l}^2{\omega }^2 \\ So\tau x=\frac{1}{6}m{l}^2{\omega }^2 \\ or\tau =\frac{1}{6x}m{l}^2{\omega }^2 \\ Thisistherequiredtension\tau =\frac{1}{6x}m{l}^2{\omega }^2. \end{gathered}$$