A note to my readers
I make a conscious effort to avoid arbitrary targets – it seems nonsensical to center your appreciation of a loved one on the 14th of February, to delay making a positive change because the Gregorian calendar has not reset or to be more charitable once a year due to a festival from a religion you are not a member of. That said, there is a relative amount of calm in the weeks surrounding the holidays that gives the opportune moment for self-evaluation. With regard to Rationalising the Universe I am pleased with the progress we have made and continue to be astonished with the number of writers from all disciplines who engage with the site. I offer no great change in direction for a site which is not broken – but rather propose some refinements.
This site was set up for a number of reasons – but the overarching aim was to bask in the glory of science for nothing but pleasure. Over the last few months I have written increasingly about the forefronts of theoretical and particle physics – something of great interest to us all, but as caveat-ed I have more questions than answers in these areas. Whilst this is a noble pursuit, I fear I am being permanently seduced while avoiding the foundations of mankind’s knowledge; popularizing popular science is actually fairly easy, it is popularizing the bread and butter which requires a certain zeal. I propose over the coming year, very occasionally we cover an essential area in any undergraduate physicists toolkit. This will, I hope, be (slightly) more interesting than the average textbook. This is not to say musings on the forefront of theoretical physics are suspended – they will remain the main event, interspersed with essential physics to ensure we do battle properly equipped. This may be a little painful I am afraid.
Thank you for your continued interest and support in the site, I hope you choose to stay reading Rationalising the Universe through 2017
Simple harmonic motion
What an arrogant title – unfortunately the name is too embedded in science for us to be able to make any changes to it, so whilst you may not find this topic simple, we will call it simple harmonic motion, or SHM for brevity. I often like to start with the dictionary definition of a word or phrase before uncovering the detail, which for SHM we have;
simple harmonic motion
oscillatory motion under a retarding force proportional to the amount of displacement from an equilibrium position.
Forces like this can be modeled very neatly using mathematics, since we have a force which depends only upon the position of the particle. We will consider the case of a spring, with one end attached to a fixed end and the other free to move with a particle attached to the end of it. The set up can be seen in the diagram below.
All I intend to do today is to derive from first(ish) principles an equation which will allow us to model the motion of the particle. Note that the above diagram is showing phases of time – the only motion we have is in the vertical direction. This is called the j-direction which in the below formulas, for now you will see denoted j with a hat (^) on it. This is not really any different to saying upwards, but what it actually means is a unit vector in the vertical direction which we have defined. Generally when you try to model anything which has classical motion you first need to consider the forces. There are two forces here – the weight of the particle and the spring force.
Dealing with the weight first, the weight of any object is the mass of the object multiplied by the acceleration due to gravity. It is this second part which changes depending on the gravitational field you find yourself in – hence why you can say you weigh less on the moon than on Earth. If we have expressed the positive j-direction as vertically downwards, then the weight of the particle can be expressed as follows;
Now we move onto the spring force – the spring force obeys a law known as Hooke’s law. Often when you see formulas written down they seem very difficult to construct – actually they are just the result of people who mess around with springs and know how to draw graphs. From experiment, it turns out the force a spring exerts is proportional to the deformation of the spring (which may be compressed or stretched) by a constant known as the stiffness of the spring (k). The force is directed towards the center of the spring – which leaves us with an initial dilemma for our model; should we take the force as being in the positive or negative j-direction, since it can be both? Spoiler – it makes no difference. So we can set up our spring force as follows, where x is the position of the particle.
Note that the term l with a subscript 0 is the natural length of the spring. That was a little dry but now we are in business, we have forces giving rise to motion of a particle which can only mean Newton. We want to express the motion of the particle on the spring as F=ma, however there is an issue in that a introduces a further variable. However let us define the x-direction to be vertical (I know this isn’t a Cartesian set up, but it’s my model so deal with it). Now the position of the particle at any moment in time can be modeled as some function of t, say x(t). Since velocity is the rate of change of position, and acceleration is the rate of change of velocity I may denote the acceleration as the second derivative of x, or in applied mathematics notation x with two dots above it. A derivative is simply the rate of change – so I am taking the rate of change twice of x. This means I can rewrite our famous F=ma as;
The next logical step from here is to replace F with the forces we have define above. All of a sudden I am going to drop my unit vector j – this is okay. What I have done is reached a point where I have realized I only have j-components so resolved in this direction (see note 1), so I can just consider these forces with the unit vector implied.
Now using the above I consider only the last two terms, which I expand out and collect all x-terms on one side with all constant terms on the other side. This gives;
This is a differential equation in x – an equation that links the derivatives of x. All other terms here are constants, so really once we have set up a real system these will just be numbers. If you are really mathematically minded, or just like jargon, what I have constructed is a second order constant coefficient in-homogeneous differential equation. The solution to a differential equation is a function which satisfies the equation – i.e some kind of mathematical expression that when we substitute it into the above, the equality is satisfied. I have illuminated more of the logic in the notes for those who are interested, but the first stage is to solve the equation as if the right hand side were to equal zero (which is called the complimentary homogeneous function). To do this I have defined a new constant omega such that;
Now an equation which will satisfy my differential equation if the right hand were to be zero is as follows (note 2);
Here B and C are arbitrary constants. Next we need to deal with the right hand side – since the right had side is not actually zero we need to add a term to the above solution to make it work. In fact what we need to add is the equilibrium position of the particle, multiplied by the stiffness of the spring. Please see note 3 for further discussion – but if you think about what is happening; a particle oscillating around an equilibrium position this hopefully will roughly”feel right”. This gives us the following final general solution.
And there we have our solution – we can define the position of a particle, oscillating in harmonic motion from a spring with the above equation. We would need to know the stiffness of our spring, the mass of the particle and the natural length of the spring in order to quantify omega, k and x-eq. Following on from this, we would need an initial condition in order to remove the arbitrary constants B and C – this would look something like initially the particle is released from rest at position x = 0.1m for example. This will be specific to the model we look at.
Why do I care?
Being able to model the motion of particles under forces is central to all of Physics; and it is entirely implausible to think that one might go on to model complex quantum mechanical systems if simple classical skills have not been gained. That is not to say the above skills will be directly used (they generally will not), but the mindset of the Physicist (and the Mathematician) is to be able to take a real world situation, express as much of this information mathematically and produce a model which is a close match to real world results. This is a most wonderful thing. This article goes into a little more depth around the applications of harmonic motion.
Finally – my apologies for this dense technical post. Please ask anything you have not understood, I can give confident answers on this topic; my next few posts will be light and much less technical.
- When you consider motion in one direction, you will only have a resultant force in that direction. If you had a resultant force in another direction you would have motion in more than one direction – so by construction we do not. That isn’t to say we don’t have forces in other directions – say a train moving along a straight track- we have the weight and the normal reaction force vertically and the thrust and resistive forces horizontally, however the vertical forces are equal and opposite, so we can consider only the horizontal to model the motion.
- In order to solve this differential equation we first consider the auxiliary equation which is found as; Clearly then lambda is equal to the square root of -omega as defined previously which gives my roots as +/- omega i. When we have complex roots a +/- bi, we can construct a solution in the following form Where in this case we have a=0, hence this term drops away to 1, with b equal to omega.
- The next step to solve an in-homogeneous differential equation is to take a particular integral and add it to the equation – generally this is done by taking a trial term which matches the right hand side in form, plugging it into the equation and equating coefficients. In this example it is actually very easy – if I take x = K, some constant since I have a constant on the right hand side, when I plug this into my differential equation I am left withWhich does not take much rearranging to give me the required solution above. The solution to an inhomogenous differential equation is the sum of the solution to the homogeneous complimentary function plus the particular integral – by the principle of superposition. In order to make the conceptual leap to this being k times the equilibrium position, simply resolve all forces where the is no resultant force. Ask me if you get stuck.
- By convention all forces are written in bold. This is to indicate they are vector quantities. When you do them by hand you should underline them
- Unit vectors are also denoted in bold, but have a ^ on them to denote their significance as being of magnitude 1 in a specified direction.
- Using these techniques one can very easily extend the model to any particle moving with simple harmonic motion – the results will have minor changes by there is no need to restrict to this spring example. An easy natural next step is to consider a horizontal spring – the results of this are quite similar, but this applies to all manner of situations. Damping, resonance and forced vibrations are all natural extension topics to increase your modelling capabilities.
- Often in textbooks you will see the solution written the other way round, with the cosine function first taking the B coefficient. I don’t know what made me write it this way – but truly it makes no difference.