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.
 A: 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, 
$$\frac{d^2\vec r}{dt^2}=\vec F.$$
A: Essentially, the justification is that the experiments confirm it! You apply two different forces at the same time and the effect is just as if you had applied a single force which would be the parallelogram addition of the original two forces. Formally, it is sometimes heard that the reason parallelogram law works is because the forces are vectors. But the physical essence of the things going on lies in realizing that it is a fact derived from experiments that the entities we call force add the way described via parallelogram law and we call them vectors. 
