I thought it was about time I gave you (yes you!) a physics riddle to go with your morning coffee. I’ll state the problem first and then give a few hints below the fold. Readers with a familiarity with ideas in physics can probably solve this without any hints. If you are unfamiliar with physics, don’t be ashamed to check out the hints. Most importantly: this riddle can be solved without any equations. Feel free to post your solutions in the comments. I’ll give the solution as a comment later on.
Okay. Here it is:
Not to scale...
You have two objects:
- A rope of length a given length.
- Two pieces of (let’s say…) wood joined by a bolt. Together these pieces of wood stretch out to be the same length as the rope.
These objects are the same length, same mass and the same mass per unit length. You now hang them by their endpoints so that they hang side by side (as shown in the picture). The horizontal distance between the endpoints on which they hang is the same for the wood pieces as it is for the rope.
Which object has the lower center of mass?
Note: If you are unfamiliar with the concept of center of mass, check out Rhett’s post on DotPhysics here. You don’t need to understand it all for the purposes of this problem, just know what center of mass is.
Now, some hints:
1) Think about what a lower center of mass means in terms of the potential energy. If an object has a lower center of mass, then it has less potential energy.
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2) Objects tend to like to lower their energy any way they can. If an object has a certain amount of energy and is able to shift its configuration in a way that immediately lowers its energy, then it will do so.
Good luck!
Interesting puzzle. The wood blocks are basically a chain of length L with 2 links. That makes me immediately try to generalize it to a chain with n links. As n-> infinity, we get the rope. The center of mass of the chain must approach that of the rope as n gets larger, then. Intuitively, I feel that the center of mass of the chain must approach the center of mass of the rope from one side or the other, rather than bouncing around the final value.
That is, (without giving away the answer), a chain with a finite number of links always has a {lower} {higher} center of mass than a rope of the same length. But which one is it? Considering a convenient value of n might help.
Which of the two arrangements will have a lower centre of mass will depend on how the particles are distributed in each arrangement. In the left hand side arrangement, a lot will depend on whether the discreet mass units are of same mass and how they are placed (a lot of arrangements are possible that will yield the same pass per length value). In the case of a rope, it is assumed that it is a ‘continuous’ system and so the individual particles will have to be of same mass.
I do not think it’s possible to give an answer without further assumptions. If we assumed that all discreet elements of the wooden chain are of same mass, the centre of mass for both arrangements will be at same ‘level’ vertically, would be my guess.
BTW, I came across your blog only twp days ago but I have been enjoying reading it.
@Vijay
The mass per unit length (discreet mass elements) of both objects are assumed to be the same (as stated in the problem)
… but I’m not saying your guess is correct
@excitedstate
You seem to have a good line of reasoning going there. (thanks for not giving away the solution if you have figured it out).
Anyone else have any ideas?
I think, because the rope is a continuous curve and the two blocks change abruptly, the rope will have more mass below any given level than the wood, giving it a lower centre of mass
In order for the rope to have its center of mass at the same level as the wood, it would have to hang in the same sharp “V” shape as the wood does. If we pull the rope to this shape and then let it go, it falls into a curve called a catenary. The only reason for it to do this is to reduce its potential energy. Therefore, the mass center of the rope must be lower than the wood.
What CTReader said. Which is pretty much what the hints were hinting at.
Except in the two limiting cases where the separation between the “hanging points” is either zero or equal to the length of the rope, in which the center of masses will be equal.
SOLUTION:
I think you all have had enough chance to ponder this question. And you’ve done well!
CTReader had it spot on with a great visualization. And Anonymous Coward piped in with some good details of the limiting cases.
The way I would phrase the solution is similar to excitedstate’s line of reasoning. The blocks of wood have one “degree of freedom”, attributed to the hinge in the middle. That is, they can change their configuration in one particular way. If you imagine the string as simply a huge number of very small wood blocks you see that it has very many degrees of freedom. This gives the rope more ways to reduce its potential energy (relative to the wood) by changing its configuration. If it lowers its potential energy, then that is equivalent to lowering its center of mass. In short, the rope’s center of mass is lower.
This raises another interesting question. AC’s comment implies that the difference in height of mass center (D) between the wood and the rope is some function of the distance between the hanging points (X):
D=f(X)
The function f is obviously continuous and, since it varies from 0 at X=0 and comes back to 0 at X=2L, it must have a maxima somewhere in this range. Can we calculate where this maxima occurs?
CTReader:
Egads.
Using brute force (doing the integrals) to solve a slightly different problem – for a given X, what length gives the largest difference in center of mass – numerically yields an answer L ~ e^(1/2) X. But not quite L = X * exp(1/2). It’s off by a percent or so.
Solving the problem you posed – for a given L what X gives the largest difference in center of mass may yield a more elegant answer, but I think that solving that turkey will involve solving a transcendental equation along the way to working some integrals.
An interesting puzzle with a beautiful solution! Unfortunately, I think your statement of the problem is super-easy to misinterpret, and I think the accompanying diagram is downright misleading!
In my opinion, the “pieces of wood” should be changed to “straight, thin, wooden rods,” and the diagram should be changed to reflect that description. As it stands, the ambiguity of the text, and the oblong shapes of the pieces of wood in the diagram, work together to suggest that the pieces of wood can be any shape you want. The puzzle then clearly has no solution, because you can design the pieces of wood to put their center of mass pretty much anywhere.
Also, why did you draw the pieces of wood so that they look like they’re connected to the rope? Sure, if you read the text carefully and think about the diagram for a while, you can figure out what’s going on… but why force the reader to go through all that?
I know that no physics problem can be stated in a totally unambiguous way, but I think a few simple clarifications would make this problem a lot more fun for future puzzlers!