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 $A \cdot d = B$.

Figure of Principle of Moments

  • $\begingroup$ "Am I wrong here' well, not from a mathematicians point of view, but in physics one thinks of forces independently of the second law (like using scales l/deformations to measure the force), and then the second law is a statement about the effect of these forces. For your.problem you crucially need to.define what stiff means in your math framework. $\endgroup$ – lalala Oct 7 '18 at 14:36

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!


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