Description of oscillation

We need a mathematical model to describe oscillation. We often use trigonometric functions. However, we can not use all of them. It is essentially because many of them are not bounded. Recall the plot of tangent function. It extends from minus infinity to plus infinity - periodically. Actually, only the sine and cosine trigonometric functions are bounded.

The plot of tangent function is shown here. Note that value of function extends from minus infinity to plus infinity.

Figure 2: The function is not bounded.

Plot of tangent function

The plots of sine and cosine functions are shown here. Note that value of function lies between "-1" and "1".

Figure 3

Plots of sine and cosine functions
Subfigure 3.1: The sine function is bounded. Subfigure 3.2: The cosine function is bounded.

Simple harmonic motion

A simple harmonic motion can be conceived as a “to and fro” motion along an axis (say x-axis). In order to simplify the matter, we choose origin of the reference as the point about which particle oscillates. If we start our observation from positive extreme of the motion, then displacement of the particle “x” at a time “t” is given by :

x = A cos ω t x = A cos ω t

where “ω” is angular frequency and “t” is the time. The figure here shows the positions of the particle executing SHM at an interval of “T/8”. The important thing to note here is that displacements in different intervals are not equal, suggesting that velocity of the particle is not uniform. This also follows from the nature of cosine function. The values of cosine function are not equally spaced with respect to angles.

Figure 4: Positions of the particles at different times are shown.

Simple harmonic motion

Amplitude

We know that value of cosine function lies between “-1” and “1”. Hence, value of “x” varies between “-A” and “A”. If we plot the function describing displacement, then the plot is similar to that of cosine function except that its range of values lies between “-A” and “A”.

Figure 5: The scalar value of maximum displacement from the mean position is known as the amplitude of oscillation.

Amplitude

The value “A” denotes the maximum displacement in either direction. The scalar value of maximum displacement from the mean position is known as the amplitude of oscillation. If we consider pendulum, we can observe that farther is the point from which pendulum bob (within the permissible limit in which the bob executes SHM) is released, greater is the amplitude of oscillation. Similarly, greater is the stretch or compression in the spring executing SHM, greater is the amplitude. Alternatively, we can say that greater is the force causing motion, greater is the amplitude. In the nutshell, amplitude of SHM depends on the initial conditions of motion - force and displacement.

SHM and uniform circular motion

The expression for the linear displacement in “x” involves angular frequency :

x = A cos ω t x = A cos ω t

Clearly, we need to understand the meaning of angular frequency in the context of linear “to and fro” motion. We can understand the connection or the meaning of this angular quantity knowing that SHM can be interpreted in terms of uniform circular motion. Consider SHM of a particle along x-axis, while another particle moves along a circle at a uniform angular speed, "ω", in anticlockwise direction as shown in the figure. Let both particles begin moving from point “P” at t=0.

Figure 6: The displacement of particle in SHM is equal to projection of position of particle in UCM.

SHM and UCM

After time “t”, the particle executing uniform circular motion (UCM), covers an angular displacement “ωt” and reaches a point “Q” as shown in the figure. The projection of line joining origin, "O" and “Q” on x – axis is :

O R = x = A cos ω t O R = x = A cos ω t

Now compare this expression with the expression of displacement of particle executing SHM. Clearly, both are same. It means projection of the position of the particle executing UCM is equal to the displacement of the particle executing SHM from the origin. This is the connection between two motions. Also, it is obvious that angular frequency “ω” is equal to the uniform angular speed, “ω" of the particle executing UCM .

In case, if this analogical interpretation does not help to interpret angular frequency, then we can simply think that angular frequency is product of “2π” and frequency “ν”.

ω = 2 π ν ω = 2 π ν

Phase constant

We used a cosine function to represent displacement of the particle in SHM. This function represents displacement for the case when we start observing motion of the particle at positive extreme. At t = 0,

x = A cos ω t = A cos 0 0 = A x = A cos ω t = A cos 0 0 = A

What if we want to observe motion from the position when the particle is at mean position i.e. at “O”. We know that sine of zero is zero. Knowing the nature of sine curve, we can ,intuitively, say that sine function would fit in the requirement in this case and displacement is given as :

x = A sin ω t x = A sin ω t

However, if we want to stick with the cosine function, then there is a way around. We know that :

cos θ = sin θ ± π 2 cos θ = sin θ ± π 2

Keeping this in mind, we represent the displacement as :

⇒ x = A cos ω t ± π 2 ⇒ x = A cos ω t ± π 2

Let us check out the position of the particle at t = 0,

⇒ x = A cos ± π 2 = 0 ⇒ x = A cos ± π 2 = 0

Clearly, cosine function represents displacement with this modification even in case when we start observing motion of the particle at mean position. In other words, cosine function, as modified, is equivalent to sine function.

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 :

[ ω T ] = [ 2 π T X T ] = [ 2 π ] = dimensionless [ ω T ] = [ 2 π T X T ] = [ 2 π ] = 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 :

⇒ x = A cos ω t + φ ⇒ x = A cos ω t + φ

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.

Figure 7: Illustration of different phase constants

Phase constant

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 ω t + φ x = A cos ω t + φ

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 π t 3 - π 6 x = sin π t 3 - π 6

Clearly,

At t = 0,

Initial position , x 0 = sin π 3 X 0 - π 6 = sin - π 6 = - 1 2 = - 0.5 m Initial position , x 0 = sin π 3 X 0 - π 6 = sin - π 6 = - 1 2 = - 0.5 m

Further,

Amplitude , A = 1 m Amplitude , A = 1 m

Angular frequency , ω = π 3 Angular frequency , ω = π 3

Time period , T = 2 π ω = 2 π π 3 = 6 s Time period , T = 2 π ω = 2 π π 3 = 6 s

Phase constant , φ = - π 6 radian Phase constant , φ = - π 6 radian

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