Hooke's Law Sign Confusion

smashX

1. The problem statement, all variables and given/known data
I am currently doing some Hooke's Law problems. While I do not have any trouble with any exercise in particular, I do have trouble with the sign in the equation. Let's say I have a vertical spring and I attached a hanging mass m to it. The string will then stretch a distance Δy. I choose downward as positive.

Therefore, applying Newton's Second Law gives: Fnet = mg - Fk (Fk is the restored force given by Hooke's Law).

Since the vertical spring is at equilibrium, Fnet = 0. Therefore, mg - Fk = 0, or mg = Fk

But Fk also equals -kΔy, where k is the spring constant and Δy is the positive displacement (I chose downward as positive).

So Fk = mg, which is a positive value. But Fk = -ky, which is negative??? I know I must have done something wrong somewhere, but I couldn't figure out where. Could someone please help explaining this for me? Thank you very much. Any help is greatly appreciated.


2. Relevant equations
Fk = -ky (Hooke's Law)


3. The attempt at a solution
I was able to solve most of the problems by "forcing" myself to choose the correct sign; however, I still don't understand what I'm doing and why I got the correct result in the first place. I understand why there is a negative sign in the Hooke's Law equation, since it's a restored force and it must be inversely proportional to the displacement. Having said that, I still don't understand why my attempt above gave Fk both a positive, and a negative value. Please shed some light for me, thanks a lot.

Also, I apologize for the inconvenient notation. I don't know Latex or any other mathematical convention used in forum and typing documents, so please don't delete this thread. Thanks again.

rock.freak667

When you had mg-Fk=0 that meant you already accounted for the spring having the restoring force with the minus sign. So Fk=ky here.

smashX

Thanks, but I thought Fk having a negative sign was just from the diagram for this case? I mean Fk is pointing upward, right? Since I chose downward as positive, mg-Fk = 0, then I substituted Fk for -ky and got the wrong answer. I kind of get your explanation, but I'm still confused... I'm really sorry, but could you please explain this problem in any way easier to understand. Thanks a lot

PeterO

Choose up as positive and see if you still have conflict. If you don't get conflict that means that the positioning of the spring had already defined the positive direction, and you were not free to arbitrarily decide.

grzz

Another point of view:

There appears to be some trouble with direction. Hence let us take the trouble to put the problem in vector form.

Let [itex]\underline{d}[/itex] be a unit vector pointing downwards.

The weight mg is downwards. Hence let mg be represented by mg[itex]\underline{d}[/itex].
The displacement [itex]\Delta[/itex]y of the spring is downwards. Hence let this displacement be represented by [itex]\Delta[/itex]y[itex]\underline{d}[/itex].

But the restoring force, of magnitude k[itex]\Delta[/itex]y, due to the spring is upwards. Hence let us represent this restoring force by
-k[itex]\Delta[/itex]y[itex]\underline{d}[/itex].

But mg[itex]\underline{d}[/itex] + (- k[itex]\Delta[/itex]y[itex]\underline{d}[/itex]) = 0
i.e. mg[itex]\underline{d}[/itex] = k[itex]\Delta[/itex]y[itex]\underline{d}[/itex]
which just shows that the weight and the restoring force are equal in magnitude but opposite in direction.

smashX

Thanks grzz, that REALLY helps. I think I finally understand now.