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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Ooops more trouble on the blogging front

Well at least this is the end of it I hope, I was flat out with work and shifting up until Easter and then after the break I come back to find that my work PC has died (actually very similarly to my laptop - I am of course considering if there is a correlation).

And so the last week has been mad trying to sort out alternate computer resources, get the problem fixed, and of course catch up on all the work i let fall by the way side before Easter.

But never fear as I am back and I will be restarting my weekly series (hmmm what is the plural of series????? I must look into that), I have some long awaited posts on the history of science in the not too distant future and a couple of new features that I will be trial, so hopefully posting will be on the increase.

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Another Temporary Blogging Hiatus

This time because we have had to move flats and it has been a somewhat stressful experience, and most of next week will be taken up with cleaning up the old place but I will be trying to get some more stuff up by Easter.

If not enjoy your chocolate eggs (the darker the chocolate the better)!

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Now I Really Feel apart of the Wider Community

What you may ask makes me feel this way, well it is the fact that I have my first ever paper to peer review for a journal.

Now you may say that this is terrible and reviewing papers takes up so much time, that you could be using to publish your own work, and I can see that, but as a young researcher I feel it is the biggest responsibility that can be laid upon me. (Well that and having younger students in the lab to mentor).

Of course I have already had a few papers published (ok only one as first author) so I have already made contributions in that regards, but this feels bigger to me. It is peer review so that officially makes me a peer.

Ok well I had better get back to it, deadlines and all.

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I wanted numbered lists... and now I do

Well you may have or may not have noticed that all my lists were coming up bulleted, this was not meant to be the case an I have finally got around to digging through my template and learning how to mess with the right parts.

As you can probably see a few other things like the colours got changed too, in fact if you visited the site this afternoon you may have seen changes like crazy as I tried to work out what did what (I apologise if this caused any upset) and if not relax and enjoy my now correctly enumerated lists

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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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The Skeptics are at it again

EoR at Second Sight has the latest and greatest of the last fortnight's skeptical blogging all in one place with the 55^th Skeptics circle, head on over and enjoy, it is a real special number.

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How to ... disobey the second law of thermodynamics

OK well this post is very late, it was due up a week or so ago, but things have been hectic and since my laptop died working from home has become impossible, but I am endeavouring to catch my self up with these posts

Well the second law of thermodynamics (SLoT) gets quite a bum rap from all sides of the anti-science. For starters there is perpetual motion, infinite amounts of free energy, but the is also a horrible use of the SLoT as an argument about evolution.

One reason I suspect as to why SLoT is so mis-understood is that while it conveys a fairly simple physical law, when it is applied to different situations it tends to need to be stated in very different ways. To quote P.W. Bridgman

There are almost as many formulations of the second law as there have been discussions of it.

For example:

The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.

Heat cannot of itself pass from a colder to a hotter body.

A transformation whose only final result is to convert heat, extracted from a source at constant temperature, into work, is impossible.

The first two statements are from Rudolf Clausius, and the last one from Lord Kelvin. And they are all equivalent, but generally phrased to high-light what the law means with reference to a particular situation. For example Lord Kelvin's statement above is a very mechanical one, essential stating that while we can use heat to do work we can never covert all the heat to work, so there is always some energy lost as heat.

The other difficulty with SLoT is that it is a statistical law, it is based on probability. Essential every system is in a certain state (macro-state) that is comprised of all the states of the atoms (micro-states) that make up the system. The more atom states that correspond to a particular system state then the more likely that system state is to exist. Naturally there are many more disordered (macro-) states than ordered (macro-) states, so systems will tend to wards disorder (an increase in entropy).

The big caveat to this statement is that all this applies only if you do not start adding energy to the system. In other words the law only applies to isolated systems.

So the Earth is not an isolated system, it receives a lot of energy from the sun. So the Earth and everything on it does not obey the SLoT, however if we take the Universe to be our system then this is clear isolated (it is all there is by definition) then the SLoT is obeyed.

And our magnetically levitated spinning top in the foyer of the lecture theatre next to the Physics department here will only spin perpetually if the power is plugged in other wise the interaction of the magnetic fields (and some air resistance) will cause to stop spinning and fall.

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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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