A heavy uniform rod is hanging vertically from a fixed support. It is stretched by its own weight. The diameter of the rod is

The correct option is: Smallest at the top and gradually increases down the rod.

Let us analyze the situation step-by-step to understand the physical reasoning behind this conclusion:
1. Force Distribution due to Gravity:
Consider a heavy uniform rod of length $L$ and mass $M$ hanging vertically from a fixed support at the top ($y=L$, where $y=0$ is the free bottom end).
At any cross-section at a distance $y$ from the bottom end, the tension $T\left(y\right)$ in the rod must balance the weight of the portion of the rod hanging below it.
The mass of the portion below $y$ is:

$m\left(y\right)=\frac{M}{L}y$

Therefore, the tension $T\left(y\right)$ at a height $y$ from the bottom is:

$T\left(y\right)=m\left(y\right)g=\frac{Mg}{L}y$

From this expression, we can see that:
- At the bottom of the rod ($y=0$), the tension is zero: $T\left(0\right)=0$.
- At the top of the rod ($y=L$), the tension is maximum and equals the total weight of the rod: $T\left(L\right)=Mg$.
Thus, the longitudinal stress (tension per unit area) is greatest at the top support and decreases gradually down the rod to zero at the bottom.

2. Elastic Deformations (Poisson's Effect):
When a material is subjected to a longitudinal tensile stress, it undergoes longitudinal extension (stretching) along the direction of the force.
According to Poisson's ratio ($\nu$), a longitudinal extension is accompanied by a lateral contraction (a decrease in cross-sectional dimensions, and hence a decrease in diameter).
The lateral strain ${\epsilon }_{\mathrm{lateral}}$ is given by:

${\epsilon }_{\mathrm{lateral}}=-\nu ·{\epsilon }_{\mathrm{longitudinal}}$

Since longitudinal strain is directly proportional to the tension $T\left(y\right)$, the lateral contraction (and thus the thinning of the rod) is directly proportional to the tension:
- At the top of the rod, where the tension (and longitudinal stretching stress) is maximum, the lateral contraction is also at its maximum. Therefore, the diameter of the rod is smallest at the top.
- As we move down the rod, the tension decreases continuously, resulting in less longitudinal stress, less lateral contraction, and consequently a larger diameter.
- At the bottom of the rod, where tension is zero, there is no stretching and no lateral contraction. Hence, the diameter remains closest to its unstretched value.

Conclusion:
Due to Poisson's ratio and the weight distribution, the diameter of the hanging rod is smallest at the top support and gradually increases as we move down the rod.