1
$\begingroup$

Let's say point $P$ is the center of mass of an irregularly shaped object.

  1. If I make a straight cut trough point $P$ and split the object in two, is it possible for the two pieces to have the same mass? This would be possible if the centers of mass of each piece was the same distance from $P$, but is this possible with an irregularly shaped object of uniform density?

  2. What about non-uniform density?

This is just a question I thought about.

Improve this question
$\endgroup$
4
  • $\begingroup$ Of course it's possible in some cases. I'm not sure if you were asking if it's always possible or not but I'm pretty sure if the object is convex, it's always possible. The center of mass is a point so a cutting plane can pivot in the $\theta$ and $\phi$ directions through that point. This should provide enough freedom to find a cut that balances the masses. $\endgroup$
    – Brandon Enright
    Commented Jan 9, 2014 at 4:26
  • $\begingroup$ @BrandonEnright It does really seem possible since the restrictions are so loose, but I would like an example or a bit of a proof since I know how wrong intuition can be sometimes. $\endgroup$
    – Ovi
    Commented Jan 9, 2014 at 4:29
  • $\begingroup$ As Brandon points out, it's unclear precisely what you're asking. More specifically, are you asking whether it's true that given any system of particles in three dimensions, there exists a plane passing through the center of mass such that the mass on one side equals the mass on the other? Or perhaps you are asking if some fact similar to this is true under suitable assumptions (like convexity as put forth by Brandon)? I think that with suitable clarification, this could be a very interesting question. $\endgroup$
    – joshphysics
    Commented Jan 9, 2014 at 4:31
  • $\begingroup$ I've thought of a proof but it's too long to type on my phone. I will post it in a few hours. Proof requires finite size, mass, and density. $\endgroup$
    – Brandon Enright
    Commented Jan 9, 2014 at 4:40

2 Answers 2

Reset to default
5
$\begingroup$

You can do even better. The ham sandwich theorem says you can simultaneously bisect any three volumes in space with a single plane. You can have a plane through the center of mass that bisects the mass and the surface area. Make one volume a tiny sphere around the center of mass, one a very thin outer skin, and the other the rest of the object. The proof looks to me like it applies to non-uniform density as well.

Improve this answer
$\endgroup$
6
  • $\begingroup$ The proof that comes to my mind does require that the distributions are sufficiently well behaved that the fraction of [whatever] on each side of the plane be a continuous function of the orientation (trivially true of physical distributions such as mass). Aside from that it should be completely general. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Jan 9, 2014 at 4:50
  • $\begingroup$ @dmckee: I think to ask the question the distribution has to be measurable, otherwise you can't define what half the mass is. If so, the fraction will be a continuous function of orientation. $\endgroup$
    – Ross Millikan
    Commented Jan 9, 2014 at 4:57
  • $\begingroup$ Nice use of the ham-sandwich theorem. It should be pointed out, however, that in case of a linear (1D) mass distribution a bisection of mass in general requires a cut that avoids the center-of-mass. $\endgroup$
    – Johannes
    Commented Jan 9, 2014 at 5:10
  • $\begingroup$ @Johannes: yes, in 1D you can only divide one volume, not two. In 3D the theorem avoids that by placing the plane through the line and declaring there is the same amount (zero) on each side of the plane. $\endgroup$
    – Ross Millikan
    Commented Jan 9, 2014 at 5:26
  • $\begingroup$ you can simulate non-uniform density by using a porous material with arbitrarily small pores. All you need is the maximum mass density to be finite. Zero volumes of infinitely dense material can be spread out to arbitrarily small volumes, and non-zero volumes of infinitely dense material are a mess both physically and mathematically. $\endgroup$
    – John Dvorak
    Commented Jan 9, 2014 at 10:33
0
$\begingroup$

Yes, this is always possible for any object of any shape. I will explain why this is the case:

Take any object and make any cut through the center of mass. The object is now divided in two parts, A and B.

Let's say A has a higher mass than B.

We can now start rotating the plane of the cut until it has made a half rotation. After this half rotation the masses of A and B have turned around, so now mass B is greater than mass A.

During this turning process the mass of A has been decreasing and the mass of B has been increasing. And since both are continuous at some point their masses must have been equal.

Even with a non-uniform density, the lines remain continuous and so they must still intersect at some point.

Thus, for any point and any object with any denisity, there is some cut through that point that divides the object in two pieces of equal mass.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.