P100 Phase Angle

Phase Difference & Phase Angle

Figure 3.4 Identical mass-spring systems

In figure 3.4 is shown the motion of two identical mass-spring systems hung from a rigid ceiling. They oscillate at the same frequency and the same amplitude. Depending on your preferred way of viewing this motion either the black system moves in advance of the blue system, or the blue system motion is delayed with respect to the black system. Both descriptions are equivalent!

Note that the vertical separation between the hanging masses is not constant. However if you observe carefully it is possible to see that the black mass reaches the blue dashed lines a fixed time before the blue mass. We know from the rotating vector description that the y height of each mass from the equilibrium position (the magenta dashed line) is described by a sinusoidal function. The question is how to account for the time shift between the two motions within this representation. This is discussed in Serway in Chapter 13, pages 390 to 395.

What you will have found out from lecture 3 (and the textbook) is that the two motions are described as being "in a phase relationship" with each other. There is a fixed phase difference between the motions (of value 50 degrees in figure 3.4). Phase difference within the rotating vector representation is an angle.

In the rotating vector representation the black and blue systems each have their own rotating vector. Since the two systems depicted above are of the same amplitude and angular frequency, it follows that the vectors are of the same length and rotate at the same angular velocity (equivalent to the angular frequency). They differ in their directions: there is a fixed separation in angle (the phase angle f) between them as shown in the time varying figure 3.5.

NB figure 3.4 and figure 3.5 are not time synchronous with each other (they are out of phase with each other!)

Figure 3.5 Rotating vector representation of oscillations with a
phase angle difference (see for example figure 3.4 above)