Simple Harmonic Motion

In nature, what’s something that you see all the time? Is it light? (don’t be a smart ass)

Is it… I’ll just tell youse what I’m getting at. It’s wiggles!

Yes, wiggling is everywhere. Your Jello in the morning, my Jello in the afternoon, and everyone and there mother’s Jello too. It just seems to keep going!

For a body to wiggle properly, it’s gotta go from side to side. That means it’s velocity will accelerate towards some equilibrium position, indicating the presence of a Force that’s always pointing towards this equilibrium. Simple Harmonic Motion (SHM) is the simplest kind of oscillation (hence the simple) – in which this “restoring force” (and hence, the acceleration) is directly proportional to the displacement from the equilibrium.

a_x = \frac{d^2x}{dt^2} = -\frac{k}{m}x  — As you can see, this acceleration (in 1D, but 1’s fun, 1’s good ’nuff) is not constant. Lemme repeat: not constant. It’s got a dependence on that there displacement “x” from the equilibrium. The further away, the greater the acceleration.

Stated another way, the situation is: \vec{F}=-k\vec{x} for some scalar constant “k”. This is the same form that we see when introduced to Hooke’s Law for ideal springs, that “k” being Hooke’s Constant of Proportionality (more on that at Elasticity and Plasticity).

Simple Harmonic Motion describes many things that oscillate back and forth, including Waves, the swinging of a pendulum, and situations of Circular Motion (like planets orbiting (really, they’ve got the same equations, different constants and stuff)).

Take a look at this GIF.

giphy
The relation between SHM and Circular Motion. SHM is the projection of circular motion. This fact indicates the two can be described similarly.

I’ll leave the more in-depth examination of the equations to the post on circular motion later on, but I’ll quote a few of the results here.

The reference circle is the circle whose radius is equal in magnitude to the amplitude of the oscillation in SHM. A point P on this circle moves around the circle with a constant angular speed ω. The Vector pointing from the center of the circle to the point makes some angle θ to the x-axis, and rotates along with the rotating point. This particular vector is called a Phasor.

From Circular Motion, we know that with a constant ω we have a constant Centripetal Acceleration \vec{a}_P pointing to the center of the circle that produces the circular motion. This acceleration is given by \vec{a}_P = \frac{v^2}{r} = {\omega}^2r or you could replace the radius there with the amplitude and get \vec{a}_P = {\omega}^2A

Now, if you think about it, the x-component of the phasor (corresponding to the x displacement in SHM) is given by x = A\cos{\theta}, and the x-component of the acceleration in circular motion is given by a_x = -{a_P}\cos{\theta} = -{\omega}^2A\cos{\theta} = -{\omega}^2x

And as this corresponds to SHM, which has a_x = -\frac{k}{m}x we find that {\omega}^2 = \frac{k}{m} -> {\omega} = \sqrt{\frac{k}{m}} for SHM.

From one view, ω is the angular speed and from another, it’s the angular frequency of oscillation.

—More on this later—

 

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2 thoughts on “Simple Harmonic Motion

  1. Pingback: Waves – Delblg

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