Mass-vectors according to Maxwell

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In summary, Maxwell defines a mass-vector as the operation of carrying a given mass from the origin to a specific point, with its direction being that of the vector of the mass and its magnitude being the product of the mass and the vector. This can be used in determining the center of mass of a system of particles, represented by the sum of all mass-vectors divided by the total mass. To find the acceleration of the center of mass, one must take the second derivative of the center of mass with respect to time.
  • #1
StephenPrivitera
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From Matter and Motion by James Clerk Maxwell, Article 59,
"Let us define a mass-vector as the operation of carrying a given mass from the origin to the given point. The direction of the mass-vector is the same as that of the vector of the mass, but its magnitude is the product of the mass into the vector of the mass.
Thus, if OA is the vector of the mass A, the mass-vector is OA*A"

I can almost comprehend the idea of a mass-vector. Simply take the "vector of the mass" and multiply it by the mass. But what is the vector of the mass? What would be the point of the mass vector anyway?
Thanks.
 
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  • #2
Originally posted by StephenPrivitera
From Matter and Motion by James Clerk Maxwell, Article 59,
"Let us define a mass-vector as the operation of carrying a given mass from the origin to the given point. The direction of the mass-vector is the same as that of the vector of the mass, but its magnitude is the product of the mass into the vector of the mass.
Thus, if OA is the vector of the mass A, the mass-vector is OA*A"

It looks like he means take a mass m (what he calles "A") and take its displacement x (what he calles "OA") from the origin and multiply them to get a new vector mx.

What would be the point of the mass vector anyway?

It would be used in determining the center of mass xCM of a system of particles.

xCM=(1/M)Σimixi

where i=index (1,2,3,...) and M=total mass.
 
  • #3
In modern terms (and not all THAT modern!) multiply the mass and the velocity vector to get the momentum vector.
 
  • #4
Originally posted by HallsofIvy
In modern terms (and not all THAT modern!) multiply the mass and the velocity vector to get the momentum vector.
And momentum is one of many kinds of mass-vectors? You could use any vector to figure out the center of the mass, such as ma? Thus, according to Tom, the center of mass could be given by the sum of all the forces acting on a system divided by the total mass of the system.
 
  • #5
Originally posted by StephenPrivitera
And momentum is one of many kinds of mass-vectors? You could use any vector to figure out the center of the mass, such as ma? Thus, according to Tom, the center of mass could be given by the sum of all the forces acting on a system divided by the total mass of the system.

No, that approach will not give you the location of the CM. It will give you the acceleration of the CM.
 
  • #6
So to find the CM you have to use the displacement vectors of the masses in the system? If you use the momenta, acceleration, velocity, etc, and apply Tom's formula, you'll get the momentum, acceleration, velocity, etc, of the CM.
 
  • #7
Originally posted by StephenPrivitera
So to find the CM you have to use the displacement vectors of the masses in the system?

Yes.

If you use the momenta, acceleration, velocity, etc, and apply Tom's formula, you'll get the momentum, acceleration, velocity, etc, of the CM.

Yes. To go back to your "acceleration" example, how can I get the sum of miai from the sum of mixi? Answer: by taking the second derivative of xCM with respect to time, like so:

(d2/dt2)xCM=(1/M)Σimi(d2/dt2)xi
aCM=Σimiai

That's the only way I can get the sum of forces from the sum of the mx vectors with the equation for the CM. As you can see, changing from x to a on one side makes it necessary to do it on the other side, so we aren't talking about the location of the CM anymore, but its acceleration.
 

1. What are mass-vectors according to Maxwell?

Mass-vectors according to Maxwell are a concept in physics that describes the movement of mass in a three-dimensional space. In simpler terms, it is a mathematical representation of the direction and magnitude of mass displacement.

2. How do mass-vectors relate to Maxwell's equations?

Maxwell's equations, which describe the behavior of electromagnetic fields, also have implications for the movement of mass. Mass-vectors are used to represent the movement of mass in relation to electromagnetic fields, as described by Maxwell's equations.

3. Can mass-vectors change over time?

Yes, mass-vectors can change over time. As mass moves and changes direction, the mass-vectors representing its movement will also change accordingly.

4. How are mass-vectors calculated?

Mass-vectors are calculated using vector algebra, which involves both magnitude and direction. The magnitude of a mass-vector is determined by the mass and its velocity, while the direction is determined by the angle between the mass and the reference axis.

5. What are some practical applications of mass-vectors according to Maxwell?

Mass-vectors are used in various fields such as fluid dynamics, electromagnetism, and mechanics to analyze and predict the movement of mass. They are also used in designing structures and machines, as well as in the study of planetary motion and celestial mechanics.

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