SHM and the
The reference circle is the primary tool for drawing and analyzing the x-t, v-t, and a-t graphs of SHM and determining the phase of any point on any of the graphs.
The reference circle for SHM has three vectors drawn on it, the position vector of size A, the velocity vector of size vmax = ωA, and the acceleration vector of size amax = ω2A. The velocity vector is 90° ahead of (or counterclockwise from) the position vector. In turn, the acceleration vector is 90° ahead of the velocity vector. Normally the reference circle is drawn at t = 0, as shown in Figure 1. The phase constant φ0 is the positive (counterclockwise) angle between the x-axis and the position vector.
Using the
We draw the x-t, v-t, and a-t graphs using only the x-components of the three vectors. For instance, in Figure 2, we show the x-components of each vector. The x-component of the position vector is only 0.8A in length and is positive. As time passes, the position vector rotates counterclockwise and the x-component decreases towards zero. The graph of this is shown in Figure 3(a). The x-component of the velocity vector is only 5.6vmax in length and is negative. As time passes, the velocity vector rotates counterclockwise and the x-component approaches a minimum. The graph of this is shown in Figure 3(b). The x-component of the acceleration vector is only 0.8amax in length and is negative. As time passes, the acceleration vector rotates counterclockwise and the x-component increases towards zero. The graph of this is shown in Figure 3(c).
In summary the t = 0 reference circle lets us locate the starting (t = 0) value of each vector and its subsequent behaviour, going to zero, or going to a maximum, or going to a minimum.
Using
a Motion Graph to Determine the Phase Constant or Phase
Conversely any of the three graphs can be used to find the phase constant φ0 at t = 0. You first read the size of quantity (A, vmax, or amax) off the given graph. Second, read the value of the quantity off the graph at t = 0 note whether it is positive or negative. Third see if the graph is going to zero, or a maximum, or a minimum after t = 0. Steps two and three tell you which quadrant your vector is in. From the maximum value of the vector and from its value at t = 0, you have the hypotenuse and horizontal side of a right triangle. It is simple trigonometry to find the angle of the right triangle. It is then usually simple trigonometry to relate this known angle to the phase constant φ0. Figure 4 is an example. You are given a v-t graph. The maximum velocity is 10. The value of the velocity at t = 0 is 3. As time goes by, the velocity goes to a minimum. Thus we know that the velocity vector is in the second quadrant. The position vector will be 90° behind in the first quadrant. The acceleration vector will be 90° ahead in the third quadrant. Since the base is 3 and the hypotenuse is 10, the angle α> for the known right triangle is α> = arccos(3/10) = 72.5°. That is the velocity vector is 107.5° from the positive x-axis. Knowing that the position vector is exactly 90° behind the velocity vector means that the position vector makes an angle φ0 = 17.5° to the positive x-axis.
The phase θ = ωt + φ0 of any point at any later time t can also be found using the reference circle. Note that the phase θ is measured counterclockwise from the positive x-axis to the position vector only. Figure 5 shows a v-t graph and we are interested in knowing the phase at the point shown by the dashed vertical line. Since ivelocity is at a maximum at the dashed line, the velocity vector must be lying along the positive x-axis. The phase θ, from the x-axis to the position vector, is thus 270°.
The following link is to an interactive webpage where you can vary the phase constant and see the motion diagrams that this produces. Note that it will only work with MS Internet Explorer with the appropriate Office Web Components plugin.