Can someone prove the Principle of Moments from Newton's Laws?

Question

Is it possible to rigorously proof the Principle of Moments from just Newton's three laws?

Here's my formulation of the laws/axioms? Someone tell me politely if I'm wrong.

1. Particles always follow inertia (constant velocity) if no force.
2. Define momentum of a particle $i$ as $V_i(t) \cdot M_i(t)$, where $M_i(t) > 0$ is a postive real number (mass) and $V_i(t)$ is an element of $\mathbb{R}^3$, defined as the velocity of $i$. Then a force acting on $i$ at time $t \geq 0$ is defined as $$F_i(t) = M_i\prime(t) \cdot V_i(t) + M_i(t) \cdot \dot V_i(t)$$ with differentiation well-defined over the real variable $t \in [0, \infty)$.
3. Let's name the particles $i \in \{1, 2, \dots, N\}$. Consider a closed system of the $N$ particles: whereby the system is defined by (i) their masses $M_i(t) \in (0, \infty)$, (ii) velocities $V_i(t) \in \mathbb{R}^3$, and (iii) some fixed positions at a given time $T_i \geq 0$, $r_i(T_i) \in \mathbb{R}^3$. The third law states that $$\forall t \forall i, \ \text{ if } F_i(t) \neq (0,0,0), \text{ then } \exists j \text { s.t. } F_j(t) = -F_i(t)$$

Actually, as a side note, it looks like the third law is the only "Law" of nature, the only empirical assumption about the empirical world. The first law is a trivial consequence of the second, and the second seems to be to be merely a deinition of "velocity", "mass", "momentum" and "force" (hence my bold face font). -- Am I wrong here?

I want to rigorously prove the Principle of Moments: A clockwise rotation caused by a (perpendicular) force of $Bg$ Newtons on a right-side of a pivot point focused at a distance of 1 metre, needs to be balanced by left-side (perpendicular) force of $Ag$ Newtons positioned $d$ metres from the pivot, such that $A \cdot d = B$

In other words, referring to my wonderful drawing, show that if the see-saw (a uniform, stiff, strong, inelastic, rod) is balanced, then $A \cdot d = B$.

Answer 1

There are a couple of different ways you can show this. Personally, the way I think is simplest is to use Newton's laws to derive the equation $$\mathbf{\dot{L}} = \mathbf{\Gamma^{ext}}$$ where $\mathbf{L}$ is the total angular momentum of the system and $\mathbf{\Gamma^{ext}}$ is the external torque acting on the system. The result you're after comes trivially after requiring $\mathbf{\dot{L}}=\mathbf{0}$. I don't have time to derive this myself right now, but you can probably find a derivation somewhere online, and I'll try to add one later when I have time. I like this approach because it doesn't require messing around with constraint forces, although it is probably a little more removed from intuition than the other way of doing it.

The other way of doing it would be to use Newton's laws as-is and introduce forces that the two masses and the fulcrum have to exert on each other in order to maintain certain geometric relationships between them. In particular, if we center a polar coordinate system on the pivot point, we want our forces to be such that the constraints $$\dot{\theta_A} = \dot{\theta_B} = 0$$ $$\dot{r_A} = \dot{r_B} = 0$$ $$F_{fulcrum} = 0$$ are always satisfied. Again, I'll try add a complete proof later if I have more time. Hopefully, these rough sketches of proofs help though!