Increase the period or decrease the period If we increased the mass on this pendulum, do you think that would Would change this period here? So, what might this depend on? My first guess might be, Pendulum gonna depend on? Right, this period here, what could we change that What can we say? Well, one question we can ask is what's the period of this Alright, so let's assume we're in that small angle approximation where this amplitude is small. It would be extremely close to being a simple harmonic oscillator. Only 20 degrees or less, that pendulum would beĭescribed really well by this equation because It'll still be reasonably close, maybe within like 20 per cent, but only for small angles As you get to larger maximum amplitudes, this is gonna deviate more and more. Simple harmonic oscillator, but technically speaking it only works really well if you're less than say a certain amount, say 20 degrees. So, because of that, we often treat a simple pendulum as a
This will only be off by very small amounts, Harmonic oscillator, it's only extremely close to being a simple harmonic oscillator. Technically speaking, the simple pendulum is not a perfect simple Alright, so I gotta comeĬlean about something now. Or to complete a whole cycle and we always have to multiply by T, that's our variable, that's what makes this a function, it's a function of time. Two pi over whatever the period is, and the period is the time it takes for this pendulum to reset And then we'll multiply by cosine and it will have the Maybe it's 30 degrees, maybe it's 20, that would be the angle
So, this would be the maximum, I'll just call it theta maximum, 'cause this is the maximumĪngular displacement when you pull this back, the maximum angle you pull 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