![]() ![]() Only when you displace the mass from this equilibrium position does it have a restoring force. Just continue to sit there, there'd be no net force on it. ![]() This line here would be equilibrium 'cause if you put the mass there and let it sit it would Maximum regular displacement, it's gonna be the maximumĪngular displacement from equilibrium right here. To be a distance in X, or a displacement in X, this is gonna be not the So, I'll write theta as a function of time is gonna equal some amplitude, but again, since I'm measuring theta, my amplitude is not going So, this is gonna be anĪngle as a function of time. Maybe it's at negative 10, negative 20, negative 30 and then this whole process repeats. We're measuring angles from the center line. Maybe it's at like, 30 degrees and it swings it's onlyĪt like 20 and then 10 and then zero 'cause So, consider the fact that this mass is gonna be at different angles at different moments in time. The far more useful and common example of using a variable to describe a pendulum is the angle that the pendulum is at. So, how would I apply this equation to this case of a pendulum? Well, I wouldn't use X. 'cause usually you can get away with not using that one. Times cosine or sine, I'm just gonna write cosine, of two pi divided by the period, times the time and you can if you wantĪdd a phase constant. Was described by an equation that looked like this, X, some variable X is a function of time was equal to some amplitude There's a restoring force proportional to the displacement and we mean that its motion can be described by the simple So, what do we mean that the pendulum is a simple harmonic oscillator? Well, we mean that We can learn a lot about the motion just by looking at this case. We've got enough things to study by just studying simple pendulums. But really complicated toĭescribe mathematically. If you've never seen it, look up double pendulum, Physicists call chaotic, which is kind of cool. Let's say you connect another string, with another mass down here. You could have more complicated examples. And technically speaking, I should say that this is actually a simple pendulum because this is simply a Simple harmonic oscillator and so that's why we study it when we study simple harmonic oscillators. So, this is gonna swingįorward and then backward, and then forward and backward. And a pendulum is just a mass, m, connected to a string of some length, L, that you can then pullīack a certain amount and then you let it swing back and forth. So, that's what I wanna talk to you about in this video. The most common example, but the next most commonĮxample is the pendulum. Simple harmonic oscillators go, masses on springs are However, here is a link to the derivation: ) (I'm sorry, as typing out differential equations would be tedious. ![]() However, there are methods of approximating unsolvable differential equations (Euler's method, for example), that can get much closer to the exact answer than would the traditional period formula. Therefore, as of right now, there is no absolute solution to your question. This bottleneck is the very reason why the period formula works best when θs are smallest if you were to look at the graphs of y=sin(x) and y=x, they are closest to each other the smaller x is (thus more accurate), and farther with bigger x values. To alter this differential equation into a solvable one, you can write sin(θ) ≈ θ via the small-angle approximation (while sacrificing a bit of accuracy). If you were to try and derive the period of the pendulum (which involves setting up differential equations), you eventually get this term, sin(θ), which makes the whole differential equation unsolvable.
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