4.4; let's talk physics! (er, harmonic motion)

Section 4.4 is entitled “Harmonic Motion.”

I don’t think you can possibly understand how exciting this is. We recently talked about this stuff in my electronics class. At least, we talked about damping (which this is a very watered down version of it). Hooray physics! :D

The motion for a vibrating spring has the following equation:

In other physics news, in our bonus section from Chapter 3 (the one about electrical circuits), we had the following equation for RLC circuits:

When you compare these two equations, they’re very similar. If we compare coefficients, the inductance L acts as the mass, the resistance R acts as the damping constant, and 1/C (where C is the capacitance) acts as the spring constant. Also, the derivative of the source voltage (NOT energy, which is what I originally interpreted it as) acts as the external force F(t).

Let’s divide our two equations by their leading coefficients (L or m). Then they become

If we change some variables, i.e.

That final equation is known as the equation for harmonic motion. Examples of this equation would be the vibrating spring and RLC circuit. Hooray!

The constant c is called the damping constant, and f is called the forcing term. Finally, this section consists of unforced harmonic motion, meaning that f(t) = 0. This would make our harmonic motion equation homogeneous.

The equation for simple harmonic motion is when there is no damping (i.e. c = 0). The equation then simplifies to be

In the previous section, we know that for characteristic equations with two complex conjugate roots, the general solution will be

In this context, ω₀ is called the natural frequency of the spring. Also, the period of the solution x(t) is denoted as T, where T is defined as 2π/ω₀.

Now, let’s talk a bit about polar coordinates.

Suppose we have a vector (a, b) in the plane. When you write this is polar coordinates and when you assume (a, b) ≠ (0, 0), then there will be a positive number A (which is the length of the vector) and an angle φ in the interval (-π, π] called the polar angle, in which

When you substitute these equations into our general solution, we see we can write the general solution as

In this context, A is the amplitude of the oscillation, and φ is the phase of the oscillation.

Your next question might concern how we find A and φ.

Since we originally defined φ to be any angle between –π and π, then we can’t just take the arctan of b/a, considering that arctan only takes values between –π/2 to π/2. More importantly, this range corresponds to points (a, b) where a > 0. So how do we compute φ when a < 0?

Now let’s consider the case when our harmonic motion is damped (i.e. when c > 0). We’ll have the following differential equation, characteristic equation, and roots:

Now we have a discriminant to worry about, which means we have three cases to consider:

1. c < ω₀; this called the underdamped case. The roots are distinct and complex. The solution would be

2. c > ω₀; this is called the overdamped case. The roots are distinct and real. Also, λ₁ and λ₂ are both positive. The general solution is

3. c = ω₀; this is called the critically damped case. In this case, the root is a double root, i.e. λ = -c. Our general solution is

Here are some websites for you: http://en.wikipedia.org/wiki/Simple_harmonic_motion and http://scipp.ucsc.edu/~haber/ph5B/sho09.pdf and http://wiki.ubc.ca/Simple_and_Damped_Harmonic_Motion and http://hyperphysics.phy-astr.gsu.edu/hbase/oscda.html.

That’s it for 4.4! I’ve been busy and lazy, and that combination leads to me not getting ahead with this blog. We just started chapter 9 in my class, so it’s not like we’re catching up in any way (knock on wood!). Hopefully I can get a lot of chapter 4 up this weekend. If not, maybe I can get one more section up.

I’ll see you in 4.5!