I was watching this awesome video about hitting golf balls until they fail. It's a team-up video with YouTubers Destin Sandlin (Smarter Every Day) and Mark Rober. In the video, Destin and Mark want to figure out how hard you can hit a golf ball. Not how hard you could hit, or even how hard the best golfer in the universe could hit it. They wanted to find the hit that was so hard that it destroyed the ball. SPOILER ALERT—they destroyed the golf ball.
But here is the cool part. If you hit a ball like a normal human, the ball gets compressed upon contact with the golf club. During this compression, the ball essentially acts like a spring. Yes, it gets compressed for a very short time—but then it returns to its original position. This is called elastic compression. All of the springs in introductory physics are (probably) in this category of elastic compression (or stretching). In fact, this is what Hooke's Law is all about. It's a model for the force exerted by a spring that says the spring force is proportional to the amount the spring is compressed or stretched. As an equation, it looks like this:
In this expression Fs is the force exerted by the spring, s is the compression amount, and k is the spring constant—a measure of the stiffness of a spring. You will often see a negative sign in this equation. Some people put it there to emphasize that the force is in the opposite direction of the stretch. But let me be clear. Everything doesn't follow Hooke's Law—it's not really a law but more like a guideline (actually it's a scientific model). There are some objects and some situations in which the object does not need to have a linear relationship between force and stretch.
But if you compress a golf ball too much, it doesn't return to its original state. Instead, it gets smashed up such that it is deformed. It still has spring-like properties, but it's just not the same as it was before. It's different. This is called plastic deformation. As an example, imagine you have some clay. If you squeeze it too hard, it will deform and be in a new shape. It won't behave the same as before you squeezed it.
Of course, an object can be both elastic and plastic; the classic example is the common paperclip. Take one and pull it apart so that it looks like this.
In the video, Destin explains the elastic vs. plastic properties of a paperclip with a graph that looks something like this.
This is a nice visual that shows the main point—that if you push the paperclip too far, it's going to move into the elastic region. This means that it won't return to the same starting position when you remove the force, it will be different. Pretty much every material does this transition to the plastic region at some point. But how about we make a graph like this in real life? Yup. That's what I'm going to do. I'm even going to use a paperclip.
It looks basic, but it should do the trick. I have a paperclip with one end held stationary using some vice pliers. The other end of the paperclip is attached to a force probe and a rotary motion sensor. The force probe will obviously measure the force—the rotary motion sensor will actually measure the displacement. By knowing the radius of a wheel, I can convert angular position into linear position. The combination of these two sensors will give me a force vs. position graph. Here's what it looks like.
It's sort of tricky looking at this data. Remember, this is force vs. position—it doesn't show time. If you use your imagination, however, you can visualize what happens. When squeezed just a little bit, the paper clip moves up into that part of the plot that I have circled as "elastic." It just goes back and forth on this same line retracing the data. That's a normal spring. But then when you squeeze it too hard, it comes back down in a different region with a different final position. Yes, it's deformed.
But the very important thing about this plot—the elastic region isn't the area under the curve (the blue stuff in Destin's example). No. The elastic part is just a line.
If you found the slope of any part of this data, that would give you the effective spring constant (k) for the paperclip. Notice that the slope during the plastic region is fairly similar to the elastic region slope. In fact, this paperclip will still be well-behaved (elastic) but with a different length.
Oh, what about a traditional physics spring? Like the kind that you use in physics lab. What happens when one of these is stretched too far? Here is a similar plot of force vs. position for a spring.
Notice that in this case, the spring was stretched much further than that paperclip. In fact, it goes from about 10 centimeters long to almost a meter. Even then it only barely went into the plastic region. Also, since the spring "behaved" it's a little bit easier to find the spring constant. From the slope of the linear fit, this spring has a constant of around 8.6 Newtons per meter—even after getting partially destroyed. Really, this is great. You know students in physics lab abuse these springs (not really on purpose). But even after getting over-stretched, they still can be modeled with Hooke's Law.
What about that golf ball in the video from Destin and Mark? Nope. That thing's gone. Even the ball that remains intact is not really going to behave like the ball it was before the hit.