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briannamorgan

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- Thread starter briannamorgan
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- #1

briannamorgan

- #2

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Originally posted by briannamorgan

Harmonic motion is where the displacement (and hence velocity and acceleration) curve, as a function of time, is a sinusoid (like a sine or a cosine).

If you take an object in uniform circular motion and consider just its horizontal displacement, or just its vertical displacement, you will get harmonic motion.

To see this, consider that if an object is at some distance R away from the origin, at an angle θ with respect to the horizontal axis, then basic trigonometry tells us that its horizontal displacement from the origin will be x = R cos(θ), and y = R sin(θ). If R is a constant, and if the angle θ increases proportionally with time (θ = ωt for some constant ω), then this describes an object in uniform circular motion (constant distance from the origin, moving around at a constant angular rate). You then get x = R cos(ωt), y = R sin(ωt), which are sinusoidal curves if you graph them versus time.

- #3

ShawnD

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Simple harmonic motion is just like the above guy said, motion where distance from a certain point (equilibrium point), velocity and acceleration all follow a sinusoidal pattern.

Let's say we have a spring moving up and down (no friction, no air, 100% efficient). Lets say we have a spring and we pull it 15cm from the equilibrium point. That spring will absolutely never go more than 15cm from the equilibrium point in either direction. If we also know the period or the rotation speed, we can figure out instantaneous distances, velocities and accelerations.

The formula for our displacement in this case is:

d = Rcos(wt)

R is the maximum displacement from equilibrium (15cm in this case), w is rotation speed and t is time in seconds.

The formula for rotation speed is w = 2pi/T

w is not actually a w, it's supposed to be omega which looks much like a w. pi is the constant pi and T is the period of oscilation (sp?).

The formula for our displacement is determined for where we consider the start point to be. Most physics teachers will want you to consider the start point to be equilibrium and think of the spring as moving up, in that case our formula is d = Rsin(wt). If it starts in the middle and is going down, it's d = -Rsin(wt). If it starts from the vertical maximum, d = Rcos(wt). If it starts from vertical minimum, d = -Rcos(wt).

No matter what our formula for distance is, the formula for velocity is always the derivative of displacement with respect to time and acceleration is always the derivative of velocity with respect to time.

If our formula for displacement was d = Rcos(wt) meaning it starts from the vertical maximum, the formula for velocity will be v = -Rwsin(wt) and our acceleration will be a = -Rw^2cos(wt).

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