Check out this giant and world's highest (it says so right there on the side) swing.
Nice video. I like the people's reactions. It looks like they were weightless for quite some time. You know what comes next, right? Here comes the physics.
First some details. The video says "Nevis Swing" in New Zealand. This site lists the swing length as 120 meters (actually, it says it is 120 metres). Really, that is the only detail I need. Well, except for the starting angle. It looks like it starts pretty close to horizontal - so I will just use that.
Modeling the swing
Now for the model. It turns out that I have looked at pendulums before. Are you surprised? And you might think that pendulums are very remedial. Well, they aren't too simple. Especially when the amplitude angle is large (like in this case).
For this example, I am going to model the motion of these swingers by using a spring instead of a string. If the spring-string has a fairly high spring constant, it will behave essentially the same way as a string (all strings are springs anyway). This way, I can use the following numerical recipe:
- Calculate the forces on the swinger. (gravity and spring force from the rope)
- Use the forces to find the new momentum of the swinger.
- Find the new position of the swinger.
- Note that the new position will change the spring force above.
- Update time.
- Repeat the above stuff until I tell the computer to stop.
Ok, let's get to it. First let me check my program to see if it is working ok. Here is the trajectory (x vs. y) for a swing starting horizontal.
Looks circular to me - so that is good. Here is a plot of the vertical position as a function of time.
Some things to note:
- The swinger returns to the same starting height. This is good since I didn't have any friction or anything in the model.
- This actually isn't a trig function. Remember you only get simple harmonic motion (with trig function solutions) for small amplitude angles.
- Looks like there is an oscillation period of about 12 seconds. Seems a little shorter than I would have guessed, but maybe I can get an estimate from the video to check this.
Luckily, the video includes this shot:
The camera isn't quite perpendicular to the motion of the swinger - but I can at least get time data from this. Oh, also I am not too sure about the scale. Here is a plot of the vertical position as a function of time using Tracker Video.
In this example, the swinger doesn't quite make it all the way to the bottom in about 4.5 seconds. This would put the period around 20 seconds (the time to go there and back). Perhaps the difference in times between the video and my calculation can be attributed to air resistance (which I will add later). Anyway, I think it is close enough to proceed.
Adding Air Resistance
It seems like I need to go ahead and consider the effects of air on these people. Here I will use the typical model for the magnitude of the air resistance force:
Where:
- ρ is the density of air (about 1.2 kg/m3)
- A is the cross-sectional area of the object.
- C is the drag coefficient (depends on the shape of the object).
For a falling person, what is A? What is C? I can get an estimate for these by looking at a skydiver. I know that if you jump out of a plane, you will fall with a constant speed of about 120 mph (54 m/s). At this speed, the air resistance and the weight have the same magnitudes (terminal velocity). I can write:
I don't really need to know A and C separately, I just need their product. Using a mass of 70 kg, I get AC = 0.39 m2. Now, this value is for a sky diver in the free fall position. I suspect that these swingers would have a lower value since they typically aren't holding their arms and legs out. Also, they are sometimes moving head first or feet first instead of belly first. Of course, there is also drag due to the rope and stuff (but I will ignore that for now). How about I use a value of AC at about 0.3 m2.
Now I can add this into my model. Here is a force diagram of a swinger at some point during the motion:
Really, the only thing I have to change in my calculation is to add the air resistance force. Simple, right? Here is my plot of vertical position with air resistance added.
And here you can see that I am an idiot. I was thinking the time was off from the data from the video. Adding air resistance is going to increase the time it takes to get to the bottom of the swing, not decrease it. Oh well. Perhaps there are other things going on - like not starting in a horizontal position. Actually, from this shot it looks like the people start about 10 degrees above the horizontal.
Apparent Weight
It looks like the swingers experience something like 'weightlessness' at the beginning of the fall. How long would this last (with my obviously incorrect model)? First, what is apparent weight? Here is a nice summary post about astronauts in space. In short, you could say that the apparent weight is the sum of all the non-gravitational forces. For this case, the apparent weight will be the sum of the air resistance force and the tension in the rope. You don't 'feel' the gravitational force because it pulls on all parts of your body.
Here is a plot of the magnitude of the apparent weight for the first few seconds.
From this, how long are the swingers in free fall? The graph shows that the apparent weight is still under 0.5 g's up to about 2 seconds. So I am going with free-fall-ish feeling for around 2 seconds. 1-g happens around 3 seconds. At the lowest point, the apparent weight gets up to about 2.25 g's.
The video says 70 meters (metres) of free fall. Is this true? Well, I it depends on what they mean by "free fall". Technically, free fall would only be the case where the apparent weight is 0 g's. That only happens for an instant when the swing starts. Ok, what if free fall is "less than 0.5 g's"? I already said this was around time of 2 seconds. So, how far did the swinger move in 2 seconds? The y-position dropped by 20 meters in this time. From this I can find the distance traveled (along the arc) using a diagram like this:
Using some trig, I can get the angle θ as 0.165 radians. This would make an arc-length of about 20 meters. So, not 70 meters. I wonder where that 70 meter value comes from? Oh well, I would still like to try this swing if it wasn't on the other side of the world.
