Animated Solution for Physics - Properties of Solids and Liquids: A wire fixed at the upper end stretches by length $l$ by applying a force $F$. The work done in stretching is

A Wire Under Tension

Imagine a sturdy wire hanging vertically from a rigid ceiling. In its natural, undisturbed state, let's say it has an original length of $L$ and a uniform cross-sectional area of $A$.

Now, suppose we attach a heavy load to its bottom end, applying a downward force $F$. Because the wire is made of an elastic material, it doesn't just sit there—it yields slightly. It stretches by an additional length, which we will call $l$. The question we want to answer is: how much work did we do in stretching this wire?

In physics, work done on a system is never lost; it is transferred. When you stretch a wire, you are doing work against the intermolecular forces that want to keep the wire in its original shape. This work is stored within the wire as Elastic Potential Energy ($U$).

The fundamental formula for the elastic potential energy stored in a stretched wire is given by:

This equation is a beautiful synthesis of the material's internal state. Let's break down its components based on our physical setup.

First, we define Stress. Stress is the internal restoring force per unit area. Since the wire is in equilibrium, the internal restoring force equals the applied force $F$. Therefore:

Next, we look at Strain. Strain is the measure of deformation, defined as the fractional change in length:

Finally, the Volume of our cylindrical wire is simply its cross-sectional area multiplied by its original length:

Now, we substitute these three fundamental definitions back into our energy equation:

Watch what happens next. It is one of those deeply satisfying moments in algebra. The area $A$ in the denominator of the stress term perfectly cancels out the area $A$ in the volume term. Similarly, the original length $L$ in the denominator of the strain term cancels out the $L$ in the volume term.

We are left with a remarkably elegant and simple result:

If you prefer a more visual approach, there is an alternative way to arrive at this exact same result using a graph.

According to Hooke's Law, within the elastic limit, the extension of a wire is directly proportional to the applied force. If we plot a graph of Force ($F$) on the y-axis versus Extension ($x$) on the x-axis, we get a straight line passing through the origin.

The work done in stretching the wire is geometrically represented by the area under the Force-Extension graph. Since the graph is a straight line, the area under it forms a right-angled triangle with a base of $l$ and a height of $F$.

Both the analytical energy method and the graphical method lead us to the same profound conclusion. The work done depends only on the final force applied and the final extension achieved, independent of the wire's original dimensions!