Yahoo Answers: Answers and Comments for Hooke's Law: F= k x. Why is there a negative in the equation? [Physics]
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Sat, 13 Dec 2008 20:44:02 +0000
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Yahoo Answers: Answers and Comments for Hooke's Law: F= k x. Why is there a negative in the equation? [Physics]
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From George S: Answer: NO  the spring constant is never neg...
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Sat, 13 Dec 2008 21:22:03 +0000
Answer: NO  the spring constant is never negative. The final result of the equation actually has the negative sign. For example, let x = 2 meters and k = 10 (m/N). Then the equation becomes
F = k x =  (10 m/N) (2 m) = 20 N.
So the answer for F has a negative sign. As the answers by climberg and Benjamin N said, this means the force F opposing the stretching of the spring by x.
If you reversed the direction of stretching, you would have x = 2 m, so that
F =  (10 N/m) (2 m) = 20 N = +20 N,
the plus sign showing that again F has opposite sign from x, and so opposes x.
Do not try to force the equation to have the same sign on both sides. The negative sign is part of the resulting F, and indicates that F is a restoring force. The equation is giving information about both the magnitudes of the variables (F and x) and their directions (restoring force F opposes stretching out the spring by distance x).
Note that the equation involves just the distance stretched x and the restoring force F. The pulling force is present to keep the spring stretched, but is not part of the equation here. (The restoring force is "equal and opposite" to the pulling force, one of Newton's laws of forces.)
Here are additional answers based on the concept of vectors and graphs.
In the equation F =  k x, both F and x are vectors. A vector has both magnitude (a single numerical value) and a direction. The coefficient k is a pure number, no direction attached. The negative sign changes the direction of a vector (makes it point in the opposite direction.) Let the spring be attached to a fixed frame at one end and be free at the other end. Let the spring be pulled and stretched by a vector distance x, where the starting point of vector x is the original position of the free end and the end point of vector x is the new (stretched) position of the the free end. That is, vector x points in the direction of the pull, and its magnitude is the distance of stretching. (And x = 0 means the spring is not stretched at all.) At this point k times x gives the magnitude of the restoring force, but without the negative sign, this would imply the restoring force vector F points in the same direction as the stretching vector x, which is not the case. The restoring force F tends to oppose the stretching pull, and tends to "restore" the spring to its original position, so it must be represented as a vector opposing vector x. So the negative sign is added to reverse the F vector's direction, making it oppose the F vector, and giving the proper direction for the F vector.
Another way to picture the negative sign in this equation is to consider an xy graph. In this case change the symbol x to d, for displacement, so it is not confused with x of the xaxis. Then F = k d. If you consider F and d as vectors lying on the x axis, where +x lies to the right and x lies to the left (the standard arrangement), and assume the pulling force is from left to right, then obviously displacement d is a positive number. Since k is also (always) positive, the equation then makes F a negative number. The negative sign means F points to the left, opposing the pulling force, and trying to reduce x back to 0.

From Benjamin N: The negative sign means that the force arrow p...
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Sat, 13 Dec 2008 20:49:08 +0000
The negative sign means that the force arrow points in the opposite direction of the displacement. Yes, there is a negativeotherwise the force is not restorative.

From Ahsan: I think the only rationale behind the negative...
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Tue, 28 Jun 2016 06:50:53 +0000
I think the only rationale behind the negative sign is the fact that whether you apply tensile or compressile force, in either case, the spring tends to oppose the force applied and tends to restore the original position.

From ?: Because the force of the spring is acting in t...
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Mon, 30 May 2016 11:13:45 +0000
Because the force of the spring is acting in the opposite direction from the motion of the spring.

From Cedric: Add your answer
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Thu, 05 Nov 2015 17:29:46 +0000
Add your answer

From Anonymous: If you stretch the spring, you are applying fo...
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Sat, 13 Dec 2008 20:49:48 +0000
If you stretch the spring, you are applying force. If it is stretched and held in place, the net force is 0. Either the force you are applying or the force the spring is applying must be negative for their sum to equal 0. In this case, the spring is applying force opposite the direction of travel when it is being stretched. Therefore, it is negative. Hope I broke it down well enough for you.

From climberguy12: its because when you compress the string in th...
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Sat, 13 Dec 2008 20:47:40 +0000
its because when you compress the string in the positive direction, the force is in the negative direction. just think about it. if you push a ball down on a spring, it throws it up.
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