Friday's Physical Law - Mexican Wave

Last time we mentioned that one simple harmonic oscillator (SHO) couple to another SHO can have interesting effects. In fact when you have one SHO coupled to another SHO coupled to another SHO... you get a oscillation that varies not only in time but also in space and what we have is a wave.

Yes you can think of it just like the Mexican wave at you favourite sporting event (ok now that is an interesting thought - can anyone tell me where that name comes from?) first one person stands up and then the next and so on. The main difference between coupled SHO and drunk sports fans is that with sports fans you get a single pulse travelling around the ground (or very occasionally two pulses), but with the coupled SHOs each one continues to oscillate so you get a series of pulses following (often quite rapidly) one after the other.

So what exactly is meant when I say the oscillations varies in time and space. We know from last time how the SHOs vary with time, and that is carried through to the behavior of waves. We see each individual point oscillating with the same equations for position and velocity and acceleration. The variation in space can be described in a very similar way, as a cos or sin function varying with respect to position rather than time. Obviously the angular frequency term of the time variation is also replaced by a term that is related to how the wave repeats in distance, this is sometimes know as the wave vector (k). Remember:

• ω = 2π/T

well similarly

• k = 2π/λ

where λ is the wavelength of the wave and is the distance over which the oscillation repeats, much like T is time in which the oscillation repeats.

But what exactly are the oscillators that we are talking about, well these can be almost anything, ok not anything but waves happen in lots of different materials. The obvious examples of waves is those on water, but almost all musical instruments make sounds with waves and then there are earthquakes and electromagnetic radiation and ...

Stay tuned for more exciting developments next time

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Friday's Physical Law - Simple Harmonic Motion

Like we saw last week simple harmonic motion (SHM) is a periodic (repetitive) motion that is descried by sine/cosine functions. There is one other requisite in the definition of SHM and that is a restoring force. What do I mean by a restoring force, well it is mostly what is sounds like, it is a force which acts to restores the system to its original or equilibrium position.

So a restoring force is at its simplest a force that acts counter to the position of the object within the system. Think of a mass on a spring - when the spring is extended it pulls the mass back, conversely if you compress the spring it pushes the mass out. So this force can be described by:

• F = -kx

where x is the displacement and k is the spring constant, and the negative sign shows that the force counteracts the displacement.

Another example of this is a pendulum where gravity provides the force and is always trying to force the bob (the mass at the end of the pendulum) to its lowest point.

But wait if there is a force that is always pushing it back to equilibrium then how does it keep repeating its motion. Well to explain this lets look at the case of the mass on a spring (the pendulum is the same but its motion tends to be 2d and so slightly harder to explain).

If you have a mass attached to a spring and it is sitting at equilibrium and you pull it down a certain distance, then when you release it the spring pulls the mass back towards the equilibrium point, accelerating it as it goes, now as the mass gets closer to the equilibrium the force and acceleration get smaller, but the velocity gets bigger (since it is being accelerated). So when it gets back to the equilibrium x=0, so F = 0 and a = 0, however v ≠ 0 so the motion continues past the equilibrium where the acceleration now acts to slow the mass down, until it stops at the opposite point the where it started, and then accelerate it back again.

So as long as friction is small (or as well like to think for our examples non existent) then this motion will go on and on and on. This gives us our simple harmonic motion:

• x = Acos(ωt)
• v = Aωsin(ωt)
• a = -Aω²cos(ωt)

Now so you don't look at me and ask where there heck did they come from, the first one is found in last time's discussion about circles is just the position of the mass relative toe the equilibrium, and the second two come from the definitions of velocity and acceleration, so are the change of position and velocity with respect to time. Strictly speaking these last two are the derivatives of position and velocity with respect to time which relies on calculus and gives us the instantaneous values for velocity and acceleration.

Now we saw last time that ω is related to the period of motion T. And so combining the restoring force equation and Newton's second law and the above expressions for x and a we can get:

• F = -kx = ma
• ω = √(k/m)
• T = 2π √(m/k)

Now since we talk about how long it takes for things to happen with T, one other factor that is related to this is how often things happen, the frequency, f.

• f = 1/T = ω/2π

The frequency of an event is the amount of occurrences in 1 second (usually measure in Hertz, Hz), it is inversely related to the period, of is something takes o.1 second to occur then it has a frequency of 10 Hz etc. OK this is only true for repetitive (oscillatory) motion other wise the frequency does not make sense.

So we can see that a mass that oscillates back and forth about an equilibrium is SHM, however circular motion does not really fit the bill since it does not have that restoring force.

Of course like the name suggests SHM is simple, but harmonic motion in general can be a lot more complicated, such things as friction can damp the motion, or something can drive the motion. Even the presence of other oscillators coupled to the first causes interesting phenomena but that is for next week.

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Friday's Physical Law - Round and Round

We saw a while ago that circular motion requires a centripetal force, but there is also another interesting feature that comes out from all that going round, and that is that you can do it all again (and again and again ... ad nauseum).

So what we have is a type of motion we refer to as periodic motion. That is motion that repeats itself after a period of time, and most commonly this period of time is constant (ie uniform circular motion, such as an orbit). So you start off at a certain point and then you move before coming back to where you started.

This is sort of obvious when you think about travelling in a circle. And as you can see this is a very simple sort of periodic motion.

Now since going around in a circle involves going through 360° we often describe the motion in terms of the rate at which the angle changes - the angular velocity (ω) if you will. The simplest way to think of this is, like I said, a velocity so we will consider the angle changed divided by the time taken (much like velocity is distance moved divided by time taken) and so we get

• ω = 2π/T

But wait a minute I hear you say what does 2π have to do with angles. Well if instead of the more common degrees as a measure of angle we use radians, which are a slightly more natural way of describing angles (it all comes from the ratio of arc length to radius), then this is a measure of the angle. In fact you probably use radians with out even knowing it, when ever you say the circumference of a circle is 2πr then you have just used radians. So just to clarify:

• 2π rad = 360°

So in other words our angular velocity is the total angle in a circle divided by the time taken to go round the circle. As we saw in the gravity post:

• v = 2πr/T

so we can combine this with above and we see that

• v = ωr

and we also get similar expressions for arc length (s, distance around the circumference of the circle) and acceleration (a)

• s = θr
• a = αr

where θ is the angle and α is the angular acceleration.

Now why is all of this remotely important, any one familiar with geometry will note that the position relative to the centre of the circle in xy-coordinates can be described in terms of the angle or indeed ω and t.

• x = r cos(θ) = r cos(ωt)
• y = r sin(θ) = r sin(ωt)

So in addition to being a simple periodic motion, circular motion is also a simple harmonic motion, harmonic meaning that it varies sinusoidally (like a sine or cosine wave). Now this term has a very specific meaning in physics and we will see next week just what it entails, and whether or not this is an accurate description.

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Friday's Physicsal Law - Gravity

Ok well yet again this is a catchup post of this series, it was due on Feb 30 March 2.

We all have a good daily experience with gravity, we can feel it pulling us down whatever we do. And our everyday experience on how it effects us is usually limited to the acceleration that we undergo or our weight

• F = mg

where g is the acceleration due to gravity and is approximately 9.8m s^-1. Now this is all well and good for describing the effects of gravity here at the surface of the Earth. But what it does not do is tell us anything about what gravity is or how we work out 'g' for other locations (ie surface of the Moon or Mars).

For our everyday experience, that is so well described by the above equation, we can derive myriad equations to describe the parabolic motion of projectiles, to determine time of flight, maximum height, distance traveled and velocity along the path but these equations are simply those that can be used for any acceleration (interestingly enough this equivalence between acceleration and gravity plays a role in the development of General Relativity too):

• d = v_it + ½at²
• v_f² = v_i² + 2ad

etc.

However to get the true experience of gravity we must leave this time and place and travel back to the time when Tycho Brahe was observing the motion of the planets. And since we are travelling back in time we might as well get our selves situated nicely above the plane of the solar system so we can see everything.

Johannes Kepler using Brahe's observations deduces three laws that govern the astronomical.

1. The orbit of every planet is an ellipse with the sun at one of the foci
2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time
3. The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits

And it was upon these, in particular the Kepler's third law that Newton formulated his Law of Universal Gravity, basically by combining Kepler's law with his Laws of Motion.

Now to do this without resorting to inventing (or just using) calculus you and I will make a handwavey assumption (and one that isn't all that bad). The ellipse detailed in Kepler's first law are rather circular so to make the maths easier we will just use circles (note that a circle is a special case of an ellipse where the two foci are in the same place).

Now remember from last time that circular motion requires:

• a = v²/r

since the velocity around a circle depends on the circumference (2πr) and the period (T) (which probably should have been mentioned in the other post):

• v = 2πr/T

which gives us

• a ∝ r/T²

and combining this with Kepler's third law, which for a circle can be written:

• T² ∝ r³

then we get

• a ∝ 1/r²

So the acceleration and hence the force are inversely proportional to the square of the radius of the orbit (as the radius increases the force decreases). So this tells us how our weight (remember this is given by mg) varies as we change our position relative to the Earth, but what about on other planets?

Well if we were to go to the moon and weigh ourselves we would discover the scales read about 16% of what they did before we left Earth, since the Moon is smaller than the Earth then if nothing else was involved in gravity then our weight would go up, so something else must be involved and this turns out to be the mass of the object we are on (be it a planet or moon or whatever).

So this gives us Newton's Law of Universal Gravitation:

• F = G Mm/r²

where G is the gravitational constant and has a value of about 6.67×10^-11 N m² kg^-2 and M and m are the masses of the two objects (sometimes written m₁ and m₂).

Now many find it counter-intuitive that because of the M and m in the equation the force of gravity on me due to the Earth is the same as the force of gravity on the Earth due to me. Since clearly the Earth moves me and not the other way around. Of course we must remember that force is not the whole story, it is the acceleration that causes the motion and since the Earth ways more than me the movement of me is much more than that of the Earth.

This can be seen better in the case of binary stars, or Pluto and Charon, or any other objects that are orbiting a spot in between them. This consequence is really just an illustration of Newton's third law, equal and opposite action and reaction.

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Friday's Physical Law - Going round in circles

Again in my vain attempt to catch up on my weekly posting due to the myriad issues some of which I mentioned in an earlier post, this post is late, it was due on February 23.

In an earlier post in this series we touched on the concept of circular motion.

And if this force is always at right angles to the motion then the object will continue at a constant speed the changes in direction will cause it to move in a circle. In this case we call the force a centripetal force, meaning center-seeking, but more about that another time.

And now then is that other time. For the motion to be circular the velocity must also follow the circle, so at each point there must be a change in velocity at right angles to the motion, this change in velocity is the acceleration that points into the centre of the circle (at right angles to the motion) and is caused by the centripetal force acting on the object.

So, to have circular motion you can see we need two things, an object travelling at velocity v and a centripetal acceleration a (and hence force F).

To get the motion to be a nice circle you need to have the correct relationship between these two. Obviously (at least I hope so) the size of the circle (to be specific its radius, r) is also going to play a role, in fact the relationship comes out as (after a bit of complicated maths - some of which can be seen here):

• a = v²/r

which with Newton's second law gives us the nett force or the centripetal force:

• F = mv²/r

Now I used the term nett force above, what I mean by that is the force that is the result of adding all the forces on the object together, for example when we are standing still the nett force on our bodies is zero, but we are experiencing at least two forces, that of gravity pulling down on us and the ground pushing back up (Note: these are not an action/reaction pair). In general when we talk about forces especially in relation to acceleration we are really talking about the nett force.

In this case the centripetal force must be the nett force, otherwise the motion would not be circular.

There are many different ways to provide the centripetal force, obvious ones include

1. Gravity (for orbits and sloped paths like on a velodrome)
2. tension in a piece of string
3. Friction (between tyres and the ground)

Now since the centripetal force is required to keep an object moving in a circle, what happens when we remove that force. Like I said above the centripetal force is the nett force, so if it is removed then there is no force acting on the object and hence no acceleration. So an object released from circular motion will travel in a straight line in the direction of its velocity when released, this can be seen when you think about how one uses a sling to propel projectiles.

There is one further issue that people have trouble with when dealing with circular motion and that is the difference between centripetal and centrifugal but that is the subject for another post.

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Friday's Physical Law - Work Work Work

As we have previously seen force impacts a lot on motion. Today I want to look at how we apply forces, or rather what allows us to apply forces, and as you may have guess by the title this involves the concept of work.

When we exert a force on an object what we are doing is work on that object. And by work we don't mean what one does 9 to 5 but rather transferring energy. To be precise work W is done on an object when ever and object is moved a distance d parallel to a force F acting on the object

• W = F d cosθ

when F and d are parallel θ = 0 and then cosθ = 1 and the work is maximum when the angle (θ) between F and d is 90° then they are perpendicular and cosθ and hence W = 0.

Well that can all be a little complicated and daunting, but essentially you can think of it in terms of a box, if I push the box across the floor I give it motion and hence energy (by doing work: force across direction across), however if I hold it while walking across the room the box itself is only in motion because I am and as such is not getting any energy from me (so no work done: force up direction across).Of course while I was lifting the box (before I carried it across the room) then I was doing work on it.

So as you an see (hopefully) work is done and energy transferred when giving an object motion or changing its position with respect to gravity. We refer to these forms of energy as kinetic energy and gravitational potential energy, respectively.

So we will now look at these a bit closer. Firstly, kinetic energy occurs when anything has velocity (is moving), and is dependant only on the mass of the object and the velocity it has:

• E = ½mv²

And gravitational potential energy, is a form of potential (or stored) energy which depends on position with a gravitational field, and it depends on height (above ground), the mass of the object, and gravity (on Earth the acceleration due to gravity, g, is about 9.8 at the surface of the Earth, but this will vary for other planets and heights, at this stage I plan to deal with this next week):

• E = mgh

There are many other types of energy that we will get to in the fullness of time, such as the potential energy of a spring, thermal energy (heat), electrical energy, nuclear energy etc, but the important thing to note is that energy can never be created or destroyed only converted (transferred) from one form to another (this incidentally is the first law of thermodynamics).

And when it comes to motion this idea of conservation of energy can simply be put as:

• ½mv²_f + mgh_f = ½mv²_i + mgh_i + W

where the _f indicates the object in a final state and the_i indicates the object in an initial state. And we can see that friction is a case of work acting against the motion of an object, reducing its kinetic energy and creating heat (thermal energy).

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Friday's Physical Law - Conservation of Momentum

Well all have experience Newton's Third Law of Motion in action, every time we push on something like a door we can feel it pushing back on us.

What we have here is action and reaction, and as it is normally put Newton's third law reads every action has an equal and opposite reaction. So if I push the door with a force of 5 N the door pushes back with the same force.

Of course the main thing to note about these forces is that they operate on different objects. Many people seem to get this confused by thinking that the forces cancel out, which can only happen when forces are acting on the same object, which is not the case here. To use my previous example: I push on the wall and the wall pushes back on me.

Of course the action and reaction occur at the same time, they start at the same time and they finish at the same time. So this gives us another measure of the motion that results from these forces. Remember that as we saw a couple of weeks ago

• F = ma,

and since we know the force applied in the action and reaction and the mass of each object and now the time over which the force acts we can work out the change in velocity (remember that a = v/t). So this gives us what we refer to as impulse:

• F t = mΔv = Δp

What we have here is another measure of motion, momentum (p) which is equal to an objects mass times its velocity. And from Newton's third law we see that when ever we have to objects interacting (action and reaction) then their is a change in momentum is equal (and opposite). That is the total (nett) momentum of two interacting objects is conserved (the same before and after the interaction).

As we progress through this series we will see many more examples of the idea of conservation of quantities. But in this case we have a very good means to analyse the interaction of objects (the idea of two objects interacting can be extended to multiple objects where momentum will still be conserved) such as in collisions.

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Friday's Physical Law - Reflections

In the first two weeks of this series I started out with simple motion as my fundamentals and then built from there, and I do intend to continue in that vein but every now and then I am going to feel like something different and this week is one of those times.

Today I am going to discussing some thing that is part of my PhD research and is also something that merits a comment when talking about Climate change. That is reflection.

To begin with we must have a wave travelling along that interacts with an object (or the boundary between two media - ie water and air). Now when this wave (and for the most part I will be dealing with electromagnetic (EM) waves such as light, radio waves etc., although this applies to all types of waves) interacts with an object any of three things can happen:

1. Reflection
2. Transmission
3. Absorption.

Which of these occurs depends on the properties of the object and the wave. And in most cases two or even three of these things happen together.

Reflection is where the light strikes the object and bounces off. Transmission is where the light passes through the object. Absorption is where the light is ... well ... absorbed.

If we start of with a wave intensity of 1 then:

• R + T + A = 1

where R, T, and A are the amounts of the wave reflected, transmitted and absorbed respectively.

The obvious examples are a mirror for reflection, a window for transmission, and a black object for absorption.

And you will notice that I mentioned a colour specifically for the absorption, in fact it is due to reflection and absorption that we see colour. Light shines on an object and if red light is reflected and the green and blue light absorbed then we see the object as being red.

Now what does all this have to do with my research, well, I study the electrical interactions of the upper atmosphere (more about this another time) and the main tool we use is by using the ionosphere and ground as a mirror (albeit and imperfect ones) and sending radio waves long distances between these two "surfaces" and so the reflective properties of particularly the ionosphere are very important. And by looking at the changes in the signals we receive we can determine the changes in the reflections along the way and hence the local (or global) changes that has caused them.

And of course I also mentioned that reflections can affect climate change, if you have snow, ice or tundra then the sunlight reflects of this quite well, as any one who has gone skiing can attest to. But as these melt less of the sunlight is reflected and hence more is absorbed (since the Earth is not very transparent). This leads to more heat being trapped in the Earth-atmosphere system leading to more warming (even if only locally) which will lead to more melting which will lead to more absorption and so on. In other words it is a positive feedback loop for the warming which is not a good thing, especially since we can quite easily measure the amount of ground covered by snow, ice and tundra and it is decreasing.

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Fridays Physical Law - Force and Accelleration

As we saw last week a change in motion of an object requires a force to act on the object. Of course with change in motion we mean that an object can speed up or slow down or simply change the direction of motion - all of these acts are covered by the term acceleration.

Technically acceleration is the rate at which the motion changes, or the change in motion divided by the time it takes for the change. So like velocity is how fast you are moving (changing position) acceleration is how fast you are changing your motion (velocity).

So when ever we have a force acting on an object we get an acceleration. Of course multiple forces causing multiple accelerations could also be acting to reinforce or cancel out each other. Two people pushing a car (from the same end) make it twice as easy to get going, whereas if one was pushing forwards and the other pushing equally backwards then these would cancel out.

If we increase the force then we increase the acceleration, but what else effects this relationship? The answer to that is mass, the heavier an object the more force is required to get the same acceleration as a lighter object. This is described in Newton's Second Law F = ma.

A force in the direction of motion increases the speed, and a force counter to the direction of motion will decrease the speed. A force at an angle to the motion will cause a change in direction of motion as well as an increase/decrease in speed.

We also have the interesting case of a force at right angles to the motion, this will not cause any change in speed only in direction. And if this force is always at right angles to the motion then the object will continue at a constant speed the changes in direction will cause it to move in a circle. In this case we call the force a centripetal force, meaning center-seeking, but more about that another time.

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Friday's Physical Law - Motion

Welcome to Friday’s physical law (FPL) - a weekly look at how physics governs everything, and as this is the first of this series I will be starting somewhere we hopefully can all understand with simple motion.

Everyday experience would seem to indicate that the natural state of motion of an object is a rest. An object in motion will slow down and stop unless a force is acting on it. This was the reasoning of Aristotle. Of course this is what we see here on Earth in the presence of what we now recognise as friction.

We can quite easily see the truth of what happens if we remove the force of friction, imagine if you will you are on the ISS and you push off one wall, you keep drifting until you hit the other wall, this is because now the friction that is acting on you in now minimal. We can see examples of reduced friction on Earth such as skating on ice, or ball bearings.

This leads us to the understanding that we now have, Newton’s first law: That an object remains in a state of steady linear motion unless a force acts on the object. This means that an object at rest remains at rest and an object moving in a straight line will continue at the same speed and direction of motion unless a force acts on it

And our experience of friction that slows down the object is recognised for what it is: a force on the object acting against its motion. It arises from the interaction of the object and the media (substance) through/on which the object is moving. We can tell definitively about the existence of friction by the heating of an object as it slows down, the friction dissipates the motion (kinetic energy) of the moving object turning it into heat.

The effect of friction itself depends of only two things:

1. The object which is moving (i.e. what it is made of, its shape and weight)
2. The media through/on which the object is moving

As you can see the former is really a combination of a number of factors - the most important being those listed in the bracket. Obviously if changing what the object is moving through makes a difference then so will changing the object, the shape is important, the less contact there is the less friction there will be (think of a ball bearing), and the greater the mass the stronger the interaction between the object and the surface it is moving on (ok so to be completely honest the weight factor only applies to motion across a solid surface and not to motion through a fluid).

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