![simple harmonic oscillator simple harmonic oscillator](https://media.cheggcdn.com/media/a58/a587ace4-7814-466c-9bd4-76254511890d/phpSeSdat.png)
To find the velocity from the position equation, we differentiate with respect to time. But sin(0) = 0 and cos(0) = 1, so we see that x0 = A. At t = 0, the value being plugged into sin and cos is 0. That's the value of the position at t = 0 when the clock started and the system started oscillating. Let's go ahead and nail the situation down. These initial conditions - the initial position and initial velocity - determine what A and B are for our specific physical situation. We might have started this oscillator off close or far from equilibrium, or we might have just let it go or given it a good shove. )Īctually the solution contains k and m, but to make the equation look simpler, I've just substituted in omega, where it's an abbreviation for a slightly clumsier expression:Īnd that's the general solution, for arbitrary constants A and B. For those who haven't seen it so much, it's the pretty much the first thing you'll learn in differential equations class, and if you don't want to take the class it's ok because the problem is not difficult and either way I'm just going to tell you the solution. Physicists usually solve this kind of equation by the method of recognition - we've seen it so much we just know what the solution is.
![simple harmonic oscillator simple harmonic oscillator](https://miro.medium.com/max/1200/0*Ds_YEh-eCH-AoJzI.png)
Now if you know about solving differential equations, we can actually find the particular function x(t) that satisfies that equation. The spring constant (the force produced by the spring per unit of stretch beyong equilibrium) is k, the mass of the object is m. Sounds bad, but that's just another name for the acceleration. Dots denote differentiation with respect to time, so x-dot-dot is the rate of change in the rate of change of position. The variable x is the position of the mass on the spring, and it's a function of time. We can write "the force is proportional to the stretch" mathematically in the following way: Move the mass farther from its resting point, and the restoring force is proportionally stronger. Its defining property is that the force acting on the spring is proportional to the displacement of the mass from equilibrium. It's ubiquitous in everything from solid state physics to quantum field theory, but when it comes right down to it, the harmonic oscillator is a spring. Let's kick it off with perhaps the most important model in physics: the simple harmonic oscillator. I can't promise I'll be super consistent about it, but here's hoping y'all will bear with me! As such I'm going to try to improve the ratio of more in-depth fare a bit. Still interesting, I hope, but there's really no shortage of that sort of thing elsewhere.
![simple harmonic oscillator simple harmonic oscillator](https://image.slidesharecdn.com/simpleharmonicoscillator-180404043109/95/simple-harmonic-oscillator-12-638.jpg)
And so there's been more soft-physics kind of posts around here. Over the last few months though, general grad student busyness has greatly reduced the time available for those kinds of posts. I was always doubtful there was much of a market for this, but of course there are at least some interested people and especially since writing is so fun I was and am I'm more than happy to fill that gap. The goal was to bridge the gap between popularization and textbook.
![simple harmonic oscillator simple harmonic oscillator](https://useruploads.socratic.org/VLJAXi3QliMgQQRHQoOe_qhar.gif)
Now on with the show:īack when I first started writing this blog, I focused mostly on problem solving. First of all, happy Thanksgiving everyone! I hope you spend the day happily with the people you care about, and remember to spend a moment or two reflecting on the things for which you're thankful this year.