Question

A uniform metal rod is rotated in a horizontal plane about a vertical axis passing through its end at a uniform rate. The tension in the rod is:

• A. Same at all points
• B. Different at different points and maximum at the centre of rod
• C. Different at different points and minimum at the axis of rotation.
• D. Different at different points and maximum at the axis of rotation

Answer 1

The correct option is C Different at different points and minimum at the axis of rotation.

We consider a small element of mass $dm$ at a distance $x$ from the center. The tensions acting on its two faces are $T$ and $T+dT$
Thus using the force balance we have
$(T+dT-T)=dm{\omega }^{2}x.$ (where $dm=\left(\frac{dx}{L}\right)m$)
$dT=dm{\omega }^{2}x.$
$dT=\frac{m{\omega }^{2}xdx}{L}$
or
$T=\frac{m{\omega }^2}{L}{\int }_0^{x}xdx$
$T=\frac{m{\omega }^2}{2L}\left(x^2\right)$
$\therefore$ $T$ is different at different points and is minimum at the axis of rotation.