# Hooke's Law: F= -k x. Why is there a negative in the equation?

In a problem where the Force(F) is positive and so is the distance(x), will the spring constant(k) be a negative?

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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 x-y graph. In this case change the symbol x to d, for displacement, so it is not confused with x of the x-axis. 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.

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• Anonymous

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.

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• The negative sign means that the force arrow points in the opposite direction of the displacement. Yes, there is a negative--otherwise the force is not restorative.

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• 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.

make it a good day

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