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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?