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| Tim and Andy measuring the force applied by a spring |
I think every physics class in the known universe does the F vs. x experiment for a spring: The force on a spring is measured with a spring scale or hanging masses, and is plotted on the vertical axis of a graph. The length of the spring (or the displacement from the resting position) is measured with a meterstick and plotted on the horizontal axis. Because F = kx, the slope of this linear graph is the spring constant k.
(As an aside, I've written up a detailed approach to this experiment for the College Board -- take a look here.)
This experiment is beautiful because the data are easy to take, and because even the worst experimenters get something resembling a line. However, occasionally you'll see something weird -- the graph will be a line most of the way, but very small displacements will give a significantly steeper slope. See the graph to the right (and click on it to enlarge if you can't quite see).
What's going on?
First of all, quash the inevitable misconception: "Oh, that makes sense because the more the spring stretched, the more force we had to use." Well, of course -- that's what F = kx means. We should need more force to stretch the spring for larger displacements.
The slope of this graph represents the spring constant k, which indicates the stiffness of the spring. What's happening here is that the spring is significantly stiffer under about 3 cm of stretch. Does that make any physical sense, though?
Well, in this case, yes. If you get this sort of data, take a careful look at the spring you're using:
See how many of the coils are touching each other? I asked the class to be very quiet... and then I began to stretch the spring a couple of centimeters. We could all hear the "poing!" sounds of the individual coils unsticking from each other. All the coils were fully separated when I had stretched the spring... about 3 cm.
How wonderful to demonstrate the physical reason why the graphed data looks the way it does!
ReplyDeleteI wonder, in the case of the spring above, in reference to the graph you included in your post, would it be a meaningful question for a physics teacher to ask a student, "Estimate the value of the spring constant k for the spring"? Or does that question not really make sense in this case because the spring is stiffer under 3 cm of stretch, meaning the spring constant k varies...meaning the spring does not obey Hooke's Law over all displacements, so there really doesn't exist one single spring "constant"? As a physics teacher, which interpretation do you find more appropriate?
I notice on the graph above that the student has drawn a best-fit line and has estimated the slope to be 16 N/m. Would you consider it acceptable then for the student to conclude that 16 N/m is a good estimate for the spring constant k for this spring? Or would you rather he or she write that overall, since the spring exhibits different behavior for displacements less than 3 cm, that the spring does not obey Hooke's Law over all displacements, so it is questionable and misleading to assign a spring constant value to the entire spring?
I'm trying to figure out where the limits of acceptability lie for experienced physics teachers and for AP graders. In your post, you write "Because F = kx, the slope of this linear graph is the spring constant k." And in the graph above, the data is indeed close to linear overall, more so if you ignore the data for displacements less than 3 cm. So if we continue down that path, we might say that k=16 N/m is a good estimate for the spring constant. One problem that would then arise is that if Student A told Student B that the spring constant for his spring was approximately 16 N/m, Student B would say, "Ok, so according to Hooke's Law F=kx, I predict that for your spring, when x=4.5 cm, the force required is approximately 0.72 N." But according to Student A's best-fit line, the force required when x=4.5 cm is something closer to 0.92 N. That means Student B's predicted force based on the k value provided by Student A is over 20% off from the actual force required.
So this all begs the question, does equating the slope of the F v. x graph with the spring constant k have any meaning when the data doesn't obey Hooke's Law over all displacements, but only over a portion of the graph? And if so, how much meaning? And how far are we expected to push it when the spring starts to disobey Hooke's Law over a greater range of displacements? What if the stiffer part of the spring ranged over the first 15 cm instead of just the first 3 cm?
I read your excellent write-up for your "Hooke's Law Lab" on AP Central, and I noticed that you include the following caveat: "For some springs, the graph may not be linear for small displacements...If this happens, it is acceptable to apply just enough force to spread out the coils, then redefine the equilibrium position to the point where the coils are barely separated." That makes sense, because you are trying to isolate the portion of the graph that does obey Hooke's Law, so that estimating a value for the spring constant k is meaningful to students. But it also suggests that perhaps you’re anticipating a problem that arises when you want students to estimate a value for a constant of direct variation when the data set as a whole clearly deviates from a direct variation. If we can advise students to disregard the data for small displacements so that the concept of a spring constant is meaningful for an isolated portion of a spring, then how are we to know in other cases, when we're asked to estimate other constants of direct variation, when it is acceptable to use the slope of the entire graph as is, when it is better to focus only on an isolated portion that fits the model, and when it is best simply to observe that the data as a whole just doesn't appear to be a direct variation?
BD,
DeleteInsightful comments, of course. A fundamental skill we much teach in first year physics is to make appropriate simplifying assumptions. If we're hung up in every experiment with what small effects might possibly happen, if we're hung up on technicalities, then we're not teaching this skill.
In most F=kx experiments, everything works exactly as designed. If something weird happens - as it did here - then it's interesting *at that point* to figure out why the data clearly deviates from expectations. Even then, it's totally appropriate in this case to say "the spring constant is 16 N/m, as long as the spring is stretched past 4 cm; until then, the spring constant is bigger."
In an introductory physics lab, we're not writing a phd thesis. Most of the time, data analysis is straightforward. We try to add a wee bit of complexity at a time. There's no need to go into "what ifs" until they happen. Assume simplicity until proven otherwise. Then when complexity *does* show up, we can talk briefly about the limits of the simple model... and leave it there.
Ok, thank you once again for a clear and illuminating response. As an AP Physics tutor, I'm trying to make sure I know what level of sophistication is expected in the responses, and how much detail is required from students so that they receive full credit. As with any standardized test, just as important as understanding the concepts is also knowing what kind of language and detail the graders are looking for when awarding points to student responses. Your replies are always very thoughtful and incisive. Thank you very much.
DeleteAs an aside, having tutored math and physics for 20 years now, in the old days I used to recommend that all my students purchase test prep books from other publishers. But over the last 5-10 years, the "5 Steps to a 5" guides have clearly surpassed all others in quality and usefulness, so I now ask all my students to purchase the "5 Steps to a 5" guides as their study bibles. That, along with practicing every AP test publicly available, is my recommendation to anyone who wants to score a 5. So kudos on creating such an excellent resource!