The math behind Gravity’s Rainbow (Part 5): Going all around on a swing

My grandmother always encouraged me to try going all around on a swing. At the time I thought this was pretty reckless of her and I was much too scared to try it anyway. But she probably had read this section of Gravity’s Rainbow and knew I would be fine:

Diverging oscillations of any kind were nearly the Worst Threat. You could not pump the swings of these playgrounds higher than a certain angle from the vertical. Fights broke up quickly, with a smoothness that had not been long in coming. Rainy days never had much lightning or thunder to them, only a haughty glass grayness collecting in the lower parts, a monochrome overlook of valleys crammed with mossy deadfalls jabbing roots at heaven not entirely in malign playfulness (as some white surprise for the elitists up there paying no mind, no . . .), valleys thick with autumn, and in the rain a withering, spinsterish brown behind the gold of it . . . very selectively blighted rainfall teasing you across the lots and into the back streets, which grow ever more mysterious and badly paved and more deeply platted, lot giving way to crooked lot seven times and often more, around angles of hedge, across freaks of the optical daytime until we have passed, fevered, silent, out of the region of streets itself and into the countryside, into the quilted dark fields and the wood, the beginning of the true forest, where a bit of the ordeal ahead starts to show, and our hearts to feel afraid . . . but just as no swing could ever be thrust above a certain height, so, beyond a certain radius, the forest could be penetrated no further. A limit was always there to be brought to.

Pynchon, Thomas. Gravity’s Rainbow (Classic, 20th-Century, Penguin) (S.238-239). Penguin Publishing Group. Kindle-Version.

But! It actually is possible to go around all the way with the help of rockets! See this video. Yes! There is not only the evil rocket for the world’s suicide but also a good rocket that will take us around the swing.

In the video, they show some blackboard calculations arriving at a required energy of 4000 Joule. But will this really be enough? Let’s double-check, so we can feel confident when we try this at home.

swing_calc

There are three things we need to consider:

1.Getting to the apex

How much energy is needed to get to the top depends on your (and your jetpack’s) weight. If we assume a total weight of 75kg and a swing radius of 2 meters, we can calculate the potential energy at the top:

m*g*h = 75kg * 9.81m/s^2 * 2*2m = 2943 kg*m^2/s^2 = 2943 Joule

(m=mass, g=gravitational force on earth, h = height)

2. Not failing straight down once arriving at the top

It would hurt very badly if you just got to the highest possible point and then crashed on the bar of the swing because you’ve lost all of your speed. So let’s add some extra push to avoid things going awry. We need to make sure that the centripetal force acting at the apex is bigger than the gravitational force:

centripetal force >= gravitational force

=> m* v^2 / r >= m*g

=> v >= sqrt(g*r)

=> v_min = sqrt(g*r)

=> v_min = sqrt(9.81m/s^2 * 2m) = 4.43 m/s = 15.95 km/h

So you need to go at least ~16km/h at the very top for the swing’s chain to stay straight.

To accelerate 75kg to 4.43 m/s you need a kinetic energy of:

0.5 * m * v^2 = 0.5 * 75kg * (4.43m/s)^2 = 735.75 Joule

3. Accounting for air resistance

Air resistance is often neglected in physics thought experiments because it depends on many factors that are often hard to estimate. But in this case, our life might depend on getting this right, so let’s try accounting for it. The drag force can be calculated this way:

To see what numbers need to be plugged into this formula, let’s go through the variables one step at  a time:

 is the density of the fluid. The air density depends on temperature, humidity and atmospheric pressure. We will assume a temperature of 20°C, dry air and the average atmospheric pressure on earth, which results in a density of 1.2041 kg/m^3
is the speed of the object relative to the fluid. We could calculate the speed at every point in time, but that would be a lot of work, so I’ll just assume 5 m/s as top speed.
is the frontal area of your body. We assume this to be 0.5 m^2
 is the drag coefficient. Lower values mean less friction. I think the drag coefficients of a human on a swing might be similar to that of a ski jumper, which is about 1.2 according to this website.

Now we can put it all together:

F_{D}\,=\,{\tfrac {1}{2}}\,\rho \,v^{2}\,C_{D}\,A = 1.2 kg/m^3 * (5m/s)^2 * 0.5m^2 * 1.2 = 18 Newton

So, there is a force of 18 Newtons slowing us down. How much energy will this cost us? It depends on how far we will travel. We only need to calculate how many meters we travel until we arrive at the top. For the way down air-resistance is mainly irrelevant because we are far away from terminal velocity, so gravity will ensure we get down without slowing down. Half of the swing’s circumference is 2*pi *2m / 2 = 6.28m. So the extra energy we need to overcome air resistance is smaller than 18N  * 6.28m = 113.4 Joule.

Result

For our example with 75kg payload and a swing radius of 2 meter, we need a total energy of 2943 + 735.75 + 113.4 = 3792 Joule

This is pretty similar to the result shown in the video. So I’m pretty confident that it will actually be enough. But still: Proceed at your own risk…

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