Defining the Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes

[tomtomtom1]

Homework Statement: Defining Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes
Homework Equations: Defining Centroid, Centre of Mass, Centre of Gravity for 2D/3Dshapes

Hello all;

I am trying to understand the terms:-

- Centroid for a 2D shape and 3D shape
- Centre of Mass for a 2D shape and 3D shape
- Centre of Gravity for a 2D shape and 3D shape

The reason why I am getting confused is that I keep getting slightly different explanations for example:-

- The centroid is the term for 2-dimensional shapes. The center of mass is the term for 3-dimensional shapes. BUT I am also told that Centre of Gravity is always at the same point as the centre of mass.

I was wondering if someone could explain what these terms mean for both 2D and 3D shapes.

Can someone help?

Thank you.

 

[mfig]

These terms are somewhat mixed in common parlance. One useful distinction is that the centroid is a geometric center, or average point, assuming equal mass distribution or density. The center of mass is the average point taking into account mass distribution. If the mass is distributed uniformly, the centroid and center of mass will coincide.

Picture a 5 cm long, right circular cylinder. Along the axis, the first 2.5 cm is aluminum and the last 2.5 cm is steel. The centroid is 2.5 cm from the end, right at the border between materials and on the axis. The center of mass, on the other hand, will be inside the steel and on the axis somewhere depending upon the relative densities.

The difference in 2 and 3 dimensions is merely the complication of adding another direction during the averaging process, which is an integration for continuous objects. Below is a link to computed 2D and 3D centroids of objects, for reference:

List of 2D and 3D centroids

 

[collinsmark]

Your given examples are pretty good: "centroid" only applies to planar (i.e., 2-dimensional) objects. The "center of mass" of a 3-dimensional object is the point at which the distribution of mass is balanced in all directions (and this property is independent of any external gravitational field).

"Center of gravity," on the other hand, depends on the external gravitational field. The "center of mass" and "center of gravity" of an object will be equivalent if the object is in a uniform gravitational field. For most objects on the surface of the Earth that you deal with on a day-to-day basis, this is the case. The acceleration due to gravity is usually assumed to be a uniform $9.81 \mathrm{\frac{m}{s^2}}$, and thus any object's center of gravity is at the same location as its center of mass. As you continue your physics courses, you will usually use "center of mass" and "center of gravity" interchangeably.

But things get more complicated when you start using Newton's universal gravitation, $\vec F = G\frac{m_1 m_2}{r^2} \hat a_r$. Because the acceleration due to gravity is not uniform, parts of one object that are closer to the other object will feel a stronger force than those parts of the object further away. This gives rise to what are called "tidal forces." It's what ultimately causes the same side of the Moon to be always facing Earth; it's tidally locked. And, if you're looking for an example, insofar as its gravitational relationship to the Earth, the Moon's center of gravity is not at the same location as the Moon's center of mass.

 

[Dr.D]

There is really no reason to limit "centroid" to planar (2D) spaces. It can be applied in 3D equally well. It is certainly true that common practice applies this to 2D, but that is not a necessary limitation.

 

[tomtomtom1]

There is really no reason to limit "centroid" to planar (2D) spaces. It can be applied in 3D equally well. It is certainly true that common practice applies this to 2D, but that is not a necessary limitation.

Dr D

Thank for your response.

In all honesty these concepts are still bothering me, if I may ask;

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) (thank you collinsmark ). I have drawn a picture of what centre of Mass means to me:-

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?

Thank you.

 

[tomtomtom1]

Your given examples are pretty good: "centroid" only applies to planar (i.e., 2-dimensional) objects. The "center of mass" of a 3-dimensional object is the point at which the distribution of mass is balanced in all directions (and this property is independent of any external gravitational field).

"Center of gravity," on the other hand, depends on the external gravitational field. The "center of mass" and "center of gravity" of an object will be equivalent if the object is in a uniform gravitational field. For most objects on the surface of the Earth that you deal with on a day-to-day basis, this is the case. The acceleration due to gravity is usually assumed to be a uniform $9.81 \mathrm{\frac{m}{s^2}}$, and thus any object's center of gravity is at the same location as its center of mass. As you continue your physics courses, you will usually use "center of mass" and "center of gravity" interchangeably.

But things get more complicated when you start using Newton's universal gravitation, $\vec F = G\frac{m_1 m_2}{r^2} \hat a_r$. Because the acceleration due to gravity is not uniform, parts of one object that are closer to the other object will feel a stronger force than those parts of the object further away. This gives rise to what are called "tidal forces." It's what ultimately causes the same side of the Moon to be always facing Earth; it's tidally locked. And, if you're looking for an example, insofar as its gravitational relationship to the Earth, the Moon's center of gravity is not at the same location as the Moon's center of mass.

Collins Mark

Thanks for the response.

I have drawn a picture of what I believe centre of mass is, I have posted this as a reply to Dr D - what are your thoughts on this?

 

[mfig]

Dr D

Thank for your response.

In all honesty these concepts are still bothering me, if I may ask;

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) (thank you collinsmark ). I have drawn a picture of what centre of Mass means to me:
...

Am I making sense?

Thank you.

These are what I said above in post #2. Centroid is independent of mass - it's a geometric average. Center of mass is an average taking into account mass distribution. Just like you said and showed...

 

[tomtomtom1]

Thanks mfig

My final question is this 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?

I ask because the centroid is based purely on the geometry and is independent of the mass/density of the material.

Would this be a correct statement?

Thank you.

 

[Dr.D]

My final question is this 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?

That looks OK to me. I would add that the centroid, being purely geometrical, is independent of any mass distribution. We commonly associate the centroid with the center of mass purely on the assumption that the mass is uniformly distributed.