Animated Solution for Physics - Properties of Solids and Liquids: The elastic limit of brass is $379\text{ MPa}$. What should be the minimum diameter of a brass rod, if it is to support a $400\text{ N}$ load without exceeding its elastic limit?

Imagine a brass rod hanging from the ceiling, supporting a heavy load of $400\text{ N}$. We are tasked with finding out exactly how thin this rod can be without undergoing permanent deformation.

To prevent this permanent stretching, the internal stress developed within the rod must strictly not exceed its elastic limit. The elastic limit is the maximum stress a material can withstand and still return to its original shape once the load is removed.

Stress is defined as the internal restoring force per unit cross-sectional area. Mathematically, it is expressed as:

For the rod to remain within its elastic behavior, we enforce the condition:

Since the rod is cylindrical, its cross-sectional area $A$ can be written in terms of its diameter $d$ as:

Substituting this area into our stress inequality gives us a direct relationship between the applied force, the elastic limit, and the required diameter:

Now, we rearrange our equation to solve for the minimum diameter squared. By cross-multiplying, we isolate the diameter term:

It is time to plug in our known values. The applied force $F$ is $400\text{ N}$, and the elastic limit ${\sigma }_{\text{limit}}$ is $379\text{ MPa}$. Remember that the prefix "Mega" stands for $10^6$, so we must convert this to standard SI units:

Evaluating this expression yields the square of the diameter:

Taking the square root gives us the final minimum diameter:

Converting this to millimeters, we get $1.16\text{ mm}$. If the rod were any thinner than this, the stress would exceed $379\text{ MPa}$, and the brass would permanently deform!