Animated Solution for Physics - Work, Energy, and Power: When a rubber band is stretched by a distance $x$, it exerts a restoring force of magnitude $F=ax+b{x}^2$, where $a$ and $b$ are constants. The work done in stretching the unstretched rubber band by $L$ is

Imagine stretching a rubber band. At first, it yields easily to your pull. But as you stretch it further, it fights back with increasing stubbornness. This is because the restoring force isn't constant; it increases with the distance $x$.

In our problem, this physical reality is beautifully captured by the equation $F=ax+b{x}^2$. The force depends on the position, making it a variable force.

Work Done by a Variable Force

When dealing with a constant force, calculating work is as simple as multiplying force by distance ($W=F\times d$). But here, the force changes at every microscopic step.

To find the total work done, we must sum up the tiny amounts of work done over infinitesimally small displacements $dx$. This is where the magic of calculus steps in. We use the integral form of work:

Geometrically, this integral represents the area under the force-displacement curve.

Let's set up our integral. We substitute our given force function, $F=ax+b{x}^2$, and set our limits from the unstretched position ($x=0$) to the final stretched length ($x=L$).

Now, we integrate term by term using the simple power rule $\int x^{n}dx=\frac{x^{n+1}}{n+1}$:

Applying the upper limit $L$ and the lower limit $0$, the zero terms vanish completely. We are left with our final, elegant expression for the work done:

Take a moment to appreciate this result. The first term, $\frac{a{L}^2}{2}$, looks exactly like the potential energy of an ideal Hookean spring ($\frac{1}{2}k{x}^2$). The second term, $\frac{b{L}^3}{3}$, is the mathematical signature of the rubber band's non-linear stiffness. Real materials are wonderfully complex, and physics gives us the exact tools to decode them!