Since Newton's laws are defined for point particles, I'd like to derive some laws of motions for rigid bodies only by considering a rigid body as a system of particles such that the distances from every particle to every other particle doesn't change with time. I think I derived that the force applied on one particle of a rigid body must be the same for every other particle of the rigid body in one dimension by the following:
Consider two particles on a line $P_1$ and $P_2$ both with masses $dm$ and positions $x_1$ and $x_2$. Let's say that a force $F_1$ acts on the particle $P_1$. By Newton's second law we get: $$F_1 = dm\frac{d^2x_1}{dt^2}$$ By the definition of a rigid body, the distance between $P_1$ and $P_2$ doesn't change with time. Define $r$ as this distance ie. $r = x_1 - x_2$. Therefore: $$\frac{dr}{dt} = 0$$ Taking the derivative of both sides we further get that $$\frac{d^2r}{dt^2} = 0$$ $$\frac{d^2(x_1 - x_2)}{dt^2} = 0$$ $$\frac{d^2x_1}{dt^2} = \frac{d^2x_2}{dt^2}$$ By Newton's second law this is the same as: $$\frac{F_1}{dm} = \frac{F_2}{dm}$$ (where $F_2$ is the force acting on $P_2$), and since $dm \ne 0$ finally: $$F_1 = F_2$$
These steps can be done for arbitrary amount of particles, and so we get that in one dimension, if a force is applied on one of the particle of a rigid body, every other particle of the rigid body experiences the same force.
The problem is that I cannot do a similar proof for two dimensions by defining the distance $r = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$, but I'm sure that it can be done, and that if done torque, moment of inertia and center of mass would arise. Can someone do a similar proof for two dimensions, if it can be done like this at all?
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