Simple Harmonic Motion

animationIn many ways, the scope of this discussion is attempting to expand on the general idea of a wave as a mechanism that transports energy `independent` of matter to a more general description of an oscillatory ‘wave’ motion. In this context, Simple Harmonic Motion (SHM) is not explicitly describing a physical wave, but rather a specific type of oscillating system, such as a pendulum or a weight on a spring. By way of a visual reference, the animation is showing a weight on an oscillating spring, which has a ‘resting or equilibrium point’ defined by the central y-axis from which the weight is displaced and initially held. As such, the weight has acquired potential energy, which is then released as the weight starts to oscillate around the resting point. Clearly, as a physical system comprising of just a weight oscillating up and down, there is no mechanical wave propagating in space [x] linked to this system. Yet, the graph on the right of the animation seems to be alluding to a cosine wave that maps to the up-down oscillation with respect to time [t]. This graph directly represents the vertical displacement or amplitude [A] of oscillation, which is a quantity that has already been related to the energy of a wave, i.e.

[1]      E ∝ A2

For initial simplicity, this system is assumed to suffer no loss of energy [E] and so the cosine waveform in this conceptual model will continue forever and, in-line with [1], suffers no loss of amplitude [A]. As such, we can mathematically predict the amplitude [A] of this waveform at any point in time [t] as follows:

[2]      A = -A0*cos(t)

The negative value of the initial amplitude [A0] simply represents the downward displacement with respect to the central resting point. Of course, we might question the use of time [t] within the cosine function, but in this context we can equate the period [P=tλ] to the time taken for one oscillation or wavelength [λ], even although there is no physical propagation through space [x], such that:

[3]      3

Therefore, time [t] may be considered to be directly proportional to the angular displacement along the x-axis and, as such, the value of [t], implied in [2], really corresponds to the definition in [3]. So, having clarified some of the mathematical assumptions, we can now proceed to consider the mechanical implications of this system, which is described by Hooke’s Law. In the context of the spring example, Hooke's law equates the force [F] exerted by a spring to the displacement [y] and a constant [k] that depends on the ‘elasticity’ of the spring. While [k] is not the focus of the present discussion, we can say that it depends on the material and nature of the spring’s construction and is assumed to remain constant while the spring operates within its ‘elastic limit ’. So, the force [F] required to stretch the spring by distance [y] is defined by Hooke’s Law as:

[4]      F = -(ky)

In this case, the negative sign is explained in the sense that the force [F] always acts in the opposite direction of the displacement [y], i.e. when a spring is stretched downwards, the force acts upwards. However, this force [F] can also be defined as the rate of change of energy [E] with the distance [y], i.e. F= dE/dy, therefore the input energy of this system can be determined by integrating force over its displacement range [0-y]:

[5]      5

This equation gives the initial potential energy [EP] of the system, where displacement [y2] is equivalent to amplitude [A2]. However, there is an inference in this equation that potential energy [EP] is always negative as the square of any oscillatory displacement [±y] must always be positive. At this point, we might also wish to consider the implications of the conservation of energy by citing the equation:

[6]      ET(t) = EP(t) + EK(t)

What this equation is trying to state is that the total energy [ET] at any time [t] must be equal to the sum of the potential energy [EP] and the kinetic energy [EK] at the same point in time [t]. It is easy to see from the animation, the initial energy state of the spring at time [t=0] reflects only potential energy [EP] as the weight is stationary at this point, i.e.

[7]      ET(t0) = EP(t0) + 0

Likewise, the animation suggests that there must be a point, when displacement [y=0], when [EP=0] and [6] becomes:

[8]      ET(t0) = 0 + EK(t0)

Without necessarily getting involve in all the ancillary parameters required to define [EP] in terms of [5], we can simply combined [1] and [2] to define the potential energy [EP] at any point in time [t] with the simplifying assumption that [A0=-1] and [k=1]:

[9]      E ∝ A2 = -k*[A0*cos(t)]2 ⇒ -cos2(t)

By the same logic and assumption, we can define an equivalent oscillatory equation for kinetic energy [EK] as a function of time [t]:

[10]    E ∝ A2 = -k*[A0*sin(t)]2 ⇒ -sin2(t)

graphFinally, [6] requires that the sum of [9] and [10], at any point in time [t], must reflect the constancy demanded by the conservation of energy. However, it is often quite informative to see the implications of the equations [9] and [10] in graphical form as shown in the diagram right. The black cosine curve is derived from [9] and corresponds to [EP], while the blue sine curve is derived from [10] and corresponds to [Ek]. As indicated, the sum of these 2 curves, as represented by the dotted line along the bottom must reflect the constancy as required by the conservation of energy. Finally, the larger red cosine curve corresponds to the spring amplitude as shown in the animation. All the energy curves are associated with the scale on the right, while the amplitude scale is on the left, with both normalised to unity.

So how do we explain the energy curves?

It would appear that the mathematics of the previous equations has led us to a picture in which both [EP] and [EK] have negative values, while the rate of change seems to be twice that of the amplitude. Therefore, it might be useful to work through the implications of the diagram in a stepwise manner using the reference points:

  • (1a)(1b): Are the starting points for [EP] and [EK] respectively. The energy [EP] corresponds to the work done in displacing the weight to offset [-1] on the amplitude scale. Hooke’s law, as presented in [4], rationalised the associated force [F] as a negative quantity because it always acted in opposition to the direction of the displacement [y]. However, the subsequent integration of [4] then led to [5] suggesting that [EP] must also be negative, while [7] and [8] indicate that this energy must conserved by a lossless system and reflective in the constancy of [ET]

Note: While there are instances in physics when potential energy can be described in terms of a meaningful negative value, e.g. gravitation potential, it is not clear that the negative value of both [ET,] [EP] and [EK] is really that helpful in visualising the energy conservation in this case. Therefore, while the negative values are maintained in the diagram in connection with the derivation of [5], the muscle power required to displace the weight to its initial offset [y] is real enough and that we might ignore the mathematical implications of negative energy for now.

  • (2a)(2b): When the weight is released at (1), the force [F] acting through the spring causes the weight to accelerate towards the equilibrium or resting point, i.e. [EP] is converted to [Ek]. At (2a), all [EP] has been converted to [EK], therefore this is a turning point in the energy curve and although this is not directly reflected in the amplitude [A], it can be seen that [A] does transition from negative to positive values. It is also useful to note that while the maximum value of [EK] corresponds to a maximum velocity [v], it also corresponds to a point of zero acceleration because the force on the weight at (2a) has also fallen to zero [F=ma].

  • (3a)(3b)(3c): Physically, when we observe this system, as reflected in the animation, we perceived (3c) as the primary turning point in amplitude, although as stated, (2a) was just as real a turning point for energy, velocity and acceleration, it just wasn’t so obvious from a casual observation. As such, we have explained what might have appeared to be an anomalous doubling of the rate of change in the energy curves.

  • (4a)(4b): While (4a) might appear to be distinct from (2a) when observing offset amplitude (A), the energy conditions are identical, i.e. max [EK], min [EP]. By [4b], the energy has again been converted back to [EP].

  • (5a)(5b): These points represent the completion of the amplitude cycle, but the second repetition of the energy cycle.

At this point we have only introduced the basic idea of Simple Harmonic Motion (SHM) by trying to highlight that while there is no specific mechanical wave associated with this motion, it does have wave-like attributes. However, by the same token, the conservation of energy implicit in the previous description of the wave-like nature of SHM might now be extrapolated to help described the energy cycles within ‘real’ mechanical waves. The previous discussion has suggested that energy can cycle between a maximum of potential energy to a maximum of kinetic energy. How this works in mechanical wave will be considered in subsequent discussions; but for now we might initially assume that any mechanical wave has to be kick-started by acquiring potential energy from outside the wave system itself. We might also assume that the energy ‘transported’ by mechanical waves can be described by some cycle of [EP] to [EK] distributed across the entirety of the mechanical wave, which unlike the SHM example described, exists in both time [t] and space [x]. Of course, we should also recognise that all practical systems are subject to energy loss.