a) Understand the presence of induced Eddy currents
b) Determine a damping coefficient
c) Determine the terminal velocity
As the cart moves along the aluminum track, the magnets on the accessory bracket induce eddy currents in the track. This causes opposing magnetic fields that result in a drag force applied to the cart. The magnitude of this force is measured by analyzing the deceleration of the moving cart. The relationship between the drag force and the speed of the cart is determined.
Equations of Motion
We will assume that the magnetic drag force, f, is proportional to the velocity, v
f = -b v
where b is the drag coefficient.
On a level track, assuming no other sources of friction, the drag force is the only force acting on the cart along its axis of motion. Therefore, by Newton's 2nd Law,
m a = -b v (1)
where m is the mass of the cart and a (the acceleration) is the derivative (a=dv/dt) of the velocity.
Thus Equation (1) can be written as
The solution to this differential equation is
NEEDED, BUT NOT INCLUDED:
Figure 1: The Magnetic Damping Accessory attaches to the end of the cart that does NOT have the plunger. The steel bracket is held in place by the magnets inside the cart. It slides up and down to adjust the amount of drag: The closer the three silver magnets are to the track, the more damping there is.
1. Set up the track as shown in Figure 1, including feet and elastic bumper.
2. Connect the Motion Sensor to the interface, and attach it to the track. Adjust the alignment knob on the side of the Motion Sensor so that it points parallel to the track. Make sure the switch on the top of the Motion Sensor is set to "cart."
3. Set the cart on the track (including the two extra masses), but do NOT attach the Magnetic Damping Accessory yet.
4. In PASCO Capstone, leave the sample rate at the default rate of 20 Hz.
5. Create the following calculations in the PASCO Calculator:
speed = abs([Velocity]) with units of m/s
mass = 0.253 (or whatever mass you’re using) with units of kg
Mag Force = -mass*derivative(6,[Velocity],[Time]) with units of N
6. Create a graph of speed vs. time.
7. Create a graph of Mag Force vs. Velocity.
1. Starting with the cart near the Motion Sensor, give it a small push away from the sensor, and click on Record. Your data will look better if you stop recording before the cart hits the bumper.
2. Using the screw feet, adjust the level of the track so that the cart travels at a constant speed when moving away from the Sensor. By setting up the track so that it is slightly downhill, you will be measuring only the magnetic drag on the cart. The unwanted frictional forces will have been compensated for.
3. Attach the Magnetic Damping Accessory to the cart (see Fig. 1) and determine the cart mass, with and without the two extra masses.
4. The amount of magnetic damping is determined by the distance between the magnets and the track (see Fig. 2). Use some type of spacer (any non-magnetic material will work) to set the distance to about 3 mm. Slide the bracket down until the magnets are flush with the spacer, then remove the spacer.
Figure 2: Magnetic Spacing
5. Place both masses in the cart. Starting with the cart near the Motion Sensor, give it a small push away from the sensor, and click on Record. Your data will look better if you stop recording before the cart stops moving.
6. Open the Calculator in the Tool Pallet. The Magnetic Force is calculated using Newton's 2nd Law:
Mag Force = ma
where m is the mass of the cart and the acceleration (a) is calculated using a derivative of velocity. Line 2 in the calculator shows the approximate mass of the cart, but you can change it to your cart mass.
7. The graph of the calculated magnetic force versus the cart's velocity should be a linear relationship, although the data may be noisy. You may have to take several runs to get a good looking graph.
8. Select a linear curve fit from the Graph Tool Pallet. Is your graph linear?
9. The slope of the graph is called the Drag Coefficient. Record this value.
Equations of Motion
The velocity vs. time graph shows your data from the previous page.
Does the velocity decay exponentially? Select an exponential curve fit from the Graph Tool Pallet.
1. Create a graph of velocity vs. time. Select the exponential curve fit for your data. The constant in the exponential for the curve fit is b/m. Record this value. What are the units?
2. Record several more runs of data, and record the values of b/m for each. Put all the data into a table column and turn on the Statistics with the Mean and the Standard Deviation.
3. Use the average value of b/m and the mass of your cart, m, to calculate the drag coefficient, b.
4. What are the units of b?
5. Use the standard deviation to calculate the uncertainty in your value.
6. How does this value for b compare to what you measured earlier from your Mag Force vs. Velocity graph?
Terminal Velocity: Inclined Track
As the cart accelerates down the incline (see Fig. 4), the magnetic drag force (f=bV) increases. Eventually, this backwards drag force grows to equal the component of the gravitational force down the plane (mgsinθ), and the cart reaches terminal velocity, (VT). Assuming no other forces (such as air friction, etc.),
bVT = mgsinθ
which can be written as
VT = (mg/b) sinθ (3)
1. Remove the two masses from the cart. You will use them to incline the track as shown in Figure 3. Be sure NOT to move the Magnetic Drag Accessory, or you will have to re-measure the coefficient.
2. Use calipers to measure the height of the two stacked mass bars, and a meterstick to measure the distance between the feet. Calculate the angle of incline for the track.
3. Use Eqn. (3) to predict the terminal velocity of the cart on the inclined track. Hint: The mass, m, has changed from the previous part of the experiment.
Figure 3: Inclined Track Figure 4. Free-body Diagram
1. Starting with the cart about 20 cm from the Motion Sensor, release the cart from rest, and click on Record. Your data will look better if you stop recording before the cart hits the bumper.
2. Examine the velocity vs. time graph. What is the terminal speed of the cart?
3. How does it compare to your predicted value?
Summarize your findings by reviewing goals in a quantitative way.
Introduction, setup, procedure
Data collection and graphs
Drag Coefficient by two methods
Terminal Velocity Analysis