Hooke's Law
Robert Hooke was an English scientist, a contemporary of Isaac Newton. He made fundamental discoveries in mechanics, optics, astronomy and biology. One of these discoveries, and the only one I know of that's actually named after him, is "Hooke's Law", namely, that the stretch of a spring is proportional to how hard you pull on it. In symbols, we write...
In other words, if you pull twice as hard on a spring, it will stretch twice as much. Pull on it five times harder, and the stretch will be five times greater. Here, the stretch is Δx, the difference between the length of the spring while being pulled on and the length when it's not being pulled on.
When you have a proportional relationship between variables, you can replace the ∝symbol with = if you also insert a constant of proportionality. In this case, we use the letter k to stand for the stiffness of the spring.
(Sometimes k is simply called the Hooke's Law constant.) Unlike another famous constant you know, π, it is not a fundamental constant of nature. Rather, the stiffness is a number that describes a particular spring. Different springs have different k's. Stiffer or 'stronger' springs have higher values of k.
The units of k are 
, so that means usually 
or 
, but in barbaric units would be 
or 
.
And even though the gram and kilogram are not units of force, people will often use 
or 
with the assumption that the spring and masses are on planet Earth.
Hooke's Law, by the way, is not a fundamental law of nature. There are exceptions. If the spring is stretched too far, it will be permanently deformed and will not return to its original length. Some materials
Activities & Practice
to do as you read
1. Play with the simulation to the left. You can move the ruler around to measure the stretch of each spring.
(a) How much does spring #1 stretch when you hang 100 grams on it?
(b) How far do you predict spring #1 will stretch if you hang 250 grams on it instead of 100 grams?
(c) Try it. Was your prediction correct?
(d) What is the spring constant for spring #1, in g/cm?
(e) Figure out the masses of the red, green and gold weights.
(f) Using the slider, make spring #3 as "soft" as it can be. What is its spring constant then?
(g) Using the slider, make spring #3 as "hard" as it can be. What is its spring constant then?
APPLICATIONS
Scales use springs to measure the weight of an object. Really, the scale is just measuring the force pulling on the spring, but if that pull is from something hanging from the spring, and it's at rest, the tension of the spring is equal to the weight of the object. For example, here's a scale in the produce section at my neighborhood grocery store. The more weight is placed in the hanging basket, the more the spring inside stretches, and that turns a needle that points along the series of numbers. The placement of the numbers is calibrated for whatever units the manufacturer chooses — in this case pounds. The zero mark is placed so the weight of the basket itself doesn't count. On a good scale, the zero position of the needle is adjustable to allow for different-weight containers to be used. Such an adjustment is called a tare.
The scales in the above discussion all had the object hanging below the scale. Many scales, however, are engineered so you place the object on top of the scale; the weight compresses the spring rather than stretching it. Most bathroom scales are of this type.
Because scales respond to force, they are not reliable for measuring mass, even if they are calibrated in mass units (kg) rather than weight units (N or lbs). A balance, on the other hand, will read the correct mass no matter what planet you are on. Balances don't use a spring at all, but instead have a pivot or fulcrum (like a see-saw) with known masses on one side and the mass to be measured on the other. By adjusting the amount of the known masses, or their distance from the fulcrum, they will balance the unknown mass.
Additional Activities & Practice
1.