Understanding the concept of Centriod and Centre of Mass.

Nov 2019
Hello all

I was hoping someone could help remove some doubt I have in my mind regarding the concept of Centriod and Centre of Mass.

I have been having great difficultly in understanding these concepts.

Would you agree with the following:-

I define Centroid as the centre of the shape or geometric centre. For me what that means is that if the shape theoretically weighed nothing i.e. had no density/mass then the shape would still have a centroid and that centroid would be the shapes centre.

I would define Centre of Mass as, "the point at which the distribution of mass is balanced in all directions (and this property is independent of any external gravitational field).

To hammer these points home I have drawn a picture of what I believe the centre of mass is:-


Here I have a red shape with the black point representing the centre of mass, this is the centre of mass because if I drew a line from one edge of the shape, passing through the black dot to the opposite edge of the shape, then the sum of the mass along the line from one edge of the shape to the black dot will be equal to the sum of the mass along the other end of the line from the block dot to the end of the shape.

I have tried to represent this with colored arrows where Mass M3 from the dot to the left side of the shape is equal to the opposite line from the black dot to the right edge of the shape.

Am I making sense?

Also am I correct in thinking that a Centroid of a shape is always fixed, so long as the geometry/dimensions of the shape remain unchanged then the shapes centroid will also be unchanged.

Finally would it be correct to say that the centre of mass and the centroid can be located at different points on the shape? it is only when the shape is made from a uniformly distributed material with uniform density then we can say that the shapes centroid and centre of mass is the same - would this be a correct statement?

Thank you all.
Jun 2016
Some of what you say is correct, but...
I would not put the centroid or the centre of mass of your red shape where indicated.
The centre of mass should coincide the the balance point of the shape,
I cant see your red shape balancing if I placed a point under the indicated black dot

The centroid is the point where if you measured the distance to the edge in all directions,
but counted measurements up from the centroid to the edge as positive, down as negative, right as positive and left as negative,
if you add up all the distances from the centroid using this rule, then the sum of all the distances from the centroid to the edge will be zero
(because the sum of the upward measurements will exactly match and cancel the sum of the downward downward measurements, as will the left and right measurements).

You are correct that in a uniform material the centre of mass and the centroid will coincide,
but that they may be in dissimilar places if the material is not uniform.
Then the centre of mass will be that point where measuring from that point to the edge in all directions
(using the positive and negative summing rule)
the amount of mass measured (rather than just the distance measured) cancels.