centroid,centre of mass and centre of gravity

monty37

may i know what is the difference between centroid,centre of gravity and centre of mass
of a body?is it necessary that all the points coincide?

Redbelly98

Those all mean the same thing.

davieddy

Take a triangle for example.

The centroid is the mean position of elements of area.
The centre of mass is the mean position of elements of mass.
In a uniform gravitaional field, the force of gravity on the triangle
can be taken as acting through the centre of mass.

But if the mass per unit area of the triangle varied, the centre of mass
would not (usually) coincide with the centroid.

Note also that the assumption that the earth's gravity is the same
as if its mass were all located at the centre is only true because of
its spherical symmetry.

HallsofIvy

"centroid" is a purely mathematical, geometric concept- the geometric center of a body. It can be calculated as the center of mass of an object with constant density. "Center of mass" and "center of gravity" are the same- although the concepts are slightly different: the concept "center of gravity" strictly speaking requires, as davieddy said, a uniform gravitational field while "center of mass" does not. But they are exactly the same point on an object.

D H

An alternative definition of the center of gravity of an object is the point at which the gravitational force on a point mass located at the point in question is equal to the total gravitational force acting on the object. The concept of center of gravity in a uniform gravitational field doesn't make sense with this alternate definition. It only makes sense in a non-uniform gravitational field.

With a gravity field that falls off linearly with distance, center of mass and center of gravity are the same. However, with a gravity field that falls off with the inverse square of distance, center of gravity and center of mass are not the same. For a small object, say something the size of the space station, the difference between center of mass and center of gravity is very small because the first order linear approximation is very, very good.

For a large object, say something the size of a space elevator, center of mass and center of gravity are very different. Lay literature on a space elevator often claim the center of mass needs to be at geostationary altitude. This is incorrect. It is the center of gravity that needs to be at geostationary altitude. The center of mass needs to be well above geostationary altitude.

monty37

iam clear about the three,but can you please explain the "geostationary altitude "?
for the earth as such ,a sphere,all the three points coincide ,right?