The Advent Calendar of Physics: The Spring's the Thing

Continuing our countdown to Newton's birthday, let's acknowledge the contributions of one of his contemporaries and rivals with today's equation:

This is, of course, Hooke's Law for a spring, which he famously published in 1660:

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Clears everything right up, doesn't it? OK, maybe not. This one's not only in Latin, it's a cryptogram, unscrambling to "ut tensio sic vis," which translates roughly to "as the extension, so the force," giving the correct proportionality between the force exerted by a spring or other elastic material and the amount that material has been stretched.

Why is this important? Well, because springs are incredibly important, as this video clearly demonstrates:

Deranged 1950's educational shorts aside, springs really are very important in physics, not least because Hooke's law is mathematically elegant. A mass hung from the end of a spring will bounce up and down in a regular way, making a simple harmonic oscillator. Plugging Hooke's law into >a href="http://scienceblogs.com/principles/2011/12/the_advent_calendar_of_physi…">Newton's second law gets you a differential equation for the position of the mass at the end of the spring that's among the easiest differential equations to solve. For this reason, a huge part of model-making in physics consists of finding ways to approximate other interactions as being like a spring, so you can use the simple harmonic oscillator solution as a starting point to understand what's going on.

This approximation is essential for understanding the behavior of materials in both classical and quantum-mechanical contexts. It's even the starting point for thinking about quantum electrodynamics, where light is treated as a collection of photons, not just a classical wave. So, Hooke's law is a critical step in the development of physics, and while Newton didn't much like him (some of Newton's most famous quotes reputedly include bitchy little swipes at Hooke), it's worth a moment to honor his accomplishment as part of our countdown to Newton's birthday.

Come back tomorrow for the next equation of the season.

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[The spring's the thing]
Wherein I'll catch the conscience of the king?