In a recent article,1 I explored how torque and the rotational second law of mechanics arise from force and the usual (translational) second law by modeling a rigid body with interacting point masses. The model failed to produce the correct rotational second law unless the body were not actually rigid. In his Comment, Dr. Knight takes issue with my model and in particular the conclusion that there is some problem with rigid bodies in Newtonian mechanics. Regarding the failure of my model, he claims that I conclude “that the mistake has arisen from the assumption that the connecting rod is perfectly rigid.” This is incorrect.

As I stated in my paper, the problem is that the forces between the masses only obey the weak third law and not the strong third law: in addition to being equal and opposite, the forces must lie along the line joining them (central forces). But this introduces a new problem since for collinear masses central forces have zero transverse component. So when the outer mass m2 is pushed transversely by an external force there can be (at that moment) no transverse force on the inner mass m1. Thus, m2 moves while m1 does not, resulting in a bent object. Only after this (in principle, arbitrarily small) bend develops can there be a transverse force on m1. Notice, then, that the problem has nothing to do with whether the connecting rod itself is rigid.

Next Dr. Knight claims that “the mistake actually arises from a misunderstanding of what forces are being applied and a misapplication of Newton's third law,” and more specifically that “the two masses in [my model] do not interact with each other in any way.” But the masses in my model do interact directly. I state in Sec. II that “The rod is essentially a model for the internal force between the mass and pivot.” I intended the same interpretation in Sec. III for the pair of rods connecting the pivot to m1 and connecting m1 to m2. The masses m1 and m2 are the rigid body and the forces between them are modeled by the rigid rods.

However, this objection to mass-mass interactions is specious because whether or not there is a literal rod between the two masses ultimately makes no difference. If there were a literal rod between the masses, then the mass-rod (contact) interaction must still be central. A consequence of the no-net-force and no-net-torque conditions on the massless rod is that the forces exerted by the rod on m1 and m2 will be equal, opposite, and along the line joining them (that is, along the rod). We can thus consider the masses to be interacting directly through a central force, taking us back to my original model and to its conclusion.

We can at this point attempt to change the model. Following Dr. Knight, we might imagine that there is a rigid massless rod extending all the way from the pivot to m2 with m1 stuck to its middle. But now we are modeling a rigid body with a rigid body, and we conclude that the body is rigid because we have assumed that it is. Not the most convincing model.

Let us then consider Dr. Knight's second suggestion where the rods fit into a socket in each mass, as illustrated in Fig. 1. This complicated geometry precludes point masses and requires multiple contact interactions (and perhaps even friction to properly balance N2, which is at an angle). But more importantly, notice that the rod, though rigid, hangs at some angle. The bend angle can be made arbitrarily small by having the socket shrink in size, but in the limit that the socket shrinks to the size of the rod the contact interactions become indeterminate, and the “rigid-body limit” is not necessarily well-defined. While the net force and torque between mass and rod may have the desired properties, their densities cannot be determined without considering the elastic properties of the rod, the socket, or both, so we cannot claim to have a truly rigid object after all. Again we have a bend that can be arbitrarily small, yet not zero.

Fig. 1.

Interaction of a rigid rod with a non-point mass with a socket. Having a pair of contact forces N1 and N2 allows for the socket and rod to exchange torque. Note the bending angle θ in the overall structure.

Fig. 1.

Interaction of a rigid rod with a non-point mass with a socket. Having a pair of contact forces N1 and N2 allows for the socket and rod to exchange torque. Note the bending angle θ in the overall structure.

Close modal

Finally, Dr. Knight does bring up one good point—it does not follow that all models fail just because my model failed; there very well could be a model immune to these considerations. This should be contrasted with the manifestly model-independent incompatibility of rigid bodies with special relativity. The limiting signal velocity and the relativity of simultaneity preclude any object appearing rigid to all observers regardless of how it is supposed to hold itself together. My conclusion that rigid bodies are incompatible with Newtonian mechanics is to be understood in the context of modeling. But it bears repeating that the incompatibility I described is not the obvious one; as Dr. Knight reminds us, “by definition, an object modeled with spring-like bonds won't be rigid.” The issue I explored was not stretching (which I in fact ignored), but rather a subtle and paradoxical bending arising not because the forces are spring-like, but because they are central. The paradox was that to prevent bending the inner mass requires a transverse force, but to develop a transverse force the structure must bend. So a bend, however small, is required to keep the structure from bending.

1.
D. J.
Cross
, “
The physical origin of torque and of the rotational second law
,”
Am. J. Phys.
83
,
121
125
(
2015
).