Question

In a situation with multiple forces you may only have one mass. (in this case this means you have opposing forces all on one object)

In this case is FN = m(a1 + a2 + a3 + a4 + ... + an) a valid way of writing the sum? Why or why not?

Answer 1

By definition, the net force acting on the body is the vector sum of all forces acting on it (bold letters denote vectors).

$$\mathbf{F}_N = \sum_{k=1}^n\mathbf{F}_k$$

According to the second Newton's law, each force causes its own acceleration:

$$\mathbf{F}_k = m\mathbf{a}_k$$

Thus:

$$\mathbf{F}_N = \sum_{k=1}^n\mathbf{F}_k = \sum_{k=1}^nm\mathbf{a}_k = m\sum_{k=1}^n\mathbf{a}_k$$

The last sum is simply:

$$\sum_{k=1}^n\mathbf{a}_k = \mathbf{a}_1 + \mathbf{a}_2+\mathbf{a}_3+...+\mathbf{a}_n$$

Finally, obtain:

$$\mathbf{F}_N = m(\mathbf{a}_1 + \mathbf{a}_2+\mathbf{a}_3+...+\mathbf{a}_n)$$

Answer. This formula matches with the suggested one, but only if quantities $\mathbf{F}_N$ and $\mathbf{a}_k$ are vectors. In this case it is valid way of writing the sum. Otherwise, if they are scalars, it is not a valid way of writing the sum.