# Center of mass and center of gravity don´t match, why?

• Catire
In summary, the center of mass and the center of gravity are not the same point for a tapered beam, with a difference of about 5%. This discrepancy is not due to a problem in computation, as multiple methods yield the same result. The conditions for both points to be equal are ## \sum_i m_ir_i = 0 ## and ## \sum_i gm_ir_i = 0 ##. It is possible that the confusion arises from using the term "center of mass" instead of the "half-way mass point", which is where the beam would balance on a fulcrum.
Catire
For a tapered beam of constant density the center of mass is calculated as the mean of the mass distribution along the central axis, the center of gravity is calculated by getting the equilibrium of the angular moments to the left and right of a fulcrum moving along the horizontal beam.
The resulting positions of those points don´t match. The difference is about 5 percent, hence not trivial.
Its not a problem of the computation, several different approaches give the same result (Integrals, sums, one or three spatial dimensions, high working precision).

I had assumed those points to match and in the literature I found no obvious reason why they should not.

Why this difference?

A Mathematica program of a sample calculation can be given.

There is no difference, the positions are equivalent. If you get different results it is a problem of computation.

The conditions are: ## \sum_i m_ir_i = 0 ## and ## \sum_i gm_ir_i = 0 ##.

As other posters have said, the center of mass and the center of gravity are the same point.

However, I suspect that when you said the center of mass, you really meant to say the half-way mass point. That is the point where, if you cut the beam, both sections would weigh exactly the same. But that is not the point where the beam would balance on a fulcrum because the mass of the two sections in a tapered beam is not distributed the same way.

billy_joule

## 1. What is the difference between center of mass and center of gravity?

Center of mass refers to the point at which the entire mass of an object can be considered to be concentrated. It is the point where all the mass of the object is evenly distributed. On the other hand, center of gravity refers to the point at which the entire weight of an object can be considered to act. It is the point where the force of gravity is acting on the object.

## 2. Why do the center of mass and center of gravity sometimes not match?

The center of mass and center of gravity do not always match because they are affected by different factors. The center of mass is affected by the distribution of mass within an object, while the center of gravity is affected by the distribution of weight and the force of gravity acting on the object. Therefore, if an object has an uneven distribution of mass or is subject to external forces, the center of mass and center of gravity will not match.

## 3. How do the center of mass and center of gravity affect the stability of an object?

The position of the center of mass and center of gravity can greatly impact the stability of an object. If the two points are aligned, the object will be in a stable equilibrium. However, if they do not align, the object may be in an unstable equilibrium and could potentially topple over. This is why it is important to consider the center of mass and center of gravity when designing structures or objects.

## 4. Can the center of mass and center of gravity change?

Yes, the center of mass and center of gravity can change based on the position and orientation of an object. If an object is tilted or rotated, the center of mass and center of gravity will also shift accordingly. In addition, if the distribution of mass within an object changes, the center of mass will also change. However, the center of gravity will remain constant as long as the force of gravity and the weight of the object remain the same.

## 5. How can the center of mass and center of gravity be calculated?

The center of mass can be calculated by finding the weighted average of all the individual masses within an object. The center of gravity can be calculated by finding the weighted average of all the individual weights within an object. Both calculations can be done using mathematical formulas or by performing physical experiments.

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