Xjyuk

I used to think that torque and force were equally “fundamental”. In other words, my understanding was that we usually use Cartesian coordinates in many common problems because it is a convenient system, so as a result instantaneous forces which act in straight lines seem “easier” mathematically but torques require some extra “baggage”. This baggage includes typically teaching that torque is defined in terms of force.

But if say we happened to choose polar coordinates for the problem the situation would appear the other way around. So it would be arbitrary if we chose to define forces in terms of torques instead.

But later on I learned that angular momentum is conserved independently of regular linear momentum (IIRC). This makes it seem much less certain that one should define either torque in terms of force or vice versa — it gives the impression they are more distinct than it first seemed.

That said, as far as I know a lot of physics is about defining & describing “fundamental forces” — not “fundamental torques”.

So is choosing to use either force or torque as the basis of laws & problems arbitrary? Or is there an actual fundamental rationale to when one or the other should be used?

No, torque is not a fundamental quantity. it's only job is to describe where in space a force is acting through (the line of action). Torque just describes a force at a distance. Given a force $boldsymbolF$ and a torque $boldsymboltau$ you can tell that the force acts along a line in space with direction defined by $boldsymbolF$, but location defined by $boldsymboltau$ as follows $$ boldsymbolr = frac boldsymbolF times boldsymboltau boldsymbolF $$

It might be easier to discuss angular momentum first, since torque is the time derivative of angular momentum, just as force is the time derivative of linear momentum.

For a single particle with linear momentum $boldsymbolp = mboldsymbolv$ located at some instant at a point $boldsymbolr$ the angular momentum is $$ boldsymbolL = boldsymbolr times boldsymbolp$$

So where is the momentum line in space? The momentum line is called the axis of percussion. It is located at

$$ boldsymbolr = frac boldsymbolp times boldsymbolL boldsymbolp = fracboldsymbolp times ( boldsymbolr times boldsymbolp) = frac boldsymbolr (boldsymbolp cdot boldsymbolp) - boldsymbolp ( boldsymbolp cdot boldsymbolr) = boldsymbolr frac^2 = r ; checkmark $$

provided that the point $boldsymbolr$ is perpendicular to the momentum $boldsymbolp$. Such a point can always be found, and it is the point on the line closest to the origin.

The conservation law for angular momentum (coupled with the conservation law for linear momentum) just states that not only the magnitude and direction of momentum is conserved but also the line in space where moment acts through is also conserved.

In summary, the common quantities in mechanics are interpreted as follows

$$ beginarrayl
textconcept & textvalue & textmoment\
hline textrotation axis & textrot. velocity, boldsymbolomega & textvelocity, boldsymbolv = boldsymbolrtimes boldsymbolomega \
textline of action & textforce, boldsymbolF & texttorque, boldsymboltau = boldsymbolr times boldsymbolF \
textaxis of percussion & textmomentum, boldsymbolp & textang. momentum, boldsymbolL = boldsymbolr times boldsymbolp
endarray $$

The stuff under the value column are fundamental quantities that give us the magnitude of something (as well as the direction). The stuff under the moment column are secondary quantities that depend on where they are measured and give use the relative location of the fundamental quantities. Hence the terms torque = moment of force, velocity = moment of rotation and angular momentum = moment of momentum. All that means is that these quantities are $boldsymbolr times text(something fundamental)$ and they describe the moment arm to this something.

In statics, for example, we learn to balance forces and moments, which should be interpreted as balancing the force magnitude and the line of action of the force.

I would argue that force is more fundamental concept than torque. This is mainly because torques is, for lack of a better term, a property of forces. Also, the torque produced by a force depends on your subjective choice of which point you are calculating the torque about. This is all captured in the definition of torque
$$boldsymboltau=mathbf rtimesmathbf F$$
where $mathbf F$ is the force vector and $mathbf r$ is the vector pointing from the point about which you are calculating the torque to the point where the force is applied.

Note that this defines torque in terms of a force, but you cannot determine a force from a torque. For a given $boldsymboltau$ and a given $mathbf r$ there is not a unique force $mathbf F$. Therefore, this also gives the impression that force is a more fundamental concept.

Also note that the definition of torque does not depend on if we are using polar coordinates or not. You can discuss forces in polar coordinates without referring to torque, and you can talk about torques in Cartesian coordinates.

So torque is not fundamental, however that doesn't mean it's not useful. It is useful in looking at how forces cause extended bodies to move (or not move), and it is useful when motion has rotational symmetry about some point (i.e. when angular momentum is conserved).

Just what is fundamental? Torque is certainly more than just force with additional “baggage”. And it’s more than just about coordinate systems. Torque and force are not a matter of either or. Both are needed for the analysis of motion and equilibrium.

Moment, which is another term for torque, is a fundamental concept in statics. For example in statics both forces and moments are needed to determine static equilibrium . Forces cause straight line motion. Moments cause rotational motion. The requirements for equilibrium are that the sum of both the moments and forces have to be zero. And it goes beyond statics. Bending moments and shear forces are fundamental to the study of mechanics of materials.

You will learn to appreciate the difference if and when you study statics , dynamics, and mechanics of materials.

Hope this helps

Transfer of angular momentum is fundamental, and does not reduce to a force being applied.

For example: "...by utilizing transfer of photon spin angular momentum, it is also possible to set objects into rotational motion simply by targeting them with a beam of circularly polarized light"

Source: https://pubs.acs.org/doi/abs/10.1021/nl4010817