The Olympics is now upon us, and as 18-year-old British hopeful Tom Daley begins his campaign for the individual and synchronised 10m platform events today, we look at the physics behind the sport, and the angular momentum that he must generate to successfully perform his most difficult dive: the famous “Front 4 1/2.” However, first a bit of news…

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As Tom Daley prepares for the start of his second Olympic Games, this time in his home country, he will know that he has done all he can to perfect all the dives he will be showing off. However, there is one particular dive that has been harder than others to get right: the notoriously difficult “Big Front,” or “Front 4 1/2.” With one of the highest difficulty tariffs in the sport, it is brutally simple: from an upright start, do 4 complete somersaults, and then another half a somersault to land in the water head-first, all without making a splash. Sounds easy, doesn’t it? Here’s a video of Tom at the Fina World Series in Tijuana in April this year, look out for his “Big Front” brilliantly executed at 1:49, with two perfect 10s from the judges, which helped him bring home the Gold.

This dive takes about 1.6s from the moment he leaves the platform to the moment he enters the water, and in that time he must rotate a staggering 1620°. This gives an angular velocity of 1012.5°/s, which requires two things: firstly, he must generate as much angular momentum as possible, and secondly, he must form the tightest tuck possible to keep the mass of his body as close to the axis of rotation as possible.

When somersaulting, his body rotates around the following axis.

His body has to be curled up tightly because of the principle of angular momentum. Angular momentum (L) is defined as the product of angular velocity (ω) and the moment of inertia (I), or as an equation, L=Iω. Moments of inertia, or just moments, are a rotational force, and the further away from the axis of rotation a force has, the higher its moments of inertia. This is the explanation behind the famous quotation from Archimedes: “Give me a place to stand and a lever big enough and I shall move the Earth.” If Archimedes was far enough from the levers axis of rotation, his small weight would have more moments of inertia and a greater rotational force than the Earth, and it would indeed move. We discover this instinctively on seesaws when heavier people can be balanced by smaller people if the small ones are on the edge and the heavier ones near the centre.

Like regular momentum, angular momentum is conserved. That is to say, once it has been gained, without any external torque (turning force), it will remain the same. For Daley, this means that after he has left the platform there is no way that he can change his angular momentum, and in order to control his angular velocity, he must change his moments of inertia, and this is done by bringing his body mass closer or further away from the axis of rotation. In the case of the Big Front, this is likely to be closer, as it is extremely difficult to complete all the somersaults necessary. However the control is of paramount importance, as one of the main parameters that the judges are looking for is a perfectly vertical body shape upon entry. An angle of 10° or 15° upon entry is poor, 25° is disastrous. Since the angular velocities we are looking at here are so large, this errors require only a very small change in moments to occur. If his angular velocity were 997° or 1027°, with just a 1% change in moments of inertia, then that would be enough for a very poor entry to occur, and the judges would likely struggle to award it more than 5 out of 10.

So how fast is Daley actually moving during this dive? Well, his head is the point that is furthest from the axis of rotation, and will therefore have the highest linear velocity as its moments of inertia are higher (This also means it is subjected to the most G-Force, hence why divers must work hard to keep their head and toes straight and controlled mid-dive). Using average human proportions, the distance from the head to the waist, the axis of rotation, is approximately 40% of the man’s height, in Tom’s case 1.77m, giving his torso, neck and head a length of 70cm. However, with the bend in his back that is necessary for the tuck, his head is actually about 40cm from the axis of rotation. His head travels in a circular shape relative to the axis, so the distance in one rotation can be found by finding the circumference of a circle with radius 0.4m, which is found with the following formulae.

Circumference of Circle = πd = 2πr = 0.8π = 2.513m

4 1/2 of these rotations are completed by his head, so the total linear distance travelled by his head is 4.5 × 2.513 = 11.31m. At the end of the dive, he stretches out to prepare for his entry, in which he does not rotate very quickly as he increases his moments of inertia, so about 11m of this rotation is done in the first 1.3s of the dive. This gives his head a linear speed of 8.46m/s, almost as fast as the 10m/s run by an Olympic sprinter. But this is not yet considering the downwards velocity! The Earth’s gravity exerts a force on mass that causes an acceleration in free fall of 9.8m/s². We can then use this equation to find Daley’s speed as he enters the water.

Velocity = Acceleration × velocity

v=at

v=9.8 × 1.6 = 15.68m/s

We can then add these two speeds to get Tom’s total speed whilst diving, and we get 8.46+15.68=24.14m/s. This is a huge speed, equal to almost 54 mph or 87 km/h.

As we can see, the strains on the body of any diver attempting the Big Front are huge, and near-perfect control of their body is needed to get the scores they need to be ranked among the best in the world and compete at the Olympics. As the Synchronised event starts today, the control in this is even more important, as you must be identical to your partner. Perhaps this is one of the ultimate sporting challenges: with such control needed at such high speeds, it is almost unparalleled in the sporting world.

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

The Physics (And Maths) of Soccer: Offsides, Angles and Backspin - The offside rule is one of the most confusing in soccer. What is it and why are there so many controversial decisions surrounding it?

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.

We have now nearly reached the climax of our Physics of Sport series with the arrival of the Olympics, so have a look at the posts in the series here.

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