In this particular problem: Is the mass of the system the mass of the person?

Question

I am doing this problem:

The upward normal force exerted by the floor is 620 N on an elevator passenger who weighs 650 N.

What is the magnitude of the acceleration?

This is how I solved it:

$$\sum F = F_1+F_2=N+(-W)=ma$$

I am saying here that the sum of all forces is equal to the $Normal \ Force + Weight$ and that all of this should equal $ma$.

I then re-arrange my equation:

$$(N-W)/m=a$$

Now here is where I am making a huge assumption, and where I am having problems. I decided to solve for $m$ by imagining that $m$ is the mass of the person. So in a way I am saying that the whole system in this problem is really the person. Is this a logical way to solve this kind of problem or will I run into problems by thinking this way?

If it is not clear what I did here is in equation format:

$$(N-W)/(W/g)=a$$

In other words: What is the proper way of thinking about the mass when applying $F=ma$? At the moment I am thinking of it as the mass of the objects that affect the problem. So if it was two people in the problem I would add up their masses and use that.

Answer 1

When Applying F = m*a, the mass used is simply the mass of the object that the forces are acting on. In this case, it is indeed the mass of the person.

You're right in that the weight of the person (650 N) is equal to the mass of the person times to force felt by him due to gravity. Dividing by gravitational acceleration will give you his mass.

Since the normal force on the passenger from the floor is 620 N (lower than the normal force he would feel from the floor in a stationary elevator, which would be equal to his weight), intuition says that the floor of the elevator is accelerating in the opposite direction of the normal force.

Yeah, everything looks good to me.

Answer 2

$m$ is the mass of the person, not of the combined system, since the normal force is exerted on the person, not the combined system (person $+$ elevator) itself.