Phase and Phase difference in simple harmonic motion : In general the solution of a simple harmonic equation is

As mentioned earlier A is known as the amplitude and as the phase. is a constant depending on the initial conditions and we call it the phase constant. Let us now see how does the motion look for different values of the phase constant . The displacement versus time plots for different signs of the phase constant are shown in figure 10.

For Φ > 0 the motion at t = 0 begin at a value or phase angle that it would have slightly later in the case. On the other hand, for Φ < 0 the motion is such that a particular displacement for the case is reached at a later time. The motion lags behind the motion. I leave it for you to figure out yourself how the corresponding velocities are related.

Let us now at the special case of . In this case I get

and for

So you see that a phase difference of 180^o, whether position or negative, means the same thing. I would like you to plot the displacement versus time graph for these particular cases. For the phases in between you should see for yourself how the displacements at t = 0 are different from case.

A good way of visualizing the simple harmonic motion is the phasor or vector diagram. I discuss that next.

Phasor or vector diagram: A nice geometric way of looking at various quantities in a simple harmonic motion is the vector or a phasor diagram. You may have seen it in your 12^th grade while studying AC circuits. Let me show you how we represent in a geometric way. You see that displacement in this case is the x component of a vector making an angle ωt from the x-axis. Thus the displacement is represented as shown in figure 11. The motion described by is thus given by the projection of a vector of length A, rotating counterclockwise at a rate ω, on the x-axis.

Let us now see how the velocity and the acceleration will be represented in this scheme? The velocity and acceleration are given as

The displacement, velocity and acceleration are shown in the phasor diagram in figure 12. A general feature that we observe from this phase diagram is that the velocity vector is always ahead (measuring counterclockwise) of the displacement vector and the acceleration vector is at π (ahead or behind?) the displacement.

So far we have discussed the simple case of . What about the general case of . This is also equally simple. All we have to do is keep the initial position of the vector at t = 0 at an angle Φ from the x-axis and start rotating it from there. The velocity vector and the acceleration vector are then going to be given at and π from it, as discussed above. This is shown in figure 13.

Recall that in the middle of this lecture I had solved a spring-mass problem with different initial conditions. I would like you to make the phasor diagram to represent the motion of the mass in many different situations like those considered above. Do not solve for x(t) to start with, just make the phasor diagram directly to see if you have got a feel for motion under different conditions.

Finally in this lecture I look at the energy of a system performing simple harmonic motion. The potential energy U(x) and the kinetic energies T are

The total energy E is of course a sum of the two. With   this gives

Since , we get

Thus the energy depends on the square of the amplitude. This makes sense because if I stretch a spring by A, the energy stored in it is . On releasing the mass it performs oscillations of amplitude A. Thus you see that amplitude A immediately implies a total energy given above.

I have now set up all the basic concepts of simple harmonic motion. In the coming lectures I will introduce damping in the system and see how it evolves.