1.1 Simple harmonic motion  (Page 4/4)

We see that adding or subtracting “π/2” serves the purpose. When we add the angle “π/2”, the position of particle, at mean position, is ahead of positive extreme. The particle has moved from the positive extreme to the mean position. When we subtract the angle “-π/2”, the particle, at mean position, lags behind the position at positive extreme. In other words, the particle has moved from the negative extreme to the mean position.

Here, we note that “ωt” is dimensionless angle and is compatible with the angle being added or subtracted :

$$\left[\omega T\right]=\left[\frac{2\pi }{T}XT\right]=\left[2\pi \right]=\text{dimensionless}$$

Representation of displacement from positions other than extreme position in this manner gives rise to an important concept of “phase constant” . The angle being added or subtracted to represent change in start position is also known as "phase constant" or “phase angle” or “initial phase” or “epoch”. This concept allows us to represent displacement whatever be the initial condition (position and direction of velocity – whether particle is moving towards the positive extreme (negative phase constant) or moving away from the positive extreme (positive phase constant). For an intermediate position, we can write displacement as :

$$\Rightarrow x=A\cos \left(\omega t+\phi \right)$$

Note that we have purposely removed negative sign as we can alternatively say that phase constant has positive or negative value, depending on its state of motion at t = 0. The concept of phase constant will be more clearer if we study the plots of the motion for phase "0", “φ” and “-φ” as illustrated in the figure below.

The figure here captures the meaning of phase constant. Let us begin with the uppermost row of figures. We start observing motion from positive extreme (left figure), phase constant is zero. The displacement is maximum “A”. The particle is moving from positive extreme position to negative extreme (middle figure). The equivalent particle, executing uniform circular motion, is at positive extreme (right figure).

In the middle row of the figures, we start observing motion, when the particle is between positive extreme and mean position (left figure), but moving away from the positive extreme. Here, phase constant is positive. The displacement is not equal to amplitude. Actually, maximum displacement event is already over, when we start observation (middle figure). The particle is moving from its position to negative extreme. The equivalent particle, executing uniform circular motion, is at an angle “φ” ahead from the positive extreme position (right figure).

In the lowermost row of the figures, we start observing motion, when the particle is between positive extreme and mean position (left figure), but moving towards the positive extreme. Here, phase constant is negative. The displacement is not equal to amplitude. Actually, maximum displacement event is yet to be realized, when we start observation (middle figure). The particle is moving from its position to the positive extreme position. The equivalent particle, executing uniform circular motion, is at an angle “φ” behind from the positive extreme position (right figure).

From the description as above, we conclude that phase constant depends on initial two attributes of the particle in motion (i) its position and (ii) its velocity (its direction of motion).

From the discussion, it is also clear that using either "cosine" or "sine" function is matter of choice. Both functions can equivalently be used to describe SHM with appropriate phase constant.

Phase

Simply put phase is the argument (angle) of trigonometric function used to represent displacement.

$$x=A\cos \left(\omega t+\phi \right)$$

The argument “ωt + φ” is the phase of the SHM. Clearly phase is an angle like “π/3” or “π/6”. Sometimes, we loosely refer phase in terms of time period like “T/4”, we need to convert the same into equivalent angle before using in the relation.

The important aspect of phase is that if we know phase of a SHM, we know a whole lot of things about SHM. By evaluating expression of phase, we know (i) initial position (ii) direction of motion (iii) frequency and angular frequency (iv) time period and (v) phase constant.

Consider a SHM equation (use SI units) :

$$x=\sin \left(\frac{\pi t}{3}-\frac{\pi }{6}\right)$$

Clearly,

At t = 0,

$$\text{Initial position},x_0=\sin \left(\frac{\pi }{3}X0-\frac{\pi }{6}\right)=\sin \left(-\frac{\pi }{6}\right)=-\frac{1}{2}=-0.5m$$

Further,

$$\text{Amplitude},A=1m$$

$$\text{Angular frequency},\omega =\frac{\pi }{3}$$

$$\text{Time period},T=\frac{2\pi }{\omega }=\frac{2\pi }{\frac{\pi }{3}}=6s$$

$$\text{Phase constant},\phi =-\frac{\mathrm{\pi }}{6}\mathrm{radian}$$

As phase constant is negative, the particle is moving towards positive extreme position.