# Negative and positive results when dealing with Newton's Laws

I think it's a pretty simple question. Let's say we're talking about a moving elevator in which a person stands on a digital balance. The textbook tells me to treat the downward direction as positive, so the value of acceleration is 9.8 m/s^2 instead of -9.8m/s^2

I'm asked to find out the acceleration of the elevator when the person's weight appears to be 0.75 times of its normal weight. I got a negative value, which is the correct answer. Then the book tells me that since it's negative, it means that the elevator is going down. But I thought that the downward direction is treated as positive!

(I think the book is correct. I just don't understand what the negative sign means.)

## Answers and Replies

Erland
Something is wrong here. Since the balance shows less then the person's normal weight, the elevator accelerates downwards (if it accelerated upwards, the balance would show more than the person's normal weight.), i.e. in the positive direction. Thus, the correct answer should be a positive number, not a negative.

Thanks for replying! I hope this doesn't get moved to the homework section, since this isn't homework. It's just from a book I like reading.

Here's the question:

A person stands on a bathroom scale in a motionless elevator. When the elevator begins to move, the scale briefly reads only 0.75 of the person's regular weight. Calculate the acceleration of the elevator, and find the direction of the acceleration.

9.8 * 0.75x = x (9.8+a)
a = -2.45 m/s^2

Doc Al
Mentor
9.8 * 0.75x = x (9.8+a)
a = -2.45 m/s^2
These equations are a bit obscure (and incorrect). Better to start with Newton's 2nd law and work it systematically, that way you won't have any sign errors:
ƩF = ma

What forces act on the person? (One of them will be the normal force, which is the scale reading.) What's the net force?

Ken G
Gold Member
Basically, there are always two ways to do signs in physics-- trust that the equations will get them right, or do the signs manually. Doc Al's way is the first one-- write the equations down following the rules for doing that (which the book doesn't), and then solve for what you want to know, complete with signs. This is usually the preferred technique because it is automatic.

The book is using the second way-- just assert the signs manually. So "a" is written as an augmentation to gravity, so is positive when the elevator is going up, regardless of the sign of gravity (note the sign chosen for gravity is outside the parentheses in your quote). You have to recognize when this "manual" approach to signs is being taken, and in fact, in many real-world applications it is actually easier to just apply the signs manually, as long as you know what your conventions are. The most important thing, either way, is to continually stop in the middle of any physics problem and ask yourself "does this make sense?" Following that practice is much more important than any sign convention you could use. Note this is what you did when you asked the question in the first place, so it's much less important how the sign of a was implicitly enforced by the equations used.

Thanks for the replies:

Doc Al mentioned that the normal force is the reading of the balance. But I thought that the force exerted on the weight is the reading?

Doc Al
Mentor
Doc Al mentioned that the normal force is the reading of the balance. But I thought that the force exerted on the weight is the reading?
What do you mean by "the force exerted on the weight"?

It would be helpful if you identify all the forces acting on the person in the elevator. Then you can quickly derive any formula you need using any sign convention you like, just by applying Newton's 2nd law.

Perhaps I should rephrase my question.

Why is the gravitational acceleration positive when I do my calculations?

I thought that it should be negative since it's pointing downwards. (I'm assuming that downwards is negative now)

Ken G
Gold Member
Look at how a is defined. It adds to 9.8. It doesn't matter that 9.8 is given a minus sign if a positive value for a will add to 9.8-- adding something to 9.8 means it is downward, regardless of whether down is positive or negative outside that parentheses. That's what I mean about doing the signs manually-- just look at the implicit assumptions made about the signs, you can't expect them to come out magically correct unless you go back to the start and just identify all the forces like Doc Al said. Every other approach is going to have a manual aspect and you have to look at the implicit ways the signs are being treated.