We all know that "Newton's second law" states that any particle or a system of particles with a constant total mass $m$ under influence of $n$ external forces will move with a constant acceleration $\vec{a}$ proportional to the sum of all forces such that we can write:
$$\sum_{i=1}^{n} \vec{F_i}=m\vec{a}$$
Now my question is if the individual external forces $\vec{F_1}$, $\vec{F_2}$, $\vec{F_3}$, …, $\vec{F_n}$ generate the corresponding accelerations $\vec{a_1}$, $\vec{a_2}$, $\vec{a_3}$, …, $\vec{a_n}$ independently, can we write the equation for an "Individual external force" as follows?
$$\vec{F_i}=m\vec{a_i}$$
I mean can we deduce the equation for each individual external force from 2nd law?
(Notice that the total acceleration should be: $\vec{a}=\sum \vec{a_i}$)
Best Answer
Well, yes of course. But bear in mind that the single $a_i$ are not observable, so is more of a mathematical/logical tool.
Then the equation becomes, in one dimension (the same applies to vector calculus):
$$\sum_i F_i = ma = m\sum_i a_i = m\sum_i {F_i\over m} = \sum_i F_i $$
it is an identity, you can use any of the forms above!
That is the strength of linear formulas: you can solve everything separately and then add everything up (linear superposition principle)
EDIT: Let's put it another way. If each force was acting without the others you would have $F_i=ma_i$ thus ${F_i\over m}=a_i$.
Then now take: $$\sum_i F_i=ma$$ and divide evrything by $m$. You get:
$$\sum_i {F_i\over m} = a$$
Then, using the formula above: $$\sum_i a_i = a$$
This proves that the decomposition in single forces leading to single accelerations make sense. You could have done another decomposition, such as, supposing you only have $F_1$ and $F_2$, $$F_1=m {a\over 3}$$ and $$F_2=m{2a\over 3}$$. Then you would still have
$${F_1\over m}+{F_2\over m}={a\over 3}+{2a\over 3}=a$$
and as long as the two forces are acting together it is a fine decomposition. Yet it has no physical sense, as if the forces were acting singularly the equations above would be wrong.
So to say, since you can decompose a sum of force anyway you want you just choose the only decomposition which would make sense if the forces were acting alone.
This kind of superposition has physical sense only if the single forces are real (e.g. one is pulling and one is pushing). This may also help you to solve problems in which several forces are acting: you can solve each one separately and then sum the results (make sure you sum the vectors in the right way..!)