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.
- Particles always follow inertia (constant velocity) if no force.
- 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 $t \in [0, \infty)$.
- 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 $A \cdot d = B$.