An Advanced Swing
By Alastair Pilgrim
Up to this point, we have modelled our flyer swinging on his trapeze as a ball on a string. While this has allowed an understanding of some aspects of trapeze, there are several shortcomings with this approach.
With the ball and string model, the only way to swing higher is to start higher. However, once we allow the flyer to move his body, this is no longer the case.
In this article we will show how a flyer can increase the height of his swing by moving his body. Our new model is therefore 'ball on a variable length string'. This is still a simplification compared with real life, but it captures the interesting physics which will allow us to gather some more interesting insights into the swing.
If you watch high level trapeze artists swinging, you will notice that while each has his or her own style. However there are basic movements common to all, which can summed up as:
"Long on the way down, short on the way up"
Let's look at what this means and why it causes the swing to increase in height. We will then consider a flyer swinging and show how this principle can be put into action.
First of all, let's define what we mean by the words 'long' and 'short'. When we modelled the flyer as a ball on a string, we showed that it was the length of the string that defined the period of the swing. When we consider a flyer hanging from a trapeze bar, the same is still true, however as before we should note that it is not the length of the trapeze cables that represent the length of the pendulum, but rather the distance from the pivot (the attachment point at the top) down to the flyer's centre of mass (generally around the hips). A long pendulum results from moving the flyer's centre of mass away from the pivot (straightening his body). A short pendulum results from the flyer bringing his centre of mass up towards the pivot, and there are a number of ways to achieve this.
Why does this help?In our simple model, once a flyer is swinging the energy in the system (flyer, bar, cables and pivot) is fixed. It just moves from one from to another, oscillating between kinetic energy and potential energy as the flyer swings back and forth. In order to increase the height of the swing, the flyer must put more energy into the system. This is because we know that at the extreme end of his swing, where he is stationary, all the energy is in the form of gravitational potential energy (with zero kinetic energy).
PE = m x g x h
(m = mass, g = acceleration due to gravity, h = height with respect to the bottom of the swing)
Therefore if h is to be increased, then the energy in the system needs to be increased (as m and g are both fixed). This energy need to come from the chemical energy stored within the flyer (from his food). In order to add this energy into the swing, he needs to do work during the swing (in this context work is technical term which means moving against a force):
Work Done = force x distance moved
Let's now look at how the flyer can do work on the swing by shortening the pendulum on the way up and thereby gain height.
'Short on the way up'From the flyer's perspective, he experiences both a centrifugal force pulling his body away from the pivot and a gravitational force pulling his body downwards. Any work that he can do against these two forces will add energy to the swing. If he wants to move his centre of mass up towards the pivot he must pull up in the opposite direction to the centrifugal force and against a component of the gravitational force (in other words he has to do work = expend some energy). Initially, we will imagine the flyer simply bending his arms and pulling himself up towards the bar.
We will look at each force separately, but of course the flyer only feels one combined force.
Centrifugal ForceMoving his centre of mass towards the pivot shortens the pendulum and reduces the moment of inertia of the pendulum (I). We know that in the absence of external forces angular momentum is conserved (if we look over a short enough time on the swing this applies here because the force the flyer exerts is along the cables and perpendicular to the direction of movement).
Angular Momentum (L) = I x ω
Therefore if I decreases, the angular velocity (ω) must increase to keep the angular momentum the same. So the flyer accelerates and so swings faster.
Since the kinetic energy of the system is given by:
KE = I x ω² = L x ω
Therefore if omega increases with constant angular momementum, then the kinetic energy increases. The flyer has used the chemical energy inside his body and added some of it into the swing. As we saw above, this means the swing will be higher once all this kinetic energy has been converted back to potential energy at the end of the swing.
The more energy the flyer can add into the swing, the more height he will gain. As we can see in the equation above, the kinetic energy depends on the moment of inertia and the angular velocity. Since the angular momentum is constant if we look over a short enough time, we can therefore say that the amount of extra energy the flyer can add depends on the amount by which he can change the angular velocity (which we know he can do by pulling himself in towards the pivot and reducing his moment of inertia). So, to increase the height by a lot, he needs to shorten his pendulum as much as he can, and do this at a time when he is moving the fastest (ie at the bottom of the swing).
Gravitational ForceAs the flyer pulls up towards the pivot, he is also lifting his body. However unlike the centrifugal force which pulls along the line of the cables, gravity always acts downwards.
The force due to gravity is given by:
Force = m x g
However, only a component of this force acts along the line of the cables:
Force (parallel to cables) = m x g x cosθ
The potential energy the flyer gains by shortening by a distance d is given by:
PE = m x g x cosθ x d
(force x distance moved)
Pulling against gravity lifts the flyer and adds to his potential energy. As we showed above, this means he will be higher at the end of the swing.
We also note that cosθ is greatest when θ=0, ie. at the bottom of the swing. So the best time to pull against gravity to gain the most energy is at the bottom.
Putting this into practice
We've seen that both forces are greatest at the bottom of the swing, so this is the best time to shorten the pendulum and add the most energy. However, pulling up to the bar at the bottom is very hard, and so the flyer makes a compromise and swings his body up to the bar, shortening his pendulum, as he is approximately halfway up to the peak. This makes up the first part of the force-out.
This explains why 'short on the way up' is a good way to increase swing height, and how the flyer achieves it by doing work against the centripetal and gravitational forces to add kinetic energy into the system (which is converted to potential energy at the ends of the swing).
'Long on the way down'Once the flyer has shortened his pendulum on the first swing, one strategy would be to continue to shorten it on the second and third swings etc. However, there is obviously a limit to the extent to which the swing can be shortened, given that the flyer needs to remain hanging under the bar in order to perform his trick. What he really needs to do is find a way to lengthen his swing again (without undoing the work he has just done to shorten it), so that he can the shorten it again on the next swing to gain yet more height. This is exactly what he does.
At the peak of the swing, the angular velocity of the flyer (ω) is zero. Therefore he has zero angular momentum, and so zero kinetic energy. All the energy in the system is stored as potential energy.
KE = I x ω²
So we can conclude that while ω is zero, he can change his moment of inertia as much as he likes without changing his kinetic energy. In other words he is free to lengthen his body without losing any kinetic energy.
What about potential energy? Potential energy is given by:
PE = m x g x h
Therefore provided the flyer keeps his body at the same height above the ground, he will not lose any potential energy. At the peak of the swing, the fly cables are rarely horizontal, therefore if the flyer just straighens his arms to lengthen the swing he will inevitably move downwards (reducing h) and decrease his potential energy.
The flyer needs to lengthen his swing away from the pivot in a horizontal direction, hence the force-out.
The flyer then swings down from the peak with a long swing, and gets ready to shorten as he swings up to the next peak.
As an aside, even if the flyer doesn't force-out horizontally and just straightens his arms, it is still possible to keep some of the extra energy. The energy that the flyer loses by lengthening his pendulum by a distance d is given by:
PE = m x g x cosθ2 x dHowever provided that θ2 is bigger than our original θ (ie. he lengthens at a higher point than he shortened) then cosθ2 < cosθ and the PE lost will be less than the PE gained.
This explains the front part of the swing and how a flyer can use the principle of 'long on the way down, short on the way up' to gain height. Now let's show how this applies to the back part of the swing.
The BackendAs the flyer swings back towards the platform, he rises up into a '7' position. Not only does this have the benefit of stopping him from hitting the platform, but it also brings his hips up towards the bar... shortening his swing on the way up (as we would expect).
At the peak above the platform, the flyer can't lengthen or he would hit the platform on the way down. He therefore lengthens as soon as he can after the platform in order to get as much of the benefit as he can.
Putting everything together, we can see the application of the 'long on the way down, short on the way up principle' applied to both the forward and backward swings.
If we imagine just following the flyer's centre of mass (and therefore the length of the pendulum) we can see how he shortens and lengthens the pendulum during the swing:
Beyond the ball and string modelSo far we have simplified the complex reality of a flying trapeze swing using two simple models: 'ball on a fixed string' and 'ball on a variable length string'. These have helped us to understand the basics of swinging. However, once you have got to grips with the explanation above, you will be able to spot that it still doesn't explain certain parts of the swing.
The next level of complexity we could consider would be to model the flyer as a 'double pendulum':
Analysis of this type of system allows us to explain why, for example, the flyer beats back hard under the platform, and why this accelerates his legs forward and upwards into the force out at the frontend.
However, this is a more complex system and well beyond the level of this article (but is still only an approximation of a swing). If you are interested in further reading, then you might wish to start at the Wikipedia page which, at the time of writing, had this interesting animation of a simple double pendulum in action: (reload the page to see it start from the beginning).
While it isn't following exactly the same movements as a flyer on a trapeze, it is easy to see that this is at least closer to reality than our 'ball on a string' model.