SHM Defined:
When a particle moves according to the equation
its motion is called simple harmonic. Here are the symbols:
You can best see these on a graph - please look at Figure 12.1 on page 333 and 12.5 on 338 in Serway's textbook (3rd Ed). Suppose a mass is attached to a spring and the assembly is resting on a frictionless table. Displace the mass in the x-direction (perhaps 12 cm) and give it an initial negative velocity as you release it. The mass will oscillate according to the above cosine formula. The cosine curve has been displaced in the x-direction because of the initial velocity (the displacement is the phase angle divided by the angular frequency). The amplitude A is the maximum displacement; because there is an initial negative velocity, the mass moves inward away from its maximum displacement position. It does not reach maximum displacement until it reaches the point of maximum compression (marked -A in the Figure). The period T is time the mass requires to move one full cycle.
Derivatives may be used to find the velocity and acceleration:
You should be able to see that the coefficients of these trigonometric functions are the maximum (and minimum) values of x, v, and a.
The frequency f (not the angular frequency) is related to the period:
It is also related to the angular frequency:
This may seem like a lot of information that needs to be taken simply on faith - where does it all come from? Well, some things are merely definitions like the amplitude and the period. Since the period is the time per cycle, its reciprocal is cycles/sec (also called the Hertz or Hz), so you can see why f = 1/T. As for the shift being the phase angle divided by the angular frequency, you have to verify that for yourself or take a good course in trigonometry. You could use a spreadsheet to plot sin t, sin 2t, and sin 3t to see what happens to the period as the angular frequency changes. You could follow this up by plotting sin (t + 30o), sin (2t + 30o), and sin (3t + 30o). Then you'll see that the shift of the curve (I used sines here but you could do this as easily using cosines) really is the phase angle divided by the angular frequency.
Example 1: Oscillatin' for fun
Spring Motion:
You should follow through the discussion in your textbook. From this, you can see that
Thus,
Because we're working with a spring, we have this additional relationship for the period. Everything else applies - the general SHM equations are still valid. Information can be obtained from a spring system using these equations and Newton's Second Law:
Example 2: A spring system
Example 3: A hypothetical spring-mass system