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Defining simple harmonic motion

Classical mechanics

Defining simple harmonic motion

The simple harmonic motion is described here, in all its simplicity, with the use of geometrical constructs as visual aids, for your easier understanding.

Physics, particularly mechanics, largely cropped up from the need to understand motions of objects around us. The motions of the Sun and the Moon across the heavens, the motions of stones tossed up into the air, the motions of boats calmly sailing across seas, and so on. If you closely look at each of these examples, you’ll notice that there’s something nice about these particular motions. The Sun and other celestial objects appear to cross the heavens periodically; stones thrown up into the air, land in places that are mostly predictable (by experience); boats sailing in calm seas mostly follow straight line paths over considerable distances; etc. The niceness of these motions basically lies in their predictability and uniformity. Mechanics did not originate from a curious attempt to explain the erratic motion of a flying bee, for example. It’s not just because the bee’s motion is complex and unpredictable, we in fact have no idea where to begin explaining a bee’s motion! That’s how humans think. Whenever we come across anything that appears tangled and unclear, we choose to first seek answers to simpler (yet related) questions, and keep the complex for another time. Following suit, we here seek to understand a very simple and nice kind of motion – the simple harmonic motion.

Motion and its types

Just like we have criteria (however vague) to classify motions as nice and not nice, we have precise criteria to describe the nature of motions accurately. We use two parameters of motion – distance and time, for this description. We speak in terms of how much distance (and in what direction) does an object cover in any length of time. A familiar type of motion is the one in which the distance of an object from a specific point, increases (or decreases) constantly in a particular direction, as time passes by – it’s the “linear uniform motion”. Mathematically, it is described by the equation: x(t)=At. Here, A represents a quantity that doesn’t change with time, and x is the distance the object covers in a time t. You may imagine the motion of a fired bullet across a distance of few meters, as an approximate example.

Another type of motion is that of an object moving so that its distance from a specific point remains fixed, and in particular, moves along the circumference of a circle. Circular motion may possibly be described by the simple equation x(t)=B; B being a constant of time and is essentially the radius of the circle along which the object moves. Imagine the motion of a capsule in a rotating giant wheel.

And of course, there’s the motion in which the distance from a point may change randomly with time – the bee’s motion. And we won’t write down the equation for this motion here, not because it doesn’t exist, but because the problem is too vague, we haven’t been told how the bee moved, in the first place. (Even if we were told how, the equation would, in general, be as complex as an equation can get!)

Simple harmonic motion equation

Following the above procedure, we now define the simple harmonic motion. An object whose distance from a fixed point changes sinusoidal with time, is said to perform simple harmonic motion. You might already have the equation thought up in your mind: x(t)=\cos{t}. But it is incorrect to write an equation so. Notice the argument of the cosine function – t here has dimensions of time, but the argument of a trigonometric function is an angle and can be either in degrees or in radians. Therefore, we correct the equation and write: x(t)=\cos{\omega t}; where \omega is just a constant of time and has units of radians per second ([\omega t]=radians/second \times second=radians). The equation still isn’t correct. Recall that a trigonometric function is dimensionless and we’re using it to represent distance, which has dimensions of length. Therefore, we introduce one more constant and write: x(t)=A\cos{\omega t}; A having the dimensions of length (meters in SI units).

What the equation says

If you look at the equation of simple harmonic motion closely, you’ll see that A\cos{\omega t}, in some way, seems to be representing the “adjacent side” of a right angled triangle whose hypotenuse is of length A. To make the picture more concrete, let’s consider a two dimensional Cartesian coordinate system, as shown below.

Cartesian coordinates

(a) 2-dimensional Cartesian coordinate system. (b) The triangle with hypotenuse A.

The coordinate system is placed so that its origin coincides with the vertex of the triangle, with angle \omega t. x(t) represents the distance along the x-axis from the origin ‘O’, and is just the adjacent side of our hypothetical triangle. The equation tells us that at t=0, that is, just when you start your stopwatch, the triangle is squished, with the hypotenuse merging into the adjacent side: x(0)=A. As time proceeds from t=0, the argument of cosine increases, while the overall value of the cosine decreases. This essentially means the adjacent side of our triangle shrinks with time, the angle of interest (\omega t) grows, and the triangle itself resurrects back, with its hypotenuse remaining at the fixed length A. We know this approach might seem a bit demanding on your imagination. So, let us give you the big picture all at once.

Simple harmonic motion

Observe the motion of point Q along the circle’s circumference in the order (a)>(b)>(c)>(d). As Q rotates, P performs simple harmonic motion, about O.

We just have been speaking of the triangle OPQ, where OP=A\cos{\omega t}, OQ=A and P\hat OQ=\omega t. A being of fixed length, is also the radius of the circle along whose circumference the point Q moves. From this figure, it must be obvious to you what we meant by the squishing of the triangle at t=0 and its subsequent resurrection immediately after t=0. Notice that when as much time has elapsed so as to make the angle \omega t=\frac{\pi}{2}, the triangle is again squished, but this time along the y-axis. Thereafter, for greater P\hat OX, the adjacent side OP becomes negative and increases until the triangle again collapses on the negative x-axis at \omega t=\pi. The revival and collapse of the triangle repeats in the same fashion as the point Q traverses the circle and reaches back at the positive x-axis. Thereon, the entire process repeats over and over again. Note that here \omega is a factor that decides how quickly the point Q goes round the circle. Larger \omega means faster circulation. And so it is rightly named the angular frequency (we’ll define it precisely, soon).

It’s SHM and not circular motion here

It is possible that one would have misunderstood the whole idea, by looking at it as the description of circular motion of the point Q, at a fixed rate \omega. Recall that we here are only interested in simple harmonic motion (SHM); and here it is the point P and not Q that is performing simple harmonic motion (the sinusoidal dependence). Therefore, our prime interest is in the variation of the length OP=A\cos{\omega t}, with time. The triangle, the circle, etc are mere geometrical constructs we’ve erected so as to gain an intuitive understanding of the motion. So, we here say that as the point Q goes around the circumference of the circle, its “projection” on the x-axis, the length OP, varies with time, simple harmonically. If the word projection sounds uncomfortable, you might view it as the shadow of the line OQ on the x-axis, due to a broad beam of light falling in the negative y-direction (between \omega t=0 and \omega t=\pi), and then in the positive y-direction (between \omega t=\pi and \omega t=2\pi).

Projection

A way to visualize “projection”.

In analogy with the giant wheel, you might view this as the shadow of the support to your capsule, on the ground, as the wheel rotates constantly, with sunlight shining exactly downwards. The shadow of your capsule, performs simple harmonic motion about the shadow of the center of the wheel!

In our next article, we’ll exploit the geometrical constructs introduced here further to gain a better understanding of simple harmonic motions and also explain alternate definitions of the motion.

Cover image by Abhijit Bhatwadekar

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Roshan Sawhil is a Physics postgraduate who rejoices both doing and explaining Physics. He also finds doing Philosophy as a leisure activity quite interesting. You can find and connect with him on Facebook and Twitter.

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