From where does the perpendicular component of force arise in this lever? Does it conflict with the strong form of Newton's third law?

Question

Newton's third law states that for every force applied by object $A$ on object $B$, there is an equal an opposite force by object $B$ on object $A$. The strong form of Newton's third law states that for every force applied by object $A$ on object $B$, there is an equal an opposite force by object $B$ on object $A$, and these forces are along the line of sight between $A$ and $B$.

By the strong form of Newton's third law, we are able to derive conservation of angular momentum and many other principles of rotational mechanics.

Now consider a lever with a fixed pivot with masses $A$ and $B$ on both ends. Let's say it rotates in the $xy$-plane and it is initially along the $x$-axis. To make things further clear, let's say the pivot is at the origin $x=0$, $A$ is at $x>0$, and $B$ is at $x<0$.

When I apply a force on mass $A$ in direction $-y$, the lever rotates clockwise, and mass $B$ initially moves in the $+y$ direction. During that initial motion, the acceleration vector has a nonzero $y$-component, and thus by Newton's second law there is a net force with a nonzero $y$-component.

The question is, how is this possible if every force involved is supposed to be along the line of sight between two objects/particles? Even if we consider the lever as part of our analysis, the lever itself is along the $x$-axis, so different parts along it are separated by an $x$-coordinate, not a $y$-coordinate, so no force along $y$ should ever arise.

What makes this more confusing/paradoxical is that the laws of rotational mechanics, which are used to calculate how the lever would rotate due to the force applied on $A$, are derived from the assumption that forces are only along the line of sight between two objects, and yet in the end this seems to be violated.

What I am particularly interested in is, how would we model the lever + masses scenario so that I can see exactly where the $y$-component of the force comes from? Even if we assume the lever has thickness or isn't fully rigid, it isn't clear to me how the forces act.

Answer 1

In your example, A and B are not separate objects. Motion of A affects motion of B through the forces felt and exerted by and within the lever itself. There is also the force exerted on the lever at the pivot that you never mention. If the lever were made of paper, then the paper would fold because the right portion could not exert enough force on the left portion to eventually affect the motion of B. If it were made of bamboo, the lever would bend, thus making the internal forces more visible. This is actually why the rotational motion properties are so important. They provide a way to talk about one solid object as having one motion independent of internal forces. Every small piece of a solid object moves differently (in meters/second) and pushes or pulls on neighboring pieces. There can be stretching/squeezing forces, twisting forces, skew (or sideways) forces. Bending is actually a combination of stretching on one side and squeezing on the other. These many internal forces are usually presented as "stress and strain".

Answer 2

Even if we consider the lever as part of our analysis, the lever itself is along the 𝑥-axis, so different parts along it are separated by an 𝑥-coordinate, not a 𝑦-coordinate

A real lever does have an extent on the y-coordinate, and it's critical for the operation exactly as you say. The torque cannot be transmitted without some difference in angle. Once you model some thickness, then the torque can be explained just through compression and tension along different lines.