# Falling Loop Model

The EJS Falling Loop Model shows a conducting loop falling out of a region of uniform magnetic field. It also plots the velocity of the loop as a function of time. Users can change the size the the loop, the orientation of the loop to the field, the size of the field and the location of the loop and field. Users can examine and change the model if they have Ejs installed.

## Exercises:

1. Run the simulation. The arrows show the uniform magnetic field. A conducting loop falls under the influence of gravity, but it also experiences a force as it goes from a region of no magnetic field to a constant magnetic field. What is causing this upward force? (Hint: There is an induced current in the loop. Why? This means there is a magnetic force in what direction(s) on the side(s) of the loop?)
2. Move the loop up and the field down so that the loop falls freely before entering the field. Describe your observations of the loop and the velocity versus time plot.
3. Reset the simulation. Observe the plot of the velocity versus time. How can you tell from looking at the plot that the force on the loop is non-constant for part of the fall? Once the loop is completely out of the field, the force is constant. Why? What is the acceleration?
4. Run the simulation again, but this time set the angle to ± 90o. Explain what you observe. Why does the angle matter?
5. When something is falling and its velocity stays the same, it is said to have reached terminal velocity. Reset the simulation and make sure that the angle between the loop and field is 0o (this is the angle between the surface of the loop and the field). Click on the wrench button below the velocity plot to open up DataTool (a data analysis tool). For your set-up, does the velocity seem to level off for part of the fall? If so, this is the terminal velocity of the loop as it is falling through the field. If not, change the size of the loop and run it again to get a terminal velocity. Record this value and along with the size of the loop. Run it again with different loop sizes. How does the terminal velocity depend on the loop height (size in the vertical direction)?
6. Run the simulation with the loop starting at the top of the simulation window but completely within the magnetic field (you may need to make the loop smaller or the field extent larger so the loops stays within the field for several time steps). What is the acceleration of the loop while it is completely in the magnetic field. Why? Explain the plot you get for this motion (specifically, the linear and non-linear parts)?
7. When the loop has reached terminal velocity, the acceleration due to gravity is just balanced by the acceleration due to the magnetic forces on the loop. (Why?) Show that, with an angle of 0o, the terminal velocity is given by gmR/(B2l2) where g is the acceleration due to gravity, m is the mass of the loop, R is the resistance of the loop, B is the magnetic field, and l is the length of the side that you can't adjust. Given the following values, what is the magnetic field in this simulation? m = .001 kg; R = 0.1 Ω, l = 0.1 m and, of course, g = 9.8 m/s2. Note that the plot gives you velocity in cm/s.
8. With your data showing in the DataTool, if you click-drag your mouse over a section of the plot, you can highlight that section and Fit it. From the fit, double-check that the free-fall acceleration is what you expect (the units of the velocity are in cm/s).
9. Data Analysis: Reset the simulation so that the loop's initial velocity is zero and it will begin by falling out of the field (non-constant acceleration as it begins its decent). With DataTool, try fiting the region of the curve for non-constant acceleration (make sure the Fit checkbox is checked). Because is it asymptotically reaching a constant velocity (terminal velocity), you will need to input your own equation. Select the region of the plot where the loop is experiencing a magnetic force. Double-click in the box showing the equation for a line and Fit Builder will open. Input the following equation for Line1: -a*(1-exp(-b*t)). Now, AutoFit the data to your newly defined Line1 and record the parameters a and b. As t gets very large in the equation for Fit1, show that Fit1 (=velocity) is equal to -a. Therefore, check that a is close to the terminal velocity (the units of velocity are in cm/s). Similarly, since the analytic expression for a particle with drag is v(t) = -vt(1-e-(g/vt)t) where vt is the terminal velocity, compare b to g/vt.

### References:

- Giancoli, Physics for Scientists and Engineers, 4th edition, Chapter 29 (2008).

### Credits:

The Falling Loop Model was created by Wolfgang Christian and Anne J Cox using the Easy Java Simulations (EJS) authoring and modeling tool. The exercises are by Anne J Cox.

You can examine and modify a compiled EJS model if you run the program by double clicking on the model's jar file. Right-click within the running program and select "Open EJS Model" from the pop-up menu to copy the model's XML description into EJS. You must, of course, have EJS installed on your computer.

Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.