Question 174 of 280

A uniform rod of length $l$ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is $T(x)$ at a distance $x$ from the axis, then which of the following graphs depicts it most closely?

ABCD

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Answer: (C)

The mass of the rod at a distance $x$ from the axis is $dm=\dfrac{M}{L}.dx$
$T=(dm){\omega }^{2}x$
Integrating the above equation from $x$ to $L$
$={\int }_{x}^{L}\dfrac{M}{L}{\omega }^{2}x dx$
$=\dfrac{M}{L}{\omega }^{2}{\left[\dfrac{{x}^{2}}{2}\right]}_{x}^{L}$
$\dfrac{M}{2L}{\omega }^{2}\left[{L}^{2}-{x}^{2}\right]$
The above equation depicts the relation between tension and length of rod as downward opening parabola.