Timeline for Making a cut trough a center of mass, can the masses of the pieces be equal?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 9, 2014 at 10:42 | comment | added | John Dvorak | This also shows that you can bisect an object with a vertical cut through its center of mass - just let the third object be a single point on the zenith. It also shows you can bisect an object with a cut of an arbitrary direction - but that's not an interesting result. | |
| Jan 9, 2014 at 10:33 | comment | added | John Dvorak | 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. | |
| Jan 9, 2014 at 5:26 | comment | added | Ross Millikan | @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. | |
| Jan 9, 2014 at 5:10 | comment | added | Johannes | 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. | |
| Jan 9, 2014 at 4:57 | comment | added | Ross Millikan | @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. | |
| Jan 9, 2014 at 4:50 | comment | added | dmckee --- ex-moderator kitten | 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. | |
| Jan 9, 2014 at 4:42 | history | answered | Ross Millikan | CC BY-SA 3.0 |