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Saturday, 4 August 2012

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Sunday, 22 July 2012

The Physics (And Maths) of Soccer: Offsides, Angles and Backspin

The offside rule is the one of the most complicated in soccer and leads to some controversial refereeing decisions, can they be justified? In light of the recent English performance in the European Championships, we will also be looking at goal-line decisions.

As Euro 2012 has just passed and we approach the Olympic Football, we thought it would be a good time to review the decision making of football referees, and the physics and maths behind these decisions, and potential technologies that could help them in the future.

Firstly, let's have a look at the offside rule. If we are to talk about the controversial decisions to do with the offside rule, we do first have to describe what the offside rule really is. The rule was designed in order to stop "goal-hanging" or a situation when a player just stands high up the pitch, waiting for an easy one-on-one with the goalkeeper.
FIFA, the world football governing body,  defines offside as when a player is in a offside position and: 
It is not an offence in itself to be in an offside position.
A player is in an offside position if:
        • he is nearer to his opponents’ goal line than both the ball and the 
          second-last opponent (usually the final outfield player).
A player is not in an offside position if:
        • he is in his own half of the field of play or
        • he is level with the second-last opponent or
        • he is level with the last two opponents
A player in an offside position is only penalised if, at the moment the ball 
touches or is played by one of his team, he is, in the opinion of the referee, 
involved in active play by: 
        • interfering with play or
        • interfering with an opponent or
        • gaining an advantage by being in that position 

In other words, the offside rule says that if a player on team A is closer to the goal he is shooting into than the ball and all but one of the team B players except one then the ball cannot be passed to him. Lets use a diagram:
                                     
At this point, player E on the blue team can pass to both player C and player B. This is because there are two of the opponent, red team players, A and D, between them and the goal.
Now player A has approached player E in order to tackle him, and player C has run forward. Player E can still pass to B, because the two red team players are still between him and the goal. However, now, player E cannot pass to player C. This is because player C is both in front of the ball, and there is only one red team player, in this case it could be the goalkeeper or an outfield player if, for some reason, the keeper is not back, between player C and the goal he is shooting towards. Now, there is one more possibility:
In this instance, Player E can in fact pass to both player B and player C. This is because, though player C has only one opponent player between him and the goal, he is behind the ball and therefore not in a offside position. 

Offsides can be very game changing as they can cause a goal to be disallowed or cause an important play to be broken up. This means that a lot of the offside decisions are very controversial.

Despite the importance of offside decisions, we still rely on a reasonably unreliable system to catch them, linesmen. Two referees that run up and down the sidelines. However, this relies purely on human perception, and if there is one thing we are good at, it is being tricked by simple illusions. Because we only view the world from one point, what we really see are two 2-Dimensional images (or a 2 spacial, 1 temporal dimensional image, if you want to nit-pic), of the 3 (spacial) dimensional (and, 1 temporal dimensional) reality. This means that linesmen, if they are not completely in line with the last defender, can find it very difficult to call, or not call, a very close offside decision. A situation when a player may only be half a meter offside, there is one main problem that occurs with the linesmen's perception. This is where maths comes into it:

When a linesman watches a match, the natural way for him to think of the pitch is a 2 dimensional, the only interactions and positioning that matters is from left to right. Therefore, the easiest way for a linesman to judge an offside is to see if a player who is shooting right is closer to the goal on the right than the closest defender:
Linesman's view
Bird's-eye view

The linesman can see in this situation that the blue player without the ball is obviously offside. He may be closer to the linesman but due to the reasonably sensible position the linesman has, there is no illusion.
However, lets look at a new situation, one where the blue player is still offside, but not by as much:
Bird's-eye view
Here you will see the linesman is not parallel to the last defender, but is instead at an angle. The black lines show where he is looking and the green lines show his sight line to the offside blue player, and the defending red player. Here emerges the problem. From the point of view of the linesman, the blue player is further to the left, therefore he sees this:
The blue player looks slightly onside, and the further the blue player is away from the linesman or the red player is towards the linesman, the more onside the player appears. This phenomena is reversed if the blue player is closer to the linesman's sideline than the red player, and he will appear to be offside when he is in a perfectly legal position.

There is only one position the linesman can take for this situation to be completely avoided: if the linesman is perfectly in line with the last defender at all times. However, football can be a very fast paced game, and at times it can be very difficult for officials to keep up with world-class athletes. It is obviously also a lot harder to make decisions when running at pace, so this is not always feasible. Perhaps a better method would be a camera that moves up and down the byline, always in line with the last defender, sending images to a screen monitored by an official who then raises their flag if they see an offence.

One of the main reasons that offsides can seem so controversial is that those watching on TV have a much better view than the linesmen a lot of the time. This makes it very obvious when they have made a mistake, and the mistakes that are made are usually made at the most critical times: when the last defender is running backwards whilst chasing after an attacker that is through on goal.





But after the attacker has successfully got the ball and is onside, he then has his next challenge: scoring a goal. This seems like it should be a fairly simple thing to officiate, but Geoff Hurst's infamous goal in the 1966 World Cup final is still remembered, and there have been several other incidents where it has been unclear whether the ball has in fact crossed the line.

The first issue with determining whether a ball has gone in, particularly after it hitting the crossbar, involves the same parallax illusion that the linesman has to deal with when they are looking at offsides. It is very rare that the linesman will be standing level with the goal line when looking at such a situation, so they don't have an angle from which they are able to determine whether the ball has gone in. To try and combat this, there have been experiments with officials that remain next to the penalty area so they are in a better position to make such decisions. However, as the incorrectly disallowed goal that Ukraine scored against England in their opening match of Euro 2012 shows, this does not always help.



In fact, in this video there was also an offside that was missed; the original pass from the Ukraine defender was offside (even though it is out of shot in this video), so perhaps these two poor decisions cancelled each other out.

The issue perhaps came from the fact that the official was quite close to the goalpost. This left him with the following view.


Bird's eye view
Linesman's view
The correct position for the linesman for this situation is actually just a little further back than the goalpost, to be precise the point where a ball would be at the moment the whole ball crosses the line and it counts as a goal. However, if the linesman was too close in this position, it could create the illusion that certain balls were across the line that weren't.

We can reduce this to a geometric problem. There is an area just behind the goalpost that the linesman wants to see. However, in an ideal world, he wants the post to at least partially block his sight of any ball that has not completely crossed the line, so that he can determine whether he has or hasn't. The following diagram shows the areas of the goal that the officials can see in different positions.


The further away the linesman is, the better he can see the critical point just over the line, and in this image, even the blue linesman can only see part of the ball, and it may appear to be behind the post. The green and red linesmen cannot see the ball at all, and definitely cannot award the goal!


From these positions, the linesmen can see almost every goal that does go in, but there is potential for them to incorrectly award a goal. The worst culprit for this is the green official, who could think that a ball has passed his near post when it is not even touching the line! Theoretically, the further back the linesman is, the more accurate a view of the goal he gets. To conclude, in order to make the most accurate decisions, the official should attempt to stand either in line with or just behind the post, as far back as his eyesight will allow him to. It then becomes a human compromise of at what distance he can no longer see effectively enough.





Before we finish off, let us first make our case for football officials to brush up on their physics!

Here is a disallowed goal from the 2010 World Cup, which Frank Lampard scored against Germany.



It is actually possible to tell from anywhere on the pitch that the ball was definitely over the line. The ball hits the bar, bounces, and then hits the bar again. When it hits the bar for the first time, it imparts backspin, a considerable amount of it considering the speed of the shot. This means that the ball will bounce backwards. For it then to hit the bar again after bouncing backwards, it must have been over the line at the point when it bounced. It seems that Frank Lampard 'didn't need to see a replay' because his physics knowledge had already given him his answer!

Ned Summers and Theo Caplan

If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter" on Facebook.


We hope you have enjoyed this post. If you have, then please check out our last two posts:

In light of the recent announcement by Cern, we have a post on what the Higgs Boson is and why we were looking for it, and what the future holds.
The Physics of Field Athletics - Hammer Throw, Angular Momentum,  why Hammer throwers spin before they throw, and what would happen if everyone in the world spun around at the same time.

If you have any requests for posts, please let us know! Contact details are above.

Monday, 16 July 2012

The Higgs: Why do we need it, why were we looking for it in the first place and what do we do now?

Last week, we talked about what the Higgs boson did and how. However, there is more to it. It is fine knowing what something does, but $13.25 billion and almost 4 years has been spent using the greatest particle physics machine to look for this particle, so why is it so important?

I saw a brilliant line in an article on this subject a couple of days ago and it sums up the whole situation almost perfectly:
The impact of the discovery for the Average Joe is not going to be huge. It is massive for physics, because it is an extra fact. And there is nothing scientists like more than an extra fact. 
I feel I can pretty much leave it there. However, it is not entirely spot-on. The Higgs has some sort of value as a discovery, especially for particle physicists. And as our knowledge of the universe increases our ability to control what goes on around us also increases. You may not see the results of this discovery directly, but I promise, it will be revolutionary in the end.

Anyway, enough faffing around, lets talk about why physicists have been getting worked up about this brilliant particle since the 1960s.

The Higgs itself is actually not very important. The only reason we needed to find it was because it signalled the existence of something else, the Higgs field. The actual thing that is interacted with by particles with mass. I can feel you slipping through my fingers already so lets get to the interesting stuff. The main reason that we looked for the Higgs is that it is the last piece of the standard model. So let me explain that one first.

The Standard Model is the most advanced physical theory we have yet. It describes three of the four forces in the universe and tells us of every particle that we can observe today, as well as a few others. It has its limitations, it can't describe gravity for example, but it is the closest we have to a unified theory of everything. It took the last century to form.  


You can think of it as a machine. If you took the state of the whole universe, the particles, forces and fields within it and put that into your machine. It would then shoot out equations that described these particles and how they acted. However, if we included mass in the original picture, the machine broke, it just straight up didn't work. So physicists played around with the machine. Eventually a few separate groups of scientists, including Peter Higgs (the others being Philip Warren Anderson; Higg's partners Robert Brout and Francois Englert; Gerald Guralnik; C. R. Hagen, and Tom Kibble) came up with the idea of the Higgs mechanism. This meant that mass was not put in at the beginning, but instead this mechanism became part of the machine, and Hey Presto! Particles had mass! (I took this analogy from MinutePhysics, a brilliant youtuber, who has done two videos explaining the Higgs.)

However, the vast majority of the particles in the Standard Model had been observed in experiments by the 1970s and by 2000 the Higgs was the last one left. This discovery was such a huge deal because it could have proven whether or not the Standard Model is, for want of a better word, right.

The Higgs field is specifically involved in another part of the standard model, in one of the forces it describes. The force is called the weak force and chances are, you don't know what it is. There is a reason for that. The force can only work over a really small distance, about the diameter of a proton. Why is this? Because the carriers of the particles decay really quickly. And why is that? Because they have a really high mass! The force is very similar to one we see everyday, the electromagnetic force, but the observations were so different. The more we know about mass, the more we know about the reason the weak force is like this.

What about elsewhere? Will this result impact anything other than the two (rather large) fields I've already talked about? Disappointingly, it doesn't seem like it. There is a chance that it could be important in the research of dark matter, and the more we learn, the more we can discover, but in the immediate future, the Higgs may have to move over. But maybe that is the best bit. Now we have a somewhat finished Standard Model and a Higgs field, we can move on. The wonders of supersymmetry, dark energy and dark matter are just around the corner and looking at writing this article, I am reminded of a quotation by that brilliant man, Richard Feynman, on which I will end.

"Physics is like sex: sure, it may give some practical results, but that's not why we do it." 


We hope you enjoyed this post. We are excited to see the many developments that come after this brilliant discovery.

Ned Summers

If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter" on Facebook.

Check out our last two posts:
The Higgs: What is it, what does it do and have we found it? - The recent announcement at CERN has gotten a lot of physicists very excited, but what does it do?
The Physics of Field Athletics: Hammer Throw, Angular Momentum and what if everyone in the world spun around at the same time? - Why does a hammer-thrower spin before they throw?



What are we posting about next:
The Physics of Cycling: Lets Talk About Drag - We are back to the sports posts! Why do cyclists, in team events, cycle so near to one another? And how big is the difference it makes?

If you have ideas for posts we would love to heard them. Contact details are above.



Sunday, 8 July 2012

The Higgs: What is it, what does it do and have we found it?

Lets start with a joke! A Higgs boson walks into a church, the priest taking the service quickly runs over and says: "We don't allow Higgs bosons in here" and the Higgs says "But without me, how can you have mass?"

Ok, the joke isn't very funny, in fact, it was almost as bad as the Faster-Than-Light-Neutrino joke made back in November.

Now, recently there has been a lot of news to do with the Higgs boson, or the frightfully named "God particle". Sadly, despite the importance of both the idea and the data so far collected, articles seem to be giving a full, complicated explanation or dumbing it down so far as to be nearly fiction.


What I am going to try to do here is to give you a full, easy description of one of the greatest scientific discoveries to happen in decades.

The most simple definition of the Higgs boson is both general and unhelpful, it is the particle that shows the existence of the Higgs field and gives particles mass. Many questions arise from this statement though. What is the Higgs field, what is mass and it does very little to actually tell us what the Higgs really is. To actually understand the whole phenomenon, we are going to have to work through this definition slowly. Firstly, what is mass?

There are a couple of ways to think of mass. The one you will be most familiar with, in a practical setting, is that it defines your weight. Put more scientifically, the larger the mass of an object, the more it is pulled towards the earth. We can tell that a car has a much higher mass than a person, as it is a lot easier to lift a person than to lift a car, or that it is a lot more difficult to stop a car that is rolling down a hill. This leads us onto another, more general and more complete definition for mass. Mass is simply the resistance of an object to a change its movement. That can be shown with an equation that has come up before:
Where F is the force, m is the mass and a is the acceleration.
The equation shows that the higher the mass of an object, the larger the force (a push or pull) that has to  be applied to it to make it accelerate even a little. This can be shown again if you try to push a car or if you try to push a person on a bike. The Higgs does not give mass to macro systems like a car, instead it is only relevant at tiny levels, giving mass to the tiny particles that make up the big object. This eventually adds up to the total mass.

Now we get to the crux of the matter. How does the Higgs give mass to the particles? Well I have heard a few good analogies for this one, I think I am going to use one I read a while back in a newspaper. Lets imagine a party. The party is quite big, there are a lot of people in one room, but they aren't squashed together:
Some of them are wearing party hats. It isn't relevant to the analogy, but what is a party without party hats?
As well as the people, there is a drinks table at the opposite end of the room from the door. Every person who enters the room really wants a drink and is trying to get to the drinks table.
Now, lets say a person that knows none of those attending comes into the room. They can walk all the way across the room as fast as they want. In fact, the only thing limiting their speed is their own ability.


Now, what about if someone who is friends with a few of the people in the room comes in?
 

 


A few of their friends clump around them, making it quite difficult for them to move across the room. They are slowed by the sheer number of people around them. They can still move fast, but it takes a lot more effort to do so.

Ok, lets have one more situation. What if a celebrity walks into the room?

 

They quickly get crowded by nearly all of the people in the room and it is very difficult for them to move across the room without exerting a large amount of effort.

Now, lets put this into actual physics terms. We can think of the party as the Higgs field. Put simply, a field is a place where a certain thing can take effect. In this case, the Higgs field, or the party is through the whole of the universe. The people originally in the room are the Higgs bosons. They are part of the Higgs field, and are the particles that actually give everything else mass. The first person that walked in is a particle like a photon. It doesn't react with any of the Higgs bosons (or the Higgs field) and therefore can move at the fastest speed possible. In the universe, this is the speed of light (about 300 million meters per second). Its mass is zero. The second particle could be an electron. It interacts quite a bit, but not too much. It can move quite fast, and its mass is quite small. Now the celebrity could be a quark. Quarks are some of the heaviest particles. They find it very difficult to move fast. The difference in mass of an electron and the heaviest quark is like comparing the weight of a 10 year old boy to that of a fully loaded Boeing 747 jet.

Now we know what the Higgs does, what about how they found it? Well last Wednesday, CERN, the European Institute for Nuclear Research, put out a press release after an announcement they made, saying that they had discovered a fluctuation in their data:


This fluctuation seemed to suggest, with little uncertainty, that there was a new particle they had discovered. Before we get to that though, lets first delve into how CERN got these readings.
It was one particular facility that led to this discovery. A huge network of tunnels underneath Geneva, called the Large Hadron Collider (LHD) for reasons we will discuss in a minute:


The whole group of experiments relies on one small, beautiful and very famous formula.
Where E is energy, m is mass and c is the speed of light.
When we look at mass in this formula, we now have to stop thinking about the Higgs, otherwise it just gets too complicated. Instead we are going to look at the basic implication of this equation. All it says is that mass and energy are interchangeable. We see mass being converted to energy in stars. It also implies that the opposite is true. Energy can become mass. In fact, this fact is so important that we don't measure the mass of a particle like we normally do, but instead we measure it in units of energy, electron volts, usually shortened to eV. However, this isn't too important.

Lets go back to converting energy to mass. It is theoretically possible, however, due to the very large square of the speed of light you need a very large amount of energy to create a small amount of mass. However, because particles like the Higgs (which does itself have mass) don't form in an observable situation normally, sometimes there is only one way to nudge it into a detector. Groups of protons, a type of particle, are shot around the LHC (the large circular tube we discussed a minute ago), one group clockwise and the other anti-clockwise. Eventually, when they are moving fast enough, they are brought together to collide in detectors.

The two main detectors are called ATLAS and CMS. Suddenly, there is a huge amount of energy in this small space and a large amount of strange particles are created. The detectors look to see if some energy may be missing, suggesting an undetectable particle has been created or there has been another strange occurrence. The detectors also look for the abundance or lack of some particles. You see, a Higgs, even when formed in this manner, is not around for long. In fact, it disappears before it is detected. Instead, the detectors look for the particles that could be created when a Higgs disappears and something formed in its place. The particles that come out of a collision can be changed by changing the energy of the colliding particles.

And here is where it starts to come together. The CMS and ATLAS detectors have, over the course of around 4 years, found that at the energy level of 125 Gev (giga-electron volts), there is particularly large amounts of certain particles produced. In fact, they have given it a 5-sigma rating. The sigma rating shows how certain they are about the discovery, 1-sigma is just expected deviation, 3-sigma is an observation and 5-sigma is a discovery. However, we can't be too rushed here. They have found something, and it seems to act like they expect the Higgs to act. However, we cannot confirm whether it is the Higgs or actually something else. Still, we live in an incredible age and no matter what this particle is, it will bring on even more brilliant and fascinating physics. I am personally on the edge of my seat for any more announcements that will be made in the immediate future and beyond!

This subject is huge. Here are a few links to help you if you want more information:
Another simple guide to the Higgs
A great, in depth and detail article on the Higgs, the implications of the find (which I could not go into) and the standard model
An article about the methods and maths behind finding the Higgs
More information on the actual experiment
A brief summery of everything to do with the Higgs
An article on the experiments finding the Higgs and similar experiments in the past
A CERN video about the search itself
A CERN video about what the Higgs is

We hope you enjoyed this post. The current situation we are in is fantastic and I hope you will stick with us to learn more every week!

Ned Summers

If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter" on Facebook.

Check out our last two posts:
The Physics of Field Athletics: Hammer Throw, Angular Momentum and what if everyone in the world spun around at the same time? - Why does a hammer-thrower spin before they throw?
Maths of the Heptathlon: Why the scoring system is flawed? - How is the Heptathlon biased towards athletes that are good at throwing? And how can we fix the system?


What are we posting about next:
The Higgs: Why do we need it, why were we looking for it in the first place and what do we do now? - Now you know what the Higgs does, now you can find out why we wanted to find it!

If you have ideas for posts we would love to heard them. Contact details are above.

Monday, 25 June 2012

The Physics of Field Athletics: Hammer Throw, Angular Momentum and what if everyone in the world spun around at the same time?

A few weeks ago, we posted on the physics of Shot-put and Javelin. This week we are continuing on that note, talking about one of the two other throwing events, Hammer Throw. This event is different to both Shot-put and Javelin due to the way the Hammer is thrown. So, lets find out how, and why they are different to each other.

The main difference with Hammer Throw is that the athlete spins before throwing their projectile. Shot-putters do spin, but not as many times. The reason that Hammer Throw athletes spin is very different so lets get started.

Ok, lets take a look at the current Hammer Throw world record. It is quite a long video, but you only need to watch the first throw to get an idea of the throwing style: In The reason for the spin is primarily to make the hammer move faster when it is released. This leads us to talking about something called momentum, specifically angular momentum. So before we talk about the Hammer Throw, lets first define what momentum and angular momentum is. Momentum of an object is shown by the formula:
Where p is the momentum, m is the mass of the object and v is the velocity of the object.
You may notice the little arrow-like lines above the p and the v. These arrows show that these quantities are vectors. We have talked about vectors before, but I will run over them again. A vector is simply a value that has both a magnitude and a direction. For example, my velocity cannot just be 10 ms (Meters per second). It has to be 10 ms Eastwards. The speed that you see on your car dashboard is not a vector, so can stay the same even when you are turning. Anyway, there is a universally defined property of momentum that is very interesting. This property is that momentum must be conserved. This sounds strange but in fact all it means is that the momentum in an isolated system always remains the same. The reason your car slows down when you take your foot off the accelerator is not because its momentum is disappearing, but instead because momentum is being taken from the car and being put into the air particles it keeps hitting and the ground below it. In fact, friction is just the name we give to something that takes momentum from an object that is trying to move. We have a general name for something that changes the momentum of an object, it is called a force. Now, lets move on. However, when the hammer spins, it still increases in velocity, but it is not in a straight line. This means that we have to make a new value for the momentum of a spinning object. This is what we call the angular momentum.Angular momentum can be expressed with either of two formulas:
Where L is the angular momentum, I is the moment of inertia and ω is the angular velocity.
Let me clear up some of weird terms. The equation is very similar to the one for normal momentum. The moment of inertia is like mass, it is just how much the object resists spinning. The moment of inertia of one object spun around another is mr2 (m is the mass of the object and r is the distance from the center). The angular velocity is the number of rotations the object makes around the central axis in a second. This angle is measured in radians and a total rotation is 2π. Ok so enough of the annoying technical language, so let me give you an example. Here we will use the actual measurements for the hammer throw. The mass of the hammer is 7.257 kg and 1.215 m long. For the example we will say that the thrower revolves four times in three seconds. Therefore our calculation goes like this:
The ∆"o with a dash" just means the change in the angle, and ∆t means that changed in the time. π is a irrational number which starts 3.141592... 

This calculation is reasonably useless, at least in this situation. However, it just means you can see how the theory works. However, once you know the theory, we can learn some interesting things specific for the hammer throw. When we work out the angular momentum, it is for the whole system, and this means the angular velocity is the same at both the center, and at the hammer. At the risk of, once again, making this too technical, even if two objects have the same angular velocity, they may have different speeds. Here I will use a diagram:




If an object moves from position A to position B in one second, and another object moves from position a to position b in one second, they have the same angular velocity, because the angle they move per second is the same. However, the distance from A to B is clearly larger than a to b. This means that the object that moves from A to B has a larger speed, or, the hammer has a larger speed than the spinning thrower. In other words, the longer the wire from the hammer to the thrower, the faster it will be moving at release and the further it will be thrown.

Hammer-throwers have to spin. We have seen in shot-put, where the shot is of a similar weight as the hammer, throwing without a spin, or with only a small spin, means that the object cannot gain enough speed to travel large distances. The spinning acts as a way of magnifying the power the thrower has, turning them from an athlete that throws twenty meters, to one that can throw over eighty meters.

So, now that we have talked about Hammer Throw, how about we have some more fun with angular momentum. I recently saw this comic on the fantastic (and ever so slightly geeky) site, xkcd:


It makes sense. Like normal momentum, angular momentum has to be conserved. This means that when you spin in a certain direction, in the northern hemisphere that is indeed anti-clockwise, you slow the rotation of the earth. This comic (and a post I did a while back on what would happen if everyone in China jumped at the same time) inspired me to do some calculations. It is interesting how much impact we humans can have in simple ways on the basic mechanics of the universe around us. What I wanted to work out was how much effect it would have if everyone in the northern hemisphere turned anti-clockwise, whilst everyone in the southern hemisphere turned clockwise. The calculations are in fact quite simple, though we will have to make a large amount of estimations here. Firstly, we know that there are around 7 billion people on the earth. The average mass of a person is 70 kg. Now, we are going to have to imagine people as simple rectangles, otherwise the calculation becomes too difficult. The average hight of a person is around 1.70 meters, average shoulder width is 46 cm and average distance between chest and back is 24 centimeters. We are going the be rotating the people around the axis of their height. Therefore, the moment of inertia is shown by the formula:
Where m is the mass, w is the width and d is the depth.
If we say that everyone spins about 1.5 times per second, a perfectly reasonable number, this is what we get:





Times that by 7 billion people and we get a total of roughly 104 billion newton meter seconds of angular momentum. So how does this compare with the earth? Well the earth is a sphere, so it's moment of inertia is a lot easier to work out with the formula:



The mass of the earth is 2 kg and its radius is 6 378.1 km, both of which are very large numbers, I think you will agree. On the other hand, it spins just 0.00001160576 times a second. So lets plug that into the formula:


If we work out the percentage of the angular momentum of the earth that our spinning people have, it is just about 0.000000000000000000936%. Unsurprisingly, the change would be minimal.


We hope you enjoyed this post, it is always fun to find out just how little an effect we can have on the mechanics of the earth!. 
Ned Summers
If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter" on Facebook.

Check out our last two posts:
Maths of the Heptathlon: Why the scoring system is flawed? - How is the Heptathlon biased towards athletes that are good at throwing? And how can we fix the system?
What is the Drake Equation? (Guest post from B.C.) - A guest post from one of our partners, about the equation used to predict our chances of discovering alien civilisations. Can maths answer the age-old question of whether or not we are alone?

Sunday, 10 June 2012

Maths of the Heptathlon: Why the scoring system is flawed

The Heptathlon is one of the greatest tests of an all-round athlete that exists in World Athletics. Women compete in seven events over two days for the title. But this sport is suffering because of a biased scoring system. Why is this, and what can be done?

Let's start with a bit about the Heptathlon. It has seven events, three running events (100 m Hurdles, 200m and 800m), two jumping events (High Jump and Long Jump), and two throwing events (Shot Put and Javelin). Certain equations are then used to turn the athletes' raw scores into points, which are then totalled, and whoever has the most points wins.

The current scoring system was developed by Dr Karl Ulbrich in 1952, and amended in 1984. He based these early scoring systems on certain benchmarks: certain results were adjudged to be worth 1000 points, and others worth 0 points. A line was then put through these, but it was an upwards curve rather than a straight line, to account for the fact that the better an athlete performs, the harder it is to better that score by a certain amount. In other words, it is much easier for an athlete to reduce a time for the 100m Hurdles from 13.5s to 13.0s than from 13.0s to 12.5s.

He used three different formulae to calculate the score, P for each event.

P = a(b-T)c
for the running events, where T is the time in seconds.
P = a(M-b)c
for the jumping events, where M is the heigh/length in cm.
P = a(D-b)c
for the throwing events, where D is the heigh/length in metres.

a, b and c are different for each event, and are given in the following table.

Eventabc
200 meters4.9908742.51.81
800 meters0.111932541.88
100 metres hurdles9.2307626.71.835
High Jump1.8452375.01.348
Long Jump0.1888072101.41
Shot Put56.02111.501.05
Javelin Throw15.98033.801.04
This system works in principle, but a look at modern results begins to reveal a problem. Here is a table of the highest and lowest scores in each individual event at the Heptathlon at the 2011 World Championships in Daegu.

Times given in seconds (minutes:seconds for 800m), and distances in metres. The difference is in points.
A first glance at this suggests that it is biased towards the the Hurdles, the scores for this event seem to be much higher than others! However, how high or low the scores are is actually irrelevant. For example, if you decided to add 100 points to everyone's javelin score, the outcome of the competition would still be exactly the same, but with everyone's scores simply 100 points higher. 

The important statistic is instead how large the spread of scores is, here shown by the difference between the highest and lowest points scored. For example, the best hurdler at the Championships only gained 201 points over the worst hurdler, but the best javelin thrower gained a huge 400 points over the worst! In this sense, with the current scoring, the Javelin is worth almost twice as much as the Hurdles, and the Shot Put is worth nearly as much, leaving specialist throwers with a great advantage.

These differences are not just due to individual weak points giving very poor, anomalous scores. In the table below are the 10th place results for each event, and the distance from first.


At the top end, the competition is even more biased towards the Javelin, which is nearly 3 times as important as the Hurdles when considering only the top 10 results.

In order to have a truly fair system, and to provide the best test of who is really the top all-rounder, the difference between any two given positions should be as close as possible for each event. 

The values for b have been carefully chosen to set the result that will score 0 points, and c has been chosen to give the graph the right shape. These values seem to work, and should not be changed. However, we can change 'a' to try and make the difference between the best and worst athletes equal. Using the data here, we can find new values for 'a' for each sport.

Let us set the difference between the best and worst results for each discipline to be 300 points. Taking the Hurdles as an example, we can form the following equation.

a(b-Tbest)c = a(b-Tworst)c + 300

a(b-Tbest)c - a(b-Tworst)c = 300
then substitute in the values.
a(26.7-12.93)1.835 - a(26.7-14.32)1.835 = 300
a(13.77)1.835 - a(12.38)1.835 = 300
Then factorise the left side of the equation to get
a(13.771.835 - 12.381.835) = 300
21.8198368a = 300
a=13.748957096

Using the same method, we can find new values of a for all other disciplines to give a difference of 300. These values turn out to be the following:


Using this scoring system, our table from before turns out like this.


The only problem with this is that the scores have radically different values awarded to them. This doesn't impact on fairness (except in the rare event of three fouls or false starts in any event), but could be very confusing to athletes and spectators, and may also affect competitors psychologically. To fix this, we can introduce a new value, d, which is added to the end of each formula, to change the highest score from each of these disciplines to 1050 and the lowest to 750.


This also improves the differences between the top ten.


The Standard Deviation of the difference between 1st and 10th place is 13.4 with the system we propose, compared to 61.3 with the old system. This is clearly a much better and fairer method, and one we hope the IAAF will consider. To sum up, here is the Proposed Scoring System in full.

For running events:
P = a(b-T)c + d
For jumping events:
P = a(M-b)c + d
For throwing events:
P = a(D-b)c + d

And the following table of values are used:

Unfortunately, with the current scoring system, British Gold hopeful Jessica Ennis is at an acute disadvantage. Her best events, Hurdles and High Jump, are two of the most under-awarded events, and her weak point, Javelin, is the most over-awarded! Jessica came 2nd at these Championships in Daegu by a margin of 127 points, with only her Javelin below par on this occasion, but with these scoring revisions would have lost by a much tighter margin of 42 points. With the current scoring system, it seems that Jessica, along with all the other athletes, should turn their focus towards the Shot Put and Javelin in hope of securing Gold.

We're all behind you Jess!


We hope you enjoyed this post, it was certainly very interesting to research and write. We don't have any Olympic Heptathlon tickets, but will be watching Jess with earnest.

Theo Caplan

If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter" on Facebook.



Check out our last two posts:
What is the Drake Equation? (Guest post from B.C.) - A guest post from one of our partners, about the equation used to predict our chances of discovering alien civilisations. Can maths answer the age-old question of whether or not we are alone?
The Physics of Gymnastics - The forces exerted on a gymnasts body during some routines are extreme, so how does it work?

Monday, 4 June 2012

What is the Drake Equation? (Guest post from B.C.)

This a guest post from the blog B.C.. The blog describes itself as "of Prehistory, Palaeontology and the Past". If you enjoy this post, go check them out here!

1969 is best remembered for the lunar landing. For the first time, we had truly left the surface of the Earth and stood upon a body which had gone unchanged and lifeless for over 4 billion years; truly one small step for man, but giant leap for mankind. What is rather interesting is that this event completely eclipsed a second NASA mission operating on a far larger scale.

For years cosmologists had been developing probes to land on the Red Planet. While their missions did not involve human cargo, the engineering challenges they faced were even greater. The craft would have to cross hundreds of millions of miles across a hostile, freezing void blasted with solar radiation, descend through a thin atmosphere without burning up and land without smashing on the red, iron rich rocks.
The first successful landing took place in 1972 by what was actually a Soviet probe, but the Americans soon followed up with many missions of their own. It had been a long held view that there were living organisms on Mars. Astronomers had seen, through telescopes, landforms which could only have been created by liquid water, an essential component for life from an Earth perspective.

Some went as far as to claim that channels on the surface of the planet were canals made by intelligent aliens. Dark areas recorded by the primitive black and white cameras of early fly-by probes were thought to have been lush, verdant forests. When the first probes reached the surface and sent back colour images, scientists found not a thriving world, but a barren wilderness of red rocks and sand under red skies.

Subsequent mission have failed to find any trace of Martian life, either in a fossil record or simple microbes under rocks or within the ice at the poles. These first images of the Red Planet forced cosmologists to rethink all their ideas about life on other planets. Today the subject is still contentious and without any solid physical evidence, but what little we have was sparked off by an equation created by a NASA scientist, Frank Drake, just a few years before the lunar mission.

Bring out the maths!

While this might look rather complex, the variables involved are actually quite simple.

  • R* is the average rate of star formation per year in a galaxy,
  • fp is the fraction of those stars which have planets,
  • ne the average number of planets that can potentially support life per star that has planets,
  • fl is the fraction of those planets which will go on to develop life at some point,
  • fi is the fractions of those again whose life will become intelligent,
  • fc is the fractions of those again which will develop a technology which can release detectable signs of their existence into space 
  • L which is the length of time for which such civilisations release detectable signals. 


When the numbers are multiplied together, they produce N, the number of civilisations in a galaxy with which communication might be possible with each other. Since its creation by one Frank Drake, this equation has been altered to take into account variables such as the time needed for life to develop to an intelligent, technologically advanced stage (in Earth’s case 3.8 billion years) and how long these civilisations may last (taken as 10,000 years).
The idea of time scales can be put into the form of:

In this Tg is the age of the galaxy in question. Assuming that R* is a constant, then this can be simplified to:

This can now be placed back into the original Drake equation to provide a more rounded version which takes into account time scales:

So what we are going to do is crunch the numbers as Drake did back in 1961 for our own galaxy, the Milky Way. 

  • R* = 10 stars formed per year, 
  • fp = 0.5 (half the stars will have planets)
  • ne = 2 (Stars with planets will have two capable of sustaining life)
  • fl = 1 (all these planets will develop life),
  • fi= 0.01 (A hundredth of all this life will be intelligent),
  • fc = 0.01 (one hundredth of those will be able to communicate with us 
  • L = 10,000 (whose civilisation will last for 10,000 years) 
  • Tg = 13.0 billion years old, (though this value appears twice in the equation, the other time within N* as a multiplication and a division, they cancel each other out and do not affect the rest of the mathematics, remaining only as a reminder of the time scale involved.)

Overall this gives us a value of just 10 detectable civilisations within our galaxy. Only 10! Considering that the Milky Way is 13 billion years old with a diameter of 120,000 light years and over 100 billion stars in the spiral arms, this is very small. While some discredit the Drake equation as a simple and inaccurate estimation tool which glosses over many other important factors, its significance extends far beyond mere statistics. 

What is interesting is just how different our estimates for the different variables can be. For example, the value of R* can be anything from the 10 stars that Drake used, down to 1 star, however, even then we are assume these are stars of solar mass. Modern figures suggest the real number could be anything from 0.14 to 7! 

In fact, the values for almost all the variables are in debate. Some research suggests that fp could be 0.4, but other research seems to find that the number tends to 1! And even if we discover the real number, we still have the fl, fi and fc. These are the variables to do to with the likelihood of life developing. Seeing as the only life we have ever observed was on earth, it is very difficult to estimate these. This means that the resulting value of the Drake Equation can be nearly anything, ranging all the way from not much above 0 up to 182 million contactable life forms!  

However, what we should remember is that the maths is not wrong. The equation uses multiplying probabilities, a reliable and common mathematical method. However, due to the number of difficult-to-predict variables, it is very difficult to get a reliable result. The maths can only help when the numbers are right. 

Because the Drake Equation has a large number of variables, even small errors in the values of the variables can lead to very large variances in N. For example, if we say that each variable in an equation is accurate ±10%, then with two variables, to find the resultant range of N, we have to find the highest and lowest possible value of N, assuming that the variables are as wrong as they can be within the error bars. So, if we let both variables be 1, then to find the upper bound we do
(1+(10%×1)) × (1+(10%×1))=1.12=1.21
And to find the lower bound, we do
(1-(10%×1)) × (1-(10%×1))=0.92=0.81
So this means that although multiplying the original figures would give 1, the errors mean that the actual result could be as low as 0.81 or as high as 1.21. This gives a variance of 0.4, which is relatively high.

However, at more variables to this equation, at it gets worse. If, like the Drake Equation, it has 7 variables, then the variance will be much higher. Again, we will take all variables to be 1 ±10%.

The upper bound is 1.17=1.949 (to 3 decimal places) and the lower bound is 0.97=0.478 (to 3 d.p.). This results in the huge variance of over 1.47, larger than the value itself! This means the actual value could be almost double or less than half the stated ones. Many variables in the Drake Equation have much less certainty than the ±10% used here, so this could be even worse. However, it does at least give us an idea: with ±10% accuracy on all variables, you will still get a result in the right order of magnitude.

Another important point, is that it only accounts for advanced civilisations. What it cannot give us an idea of is of any kind of life. To do this requires looking at the conditions and elemental composition of other solar systems. This is something which mathematics simply cannot do as the variables involved would simply be too complex to reduce to single, simple terms. For life to form on a planet, three different things are needed. 

The first are the right elements. From an Earth view, these are Carbon, Hydrogen, Nitrogen and Oxygen. The Carbon is the linchpin of the quartet as its chemical diversity facilitates the intricate and precise reactions which fuel living organisms. The second factor is an abundant energy source. This could be anything from heat or ultraviolet radiation to lightning strikes. All have their merits and their abundance will be decided by the third factor: the conditions present on the planet. The ability to analyse reflected light from far-off worlds can give us some idea of its elemental composition, but until we develop the technology to travel across interstellar space or an asteroid brings hard evidence of alien life to Earth, we will have to console ourselves with this equation and Bowie’s famous words ringing in our ears ‘is there really life on Mars’? 

If you are still confused, here is Carl Sagan going through the equation:



We hope you liked this guest post. B.C. posts about historical biology and the world when it was very young. If you like the post, check out the blog!

Check out our last two posts:
The Physics of Gymnastics - The forces exerted on a gymnasts body during some routines are extreme, so how does it work?
The Physics of Field Athletics: Why do the shot-put, the javelin, the discus and the hammer fly differently? - All these events have the same aspects, but they all get different distances. How does this work?