Magnetic Brakes: Terminal Velocity & Lenz’s Law
Introduction
If you haven’t read my last post covering Lenz’s law and eddy currents, I recommend doing so. That post covers all the background information for this experiment, along with details about the setup.
Okay, now that you’ve done that (or are rebellious), let’s get back into the project. In this post, I’ll predict and experimentally verify the drag force exerted on the magnet by the eddy currents at terminal velocity. This will be done for both the copper and aluminum pipes.
Predicting Drag Force at Terminal Velocity
Terminal velocity occurs when an object in freefall ceases further acceleration, proceeding at a constant velocity. The net force on any object is simply the sum of all forces acting upon it, so the net force acting on an object at its terminal velocity is zero. Therefore, the gravitational force (Fg) drawing the object to the ground is equal to the forces opposing the fall. I’ll be notating the eddy current-imposed drag force on the magnet as Fec from now on. To calculate what Fec will be at terminal velocity, I did the following:
The predicted value for Fec is 0.0588 N for the magnet at terminal velocity in both the copper and aluminum tubes. So now, we can move to verify this prediction.
Finding Terminal Velocity Experimentally
To find the terminal velocities experimentally, I’ll need another set of drops from a different height. Since I don’t have more tubes of the same diameter and thickness, I’ll put a 60.5 cm rod inside each 104.5 cm tube and record the time the magnet takes to fall 44 cm to land on top of the rod. Then, the following equation can be used to calculate the terminal velocity of the magnet in each pipe:
L1 and t1 are the lengths and average time of the runs with no rod inserted (from my previous post), and L2 and t2 are the lengths and average time of the runs with the rod inserted, respectively. We will be left with a length of tube where the magnet is at terminal velocity divided by the average time it takes to travel that length, resulting in the average terminal velocity. To recap, the average times in 104.5 cm of each tube over five runs is as follows:
Aluminum tube: 20.95 s | Copper tube: 21.56 s
Thus, the average velocity for each is:
Aluminum tube: 0.049 m/s | Copper tube: 0.048 m/s
Moving on to the 44 cm tests, the average results over five runs are as recorded below (accounting for a .2 s reaction time).
Aluminum tube: 8.85 s | Copper tube: 9.23 s
Using the experimental formula for terminal velocity, we find the following for the aluminum tube:
… and the following for the copper tube:
The calculated terminal velocities of each metal are about .001 m/s higher than the recorded average velocities over the whole length of the tubes. Since the magnet reaches terminal velocity very quickly, this result is expected, but I refuse to believe my measurements are so precise as to uncover a 1 mm/s difference in velocity. Researcher’s bias, I guess, but I’ll roll with it.
Calculating Eddy Current Force using Terminal Velocity
Now that I have the terminal velocities, I should be able to calculate Fec with the results using the drag force equation:
In this formula, p is the density of the surrounding fluid, v is the object’s velocity, Cdrag is the drag coefficient, and A is the cross-sectional area of the object. This is usually used to calculate the drag on objects inflicted by the fluid surrounding it (most often air), but it can be adapted for what we want to use it for. As I said earlier, the air drag on the magnet is negligible; this is because Fec is so much greater than the air drag. Therefore, p and A will be disregarded.
Ah, that looks much better. But wait, what’s that? There’s still an unknown coefficient in the formula!
Fear not, and behold the terminal velocity equation.
Once again, we will disregard p and A. Let’s start with the aluminum tube.
Now we may return to the drag force equation with our newly found coefficient:
Now for the copper tube:
And there we have it: a Fec of 0.0587 N for both tubes. This is only off by 0.0001 N from our prediction, presumably from ignoring both friction and air resistance.
Discussion and Conclusion
This was just a fun little exercise to essentially test the accuracy of my measurements, which turned out quite nicely. I don’t currently plan on making another post about Lenz’s law and eddy currents, but if you’ve got an idea, shoot me a message using the contact page. I’m going to be doing something more theoretical for my next post, but you’re just going to have to wait to find out what it is.
