Question

Do all objects have a center of mass. If yes why? (A mathematical proof is appreciated) If no, which objects don't?

A) Yes they do. Suppose you have two objects. This is a little like asking if there must be a point halfway between them.

B) Every object must have a point where it balances.

C) In one dimension, imagine a ruler. If you pick a point and the ruler falls over to the right, choose a new point a little farther right. It will tilt less strongly. If you go too far, it will tilt to the left. In between there must be a point where it doesn't tilt either way.

D) You can cut a shape out of cardboard and do the same thing to find a point where it balances. Likewise for a 3 D object.

Answer 1

All objects have a center of mass, which is the average location of the total mass of the object, and it can be calculated mathematically even for objects without physical mass at that point, such as hollow spheres or doughnuts.

Step-by-step explanation:

Yes, all objects have a center of mass. The center of mass can be thought of as the average location of the total mass of an object. For any extended object, whether it is a simple geometric shape or a complex system, we can calculate the center of mass by using a mathematical equation that involves summing over all tiny pieces (infinitesimal elements) of the object and their distances from a reference point, weighted by their respective masses.

Let's consider a two-particle system as an example. The center of mass (CM) can be found by taking the weighted average of their position vectors, with the weights being their masses. For two point masses m1 and m2 at positions r1 and r2, the CM position vector (R_CM) is given by:

(m1*r1 + m2*r2) / (m1 + m2)

This concept holds true for any number of particles and even continuous mass distributions, as each infinitesimal mass element dm at position r contributes to the CM. Even if a physical mass isn't present at the center of mass (e.g., a hollow sphere or a doughnut), the concept still applies, and the center of mass represents the point where the object can be balanced and around which it rotates when subjected to external forces.