I Dynamics of systems of material points

[Hak]

I have difficulty understanding the extension of the fundamental laws of material point dynamics to systems.

Example 1:
Consider a system consisting of two material points. Suppose that the two forces acting on the two constitute a pair of forces of nonzero arm. The resultant of the forces acting on the system is zero. The resultant moment is not! Why is it that to calculate the resultant moment we add up the moments of the two forces calculated separately, thus obtaining a different result than if we calculated the moment of the resultant of the forces (which would result in a null moment)?

Example 2:
Same system, but this time the pair of forces has zero arm. Assuming that due to the effect of the two (constant) forces, the two points move by a stretch $s$, why is the total work of the two forces derived by summing the work of the individual forces, resulting in $2Fs$, instead of calculating the work of the resultant of the forces (which would give work equal to $0$)?

 

[A.T.]

Why is it that to calculate the resultant moment we add up the moments of the two forces calculated separately, thus obtaining a different result than if we calculated the moment of the resultant of the forces (which would result in a null moment)?

Because we want the resultant moment to represent the rate of change of angular momentum, which can be non-zero, even if the net force is zero.

why is the total work of the two forces derived by summing the work of the individual forces, resulting in $2Fs$, instead of calculating the work of the resultant of the forces (which would give work equal to $0$)?

Because it takes a non-zero amount of energy to do this, so your approach would not make sense.

 

[Orodruin]

Consider a system consisting of two material points. Suppose that the two forces acting on the two constitute a pair of forces of nonzero arm. The resultant of the forces acting on the system is zero. The resultant moment is not! Why is it that to calculate the resultant moment we add up the moments of the two forces calculated separately, thus obtaining a different result than if we calculated the moment of the resultant of the forces (which would result in a null moment)?

A force by itself does not have a moment. Not even a zero one. In order to compute a moment you need the force and a reference point with respect to which to compute the moment. For a single particle it would be relatively natural to pick the particle itself as the reference point - thereby obtaining zero moment for any force with a line of action through that particle.

Now here is the thing: In order to add moments you must compute them relative to the same point, so once you picked a reference point you need to stick with it for all forces. Hence, in your two particle case, at least one of the forces will provide non-zero moment.

Note: The total moment generally depends on the point of reference. However, this is not the case if the net force is zero as in your case.

Same system, but this time the pair of forces has zero arm. Assuming that due to the effect of the two (constant) forces, the two points move by a stretch s, why is the total work of the two forces derived by summing the work of the individual forces, resulting in 2Fs, instead of calculating the work of the resultant of the forces (which would give work equal to 0)?

Because the work done by a force is the force multiplied by the displacement of what it is acting on - the particles in this case. The forces are opposite but so are the displacements of the particles. Hence 2Fs.