Oscillations 1

 

You know what an oscillating body is  it moves in a periodic manner through a series of repeating cycles. When something oscillates it has certain characteristics: The amplitude of the oscillation, the period of the oscillation (how long it takes to perform one complete cycle in its motion), its frequency (how often the cycle repeats every second) and more. We look at a particularly important kind of oscillation  simple harmonic motion (SHM).

 

Simple harmonic motion is sinusoidal motion, i.e. it is described by sine (and/or cosine) functions. You may recall from trigonometry that the sinusoidal functions are also called the circular functions. That’s because they are derived from the “unit circle”. In the picture, think of the arrow that represents the radius of the circle as rotating counterclockwise, having started from the horizontal (i.e. like a second hand on a clock starting from the 3 and rotating backwards). As the arrow goes around, the length of a vertical line from the horizontal to the tip of the arrow draws out the curve shown, where the vertical axis is the length of the vertical line and the horizontal line is a measure of the angle through which the arrow has rotated. The result is a map of the sine of the angle (Sin θ). If instead of the length of the vertical line you took the length of the line along the horizontal axis from the center of the circle out to a point directly under the tip of the arrow you would have the cosine of the angle (Cos θ). Looking at the curve, the height (1 here, because we use the “unit circle”) is the amplitude of the sine wave. If the arrow rotates at an angular rate ω (measured in radians per second), then  and we describe the wave as . If the amplitude were A instead of 1 we would write . Finally, if the arrow starts at some (positive) angle , then we have to add this to the position of the arrowhead and we have   is called the phase angle.

 

This is the general expression for the sine wave generated using the circle (now of radius A). Of course we could get a similar expression for the cosine wave, since one is just the other shifted by π/2 (i.e. 90^o). I’ve shown some curves to illustrate this.  The labels are hard to read, but they are, from left to right, top to bottom, plots of  vs. t for phase angles  = 0, π/4, π/2, 3π/4, π and 2π. Notice that when  = π/2 we have a cosine wave and when  = 2π we are back to our sine wave  we have phase shifted through one complete cycle.