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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
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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?

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