Why is there confusion with the sign in Hooke's Law equations?

In summary: F}k\underline{d} = mg\underline{d} - \underline{F}k\underline{d} = mg - k\Deltay = -ky.orFk = mg - k\Deltay.
  • #1
smashX
12
0

Homework Statement


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.


Homework Equations


Fk = -ky (Hooke's Law)


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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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
 
  • #4
smashX said:

Homework Statement


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.


Homework Equations


Fk = -ky (Hooke's Law)


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.

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.
 
  • #5
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
Thanks grzz, that REALLY helps. I think I finally understand now.
 

What is Hooke's Law?

Hooke's Law is a principle in physics that describes the relationship between the force applied to an elastic object and the resulting displacement. It states that the force applied is directly proportional to the amount of displacement produced, as long as the object remains within its elastic limit.

Who discovered Hooke's Law?

Hooke's Law was discovered by the English scientist Robert Hooke in the 17th century. He first described this principle in his book "Micrographia" in 1660.

What is the equation for Hooke's Law?

The equation for Hooke's Law is F = -kx, where F is the force applied, k is the spring constant, and x is the displacement produced. This equation shows that the force applied and the displacement are directly proportional to each other.

What is the unit for the spring constant in Hooke's Law?

The unit for the spring constant in Hooke's Law is Newtons per meter (N/m). This unit represents the amount of force required to produce a displacement of one meter in an object with a spring constant of 1 N/m.

How does Hooke's Law apply to real-life situations?

Hooke's Law can be observed in many real-life situations, such as the stretching of a rubber band, the compression of a spring, or the bending of a diving board. It helps us understand the behavior of elastic materials and how they respond to external forces.

Similar threads

  • Introductory Physics Homework Help
Replies
19
Views
1K
  • Introductory Physics Homework Help
2
Replies
35
Views
2K
  • Introductory Physics Homework Help
Replies
3
Views
408
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
59
  • Introductory Physics Homework Help
Replies
8
Views
140
  • Introductory Physics Homework Help
Replies
5
Views
2K
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
8
Views
2K
  • Introductory Physics Homework Help
Replies
1
Views
764
Back
Top