When I was a kid we made a lot of “forts”. I mean a lot of forts. Forts all – the – time.
Of course as a kid a fort was as simple as a piece of wood leaned against a fence, a pile of broken up concrete from a sidewalk, an “igloo” of tumble weeds nestled over a dirt depression or even just a camper shell on the front lawn (yes, all real examples from my childhood).
As I’ve grown older I still love forts but they have become much more extravagant and complicated. Along with this “natural” evolution of fort building comes the need for actual engineering. I can stack tumble weeds all day with very little safety concern but when I’m building a 4,000+ cubic foot fort under my back lawn or a 120 square foot treeless treehouse I need to start doing some real life, grown-up math.
One such “adult fort” that I built with my brother has gained a bit of popularity: The Underwater Bubble Room. With more than 1.5 million views on YouTube you can imagine I get a LOT of comments. Not surprisingly most of the comments reveal the fact that the world (at least the part of the world that comments on this YouTube video) is largely comprised of three main groups:
- Potheads
- People who know everything
- Idiots
I realize that this post will probably bore everyone in group 1, offend everyone in group 2, and confuse everyone in group 3. Therefor I’ve written this post primarily for the underrated “group 4” which is comprised of normal, intelligent people like you and me.
I want to talk about some of the forces involved specifically in The Underwater Bubble Room.
How strong do the tie-down cables need to be for the Underwater Bubble Room?
I’m glad you asked! It’s actually pretty easy to figure out with just a little help from my buddy Archimedes. Archimedes was a Greek dude who lived about 200 years before Jesus. He was a smart fellow and among other things he wrote two books collectively called On Floating Bodies in which he succinctly explains buoyancy:
“Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.”
- Archimedes
In other words the buoyant force (upward “lifting” force) of the Bubble Room is equal to the weight of the water displaced by the room. Since the Bubble Room is basically half a sphere the simplest way to figure out the buoyant force is to calculate the volume for a sphere (in cubic feet) then divide by 2 and multiply by the weight of 1 cubic foot of water.
The math
Volume of a sphere:
V=(4/3) x π x r³
Since our room is about 7′ in diameter that means r = 3.5
If we plug in our numbers we get:
V=(4/3) x π x 3.5³ = 179.59 ft³
But that is the volume of a sphere, since our room is a hemisphere (half a sphere) we need to divide that by 2:
179.59 ft³/ 2 = 89.8 ft³
1 cubic foot of water weighs 62.43 lbs, so we multiply that by our volume (displacement) to get pounds of lift:
89.8 x 62.43 = 5,606.21 lbs (of lift!)
What about the cables?
That’s a lot of lift, but we are using three cables to hold the room to the lake bed, it would be easy to think that each cable only shares a third of the lifting force right? If our tethering cables were perfectly vertical that would be the case but they are not so now we need to calculate the force on each cable based on their angle to the buoyant force to make sure they are strong enough.
To help envision why the force on a cable increases as the angle of the cable increases imagine tying a rope to a bucket of water then lifting the bucket to the height of a door knob. The force (tension) on the rope and the force your muscles need to exert to lift the bucket are each equal to the weight of the bucket.
Adding angles
Now imagine if you pass the same rope through the bucket handle and tie it to the door knob. Stand back holding the other end of the rope with the bucket between you and the doorknob. Keeping your hand at the level of the door knob you could pull the rope and lift the bucket but it would take quite a bit of work to raise the bucket even a foot. Try raising the bucket so the handle is at the same height as the doorknob – you can’t do it!! In fact, the force it takes to raise the bucket to that height spikes to infinity. It literally can not be done in our physical world.
More math!
The equation to find the force on two cables under load at a given angle is:
F = (m x 0.5) ÷ cos(a x 0.5)
where:
F is the calculated force on each cable
m is the mass or in our case the lift load
a is the internal angle between the two cables
Our cables have an approximately 110˚ internal angle. So we plug in our numbers and we get:
The dividend:
(m x 0.5) = (5,606 x .5) = 2,803
The divisor:
cos(a x 0.5) = cos(110 x .5) = cos(55) = .573576
Divide for quotient:
2,803 ÷ .573576 = 4,886.88 lbs
(hint: at 120˚ the force shared on two cables is exactly double the mass of the load.)
This math is for two cables, but we have three so we simply add the two together then divide by 3.
4,886.88 + 4,886.88 = 9,773.76 lbs
9,773.76 ÷ 3 = 3,257.92 lbs
Conclusion
Each cable will have 3,257.92 pounds of force! That’s more than 1.5 tons EACH!
High five Archimedes, thanks for being awesome.
Interestingly if we increase our angle to 120˚ the math comes out to 3,737 pounds on each cable (11,212 total). That 10˚ increase added 1,438 pounds of force to our system! It’s amazing to think that 5,606 pounds of lift can have 11,212 pounds of force!
