newtonian mechanics - Parallelogram Law of Force Addition

In most of the mechanics textbooks , the Parallelogram Law of Force Addition is stated without any justification or an experiment to confirm this law. Also after introducing this law forces are also decomposed without any justification. Please suggest any links or books related to this. Any help will be appreciated. Thanks in advance.

Answer

The justification for Parallelogram Law of Force Addition is that second Newton's Law is a vector equation linear in force.

Let us suppose we have a particle which can possibly acted by two forces $\vec F_1$ and $\vec F_2$. Newtonian Mechanics assumes that if $\vec r_1(t)$ and $\vec r_2(t)$ are the position of the particle when only $\vec F_1$ and $\vec F_2$, respectively are non vanishing, then the position when both forces are acting together is just $\vec r(t)=\vec r_1(t)+\vec r_2(t)$. This is known as the Superposition Principle and as we can see it just means we can sum the vectors $\vec F_1+\vec F_2\equiv \vec F$ and plug this into the equation of motion, $$\frac{d^2\vec r}{dt^2}=\vec F.$$