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?

$$\text{Let } m \text{ be the total mass of the uniform rod of length } l. \text{ Linear mass density, } \lambda = \frac{m}{l}.$$

$$\text{Consider a small element of length } dr \text{ at a distance } r \text{ from the axis } (r \ge x):$$

$$dm = \lambda \, dr = \frac{m}{l} \, dr$$

$$\text{The centripetal force required by this small element rotating at angular speed } \omega \text{ is:}$$ 

$$dF = (dm) \omega^2 r = \frac{m}{l} \omega^2 r \, dr$$

$$\text{Integrating from } r = x \text{ to the free end } r = l \text{ (where tension } T(l) = 0\text{):}$$ 

$$-dT = \frac{m}{l} \omega^2 r \, dr \implies \int_{T(x)}^{0} dT = -\int_{x}^{l} \frac{m}{l} \omega^2 r \, dr$$

$$0 - T(x) = -\frac{m\omega^2}{2l} \left[ r^2 \right]_x^l \implies T(x) = \frac{m\omega^2}{2l} (l^2 - x^2)$$

$$\text{This represents a downward-opening parabola with } T(0) = \frac{m\omega^2 l}{2} \text{ and } T(l) = 0.$$