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). The 1 in 60 rule used in air navigation has its basis in the small-angle approximation, plus the fact that one radian is approximately 60 degrees.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. This leads to significant simplifications, though at a cost in accuracy and insight into the true behavior. The small-angle approximation also appears in structural mechanics, especially in stability and bifurcation analyses (mainly of axially-loaded columns ready to undergo buckling). 'fringe spacing' = 'wavelength' × 'distance from slits to screen' ÷ 'slit separation'. ![]() The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to simplify equations, e.g. In optics, the small-angle approximations form the basis of the paraxial approximation. When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing simple harmonic motion. The second-order cosine approximation is especially useful in calculating the potential energy of a pendulum, which can then be applied with a Lagrangian to find the indirect (energy) equation of motion. Sin θ ≈ θ cos θ ≈ 1 − θ 2 2 ≈ 1 tan θ ≈ θ Īnd the above approximation follows when tan X is replaced by X. The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians: ![]() Simplification of the basic trigonometric functions Approximately equal behavior of some (trigonometric) functions for x → 0
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