The equation of movement for a pendulum is the differential equation
the place g is the acceleration resulting from gravity and ℓ is the size of the pendulum. When that is offered in an introductory physics class, the trainer will instantly say one thing like “we’re solely within the case the place θ is small, so we are able to rewrite the equation as
Questions
This raises a whole lot of questions, or a minimum of it ought to.
- Why not depart sin θ alone?
- What justifies changing sin θ with simply θ?
- How small does θ should be for this to be OK?
- How do the options to the precise and approximate equations differ?
First, sine is a nonlinear operate, making the differential equation nonlinear. The nonlinear pendulum equation can’t be solved utilizing arithmetic that college students in an introductory physics class have seen. There’s a closed-form resolution, however provided that you prolong “closed-form” to imply greater than the elementary capabilities a scholar would see in a calculus class.
Second, the approximation is justified as a result of sin θ ≈ θ when θ is small. That’s true, however it’s kinda delicate. Right here’s a submit unpacking that.
The third query doesn’t have a easy reply, although easy solutions are sometimes given. An teacher might make up a solution on the spot and say “lower than 10 levels” or one thing like that. A extra thorough reply requires answering the fourth query.
I tackle how nonlinear impacts the options in a submit a pair years in the past. This submit will broaden a bit on that submit.
Longer interval
The first distinction between the nonlinear and linear pendulum equations is that the options to the nonlinear equation have longer durations. The answer to the linear equation is a cosine. Fixing the equation determines the frequency, amplitude, and part shift of the cosine, however qualitatively it’s only a cosine. The answer to the corresponding nonlinear equation, with sin θ relatively than θ, isn’t precisely a cosine, however it appears loads like a cosine, solely the interval is just a little longer [1].
OK, the nonlinear pendulum has an extended interval, however how for much longer? The interval is elevated by an element f(θ0) the place θ0 is the preliminary displacement.
You will discover the precise reply in my earlier submit. The precise reply is dependent upon a particular operate known as the “full elliptic integral of the primary sort,” however to an excellent approximation
The sooner submit compares this approximation to the precise operate.
Linear resolution with adjusted interval
For the reason that nonlinear pendulum equation is roughly the identical because the linear equation with an extended interval, you’ll be able to approximate the answer to the nonlinear equation by fixing the linear equation however growing the interval. How good is that approximation?
Let’s do an instance with θ0 = 60° = π/3 radians. Then sin θ0 = 0.866 however θ0, in radians, is 1.047, so we undoubtedly can’t say sin θ0 is roughly θ0. To make issues easy, let’s set ℓ = g. Additionally, assume the pendulum begins from relaxation, i.e. θ'(0) = 0.
Right here’s a plot of the options to the nonlinear and linear equations.
Clearly the answer to the nonlinear equation has an extended interval. In truth it’s 7.32% longer. (The approximation above would have estimated 7.46%.)
Right here’s a plot evaluating the answer of the nonlinear equation and the answer to the linear equations with interval stretched by 7.32%.
The options differ by lower than the width of the plotting line, so it’s too small to see. However we are able to see there’s a distinction once we subtract the 2 options.
Right here’s a plot of the options to the nonlinear and linear equations.
Replace: The plot above is deceptive. A part of what it reveals is numerical error from fixing the pendulum equation. Once we redo the plot utilizing the precise resolution the error is about half as giant. And the error is periodic, as we’d count on. See this submit for extra on the precise resolution utilizing Jacobi capabilities and the differential equation solver that was used to make the unique plot.
Associated posts
[1] The interval of a pendulum is dependent upon its size ℓ, and so we are able to consider the nonlinear time period successfully changing ℓ by an extended efficient size ℓeff.
