Mass-vectors according to Maxwell

[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"

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.

 

[quantumdude]

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.[/color]
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 x_CM of a system of particles.

x_CM=(1/M)Σ_im_ix_i

where i=index (1,2,3,...) and M=total mass.

 

[HallsofIvy]

In modern terms (and not all THAT modern!) multiply the mass and the velocity vector to get the momentum vector.

 

[StephenPrivitera]

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.

 

[quantumdude]

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.

 

[StephenPrivitera]

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.

 

[quantumdude]

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 m_ia_i from the sum of m_ix_i? Answer: by taking the second derivative of x_CM with respect to time, like so:

(d²/dt²)x_CM=(1/M)Σ_im_i(d²/dt²)x_i
a_CM=Σ_im_ia_i

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.