 The greatest physicist of the early 18th century was Robert Hooke. At least, according to Robert Hooke. Hooke fought constantly with Newton over priority issues. Hooke is best known for Hooke's law. The restorative force on a spring is proportional to the displacement from the spring's natural length. For example, suppose it takes 25 Newtons to stretch a spring 0.2 meters beyond its natural length. How much force will be required to stretch the spring 0.6 meters? If force is proportional to the distance, so the force is to 25 Newtons as the distance 0.6 meters is to 0.2 meters. And note, we're including the units so we can determine the units of our final answer. Treating our units like algebraic variables and solving for F, we get 75 Newtons. Since work is the product of force and distance, we can compute the amount of work needed to stretch or compress a spring. Because the force varies, we need to sum over short intervals where the force is essentially constant. For example, suppose it takes 25 Newtons to stretch a spring 0.2 meters beyond its natural length. How much work will be required to stretch the spring from its natural length to a length of 0.6 meters? Now, since the force is proportional to the distance, there's a constant of proportionality. And so we can write force equals K times X. We know that the force is 25 Newtons when X is 0.2 meters. And so we can substitute those in and solve for K, which will have units of Newtons per meter. And so this gives us the amount of force required to stretch a spring to a length of X. So the amount of work is force times distance. So we'll take that force, 125X, and multiply it by a small distance, dx, and then sum those amounts. We go from 0 meters beyond the natural length to 0.6 meters beyond the natural length. And we can do the calculus to get a numerical value. And let's check out our units. So remember, an amount, its change, and its sum all have the same units. So notice that a unit will be Newtons, because that's our measure of force, times meters, because that is our measure of X. And so we're summing up Newton meters, which means that our value will be in Newton meters. And since Newton meter is a measure of work, then this looks like it could be our answer.