The use of linear approximations in physics dates back to the 17th century, when America still consisted of minor colonies of Great Britain, when Newton’s Laws of mechanics were exciting and new, chemistry had yet to be fully distinguished from alchemy. In this time of scientific revolution, every idea in science was a new one. One idea that has stood the test of time is that of the linear approximation. Below is an account of one of the first famous ones.
In 1660, Robert Hooke made some stunning observations:
- If you pull on a spring to stretch it, the spring pulls back.
- If you compress a spring, it pushes back.
- The more you stretch/compress a spring, the more it pulls/pushes back.
Hooke then made the bold statement that the force the spring pulls/pushes with (the restoring force) is proportional to the distance it is stretched/compressed from its natural length. Thus, Hooke’s Law was born!
F = -kx
Here F is the restoring force of the spring, x is the amount the spring is stretched (negative x corresponds to the spring being compressed), and k is a proportionality constant to be determined experimentally for an individual spring.
Somehow, this statement has since attained the status of a “Law” in physics, even though it is no such thing! It is simply a linear approximation to the wildly complicated behavior that materials really exhibit when placed under stress. It is easy to show that Hooke’s Law must break down if x is sufficiently far from zero. Here are a couple extreme cases:
A real spring cannot be compressed to zero length no matter how hard you try. If the natural length of the spring is given by L, then the force required to compress the spring should asymptotically approach infinity as x → -L.
If you stretch a spring too much, it will eventually break and will not return to its natural length (i.e. it will lose its elasticity.)
Based on these two facts alone, we would expect the graph of force vs. displacement to look more like this:
Based on this little bit of evidence, it would seem that Hooke’s Law is a terrible way to describe the way that springs behave. So why has it been so successful in the 350 years since its inception? Why do watches (with little springs inside assumed to obey Hooke’s Law) tell time so well? Why is it such an integral part of first year physics courses? To paraphrase Eugene Wigner, why are linear approximations so unreasonably effective at describing physical phenomena?
In the case of Hooke’s law, one reason for this usefulness is the observation that, as long as x is relatively small, the graph of force vs. displacement is “close enough” to being a straight line. This is not only true for my made up graph above, but for actual data that has been recorded for various elastic materials. As long as x stays within this approximately linear region of the graph, we can use Hooke’s Law as much as we want.
A second reason we would prefer the approximation over the exact relationship is that linear equations are much, much easier to solve than any other type of equation! In fact, there is an entire branch of mathematics called linear algebra, which could probably be defined as the study of equations that we can actually solve.
Hooke’s Law provides an easy to state equation, an easy formula for energy ( U =½kx^2) , and when it is combined with Newton’s 2nd Law, an easy to solve differential equation. On top of all this, these formulas are actually empirically accurate provided x is relatively small. None of these things would be easy if we used even a slightly better approximation to the “real” law of springs.
The fantastic success of Hooke’s Law has spawned other physical theories based upon linear approximations. Ohm’s Law, the theory of Thermal expansion, and most of the equations involving pendulums all use linear approximations as their basic equations. Linear approximations are even the basis for a whole field of mathematical physics: perturbation theory.
In short, linear approximations are what make the intractable tractable, the unknowable (approximately) knowable, and we should all celebrate this grand tradition by boldly approximating things that no one has approximated before.