recall that \(F\) is the force deforming the spring, \(x\)is the distance it deforms, and \(k\) is a spring constant which is a characteristic of the spring (meaning it does not change). We can trust that this law is true so let’s go ahead and determine what the units of \(k\) should be.
Remember from our units chapter that we can use dimensional analysis to determine the units of \(k\) since we know that for this equation to be true, the dimensions on each side of the equal sign must be equal. It follows that the units must be equal on each side as well. I’ll be honest, I tell you what the units are for \(k\) below. You will only be cheating yourself if you skip this quick brain workout.
Incorrect. Hooke’s constant is not measured in units of mass.
\(\frac{m}{N}\)
Incorrect. This is the reciprocal of the correct units.
\(N \cdot m\)
Incorrect. These are units of work or energy, not a spring constant.
\(N\)
Incorrect. Hooke’s constant relates force to displacement, so length must appear in the denominator.
\(\frac{N}{m}\)
Correct. From Hooke’s Law, \(F = kx\text{,}\) so \(k = \frac{F}{x}\text{.}\) Since force is measured in newtons and displacement in meters, the SI units of \(k\) are \(\frac{N}{m}\text{.}\)
Hopefully you actually tried this on your own and found that the SI units for \(k\) are \(\frac{N}{m}\text{.}\) If you google search “buying springs” you can see that this specification is always listed for springs that are for sale. It may be in US units (\(\frac{lbs}{in}\) …gross) but you get the idea.
