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Hooke's Law Sign Confusion 
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#1
Nov2711, 01:53 AM

P: 12

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. 


#2
Nov2711, 01:59 AM

HW Helper
P: 6,208

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



#3
Nov2711, 02:14 AM

P: 12

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, mgFk = 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



#4
Nov2711, 02:40 AM

HW Helper
P: 2,316

Hooke's Law Sign Confusion



#5
Nov2711, 02:43 AM

P: 950

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. 


#6
Nov2711, 03:10 AM

P: 12

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



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