Spring and Orbit, Second-Order ODE

simple_harmonic_motion_orbit

Surprised that the red block moves in sync with the green dot in a orbit?

After dealing with the basics of first-order ODEs (ordinary differential equations), it was time to move to second-order. Second order here means that in such ODEs, there exist a second derivative of the function in question. For example, the general form of a linear second-order differential equation looks like this:

2deq

where p(t), q(t), g(t) can be functions of a third variable t, or constants. 

For the red block in the gif picture above, we can analyze its motion by establishing such a second-order differential equation. We know by Hooke’s Law, that the force restored in the spring is given by F = kx, where k is a positive spring constant, and x is the displacement. We also know, that acceleration is no more than the second derivative of displacement in respect to time, thus by Newton’s second law, F=ma, we have:

2deq2.PNG

Note that the negative sign designates that the pulling force is in the opposite direction in which the spring pulls.

It turns out, for the homogeneous equation here, where the coefficients are all constants, we can always use the characteristic equation method to solve the problem. [1]

The characteristic equation for this specific case is:

chareq.PNG

Notice that for this quadratic, r does not have real roots, since k and m are positive constants. The imaginary roots are  r=±√k/m i. Thus the general solution is given as the following:

sol

To simplify the expression, we get:

sol2

where the lower-case Greek letter omega is the imaginary coefficient r, in other words, ω=√k/m , which turns out to be the frequency of a simple harmonic motion, because the product (ωt) cycles its value for every 2π . Now it is time to explain why the red block moves in sync with the green dot in orbit.

The general solution looks similar to a parametric equation of an ellipse. If I were to plot the cosine part of the general solution as the horizontal axis, and plot the sine part as the vertical axis, the graph is going to end up as an ellipse. The ellipse would have c1 and c2 as the lengths of its semi axes. Here is an animation of such process:

parametric_ellipse

Now, if c1 = c2, the two circles in the animation above will merge into one, and thus the animation will become a dot circling in orbit, the exact same motion as shown at the top of the page. This is called the Phase Space because the orbiting motion doesn’t happen in real life, it expands the real life one-dimensional movement into a two-dimensional circular movement. But it is extremely useful in that for every point on the circle, (that means every angle) it represents a unique point in the real-life linear motion, and tells whether if the object is pulling the spring or the spring is pulling the object.


Works Cited:

  1. Edwards, C. Henry, David E. Penney, and David Calvis. Elementary Differential Equations with Boundary Value Problems. Upper Saddle River, NJ: Pearson/Prentice Hall, 2008. Print.

Images Used:

All the equations are created via http://www.hostmath.com

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