My coworkers and I go to a cafeteria for lunch every day, and along the way we pass under this large length of hose draped over the road. Every time, it reminds me of the classic physics problem: What shape does a massive (as opposed to massless) rope make under the force of gravity?
You might imagine it would be a parabola, since that's the shape of a projectile trajectory under a constant force. It turns it it's a similar, but slightly different shape called a
catenary. The equation is
where a is a constant that depends on the mass of the rope. Over a short distance, it's fairly close to a parabola, but diverges as it gets away from the minimum:
 |
| Catenary (red) and parabola (blue) |
If you look at the Wikipedia link up there, you can see a derivation based on the forces on the rope, but it gets a bit involved. Often in cases like this, it's easier to consider the energy of the system. Given enough time, any system left alone should eventually reach a minimum-energy state. In this case, the only potential energy involved is gravity, which has a simple form near the surface. For each little bit of mass
dm, we can write
If the mass is evenly distributed along the rope, we can write it in terms of the density,
λ:
Here's where it gets complicated though. Now we need to express the length of a little bit of the rope in terms of
x and
y,
Putting all that together gives the total energy,
Unfortunately, I can't find any way to solve this, let alone minimize it. It seems this is a case where it's best to stick with forces.
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