Question

If $\rho$ is the density of the material of a uniform rod and $\sigma$ is the breaking stress, and the length of the rod is such that the rod is just about to break due to its own weight when suspended vertically from a fixed support, then

• A. Length of the rod $\frac{\sigma }{\rho g}$
• B. Stress at a cross section perpendicular to the length of the rod, at one fourth the length of the rod above its lowest point $\frac{\sigma }{4}$
• C. Stress at all horizontal sections of the rod is same
• D. The rod is about to break from its mid point

Answer 1

Answer: (A), (B)

The correct options are
A Length of the rod $\frac{\sigma }{\rho g}$
B Stress at a cross section perpendicular to the length of the rod, at one fourth the length of the rod above its lowest point $\frac{\sigma }{4}$

The stress is max at the uppermost point and is equal to the weight of the rod divided by the area. At the highest point
stress = breaking stress $=\sigma$ = weight of rod/area of cross section of the rod
$=\frac{AL\rho g}{A}=L\rho g$.
$\therefore L=\frac{\sigma }{\rho g}$
The stress decreases linearly to zero at the lowest end.