1. Attach a piece of cardboard to the top of your glider, the cardboard should be attached throughout the experiment.
2. Weigh: the glider
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(with cardboard surface attached), two light springs, and magnets. Add the mass of the glider and the magnets to 1/3 the mass of the spring, which we will call msys.
3. Measure k for both springs by measuring the length of the spring at one weight, and another weight, and than using the formula: T^2 = (4PI^2/k)*m +
[(4/3)PI^2/k]
*mspring, and add the two k values (spring constants) together, which we will call keff.
4. Set up the sonic ranger as required. *Note: Be sure to align the sonic ranger with the air track
5. Calibrate the sonic ranger by first turning on the air track, then placing the glider on the track. Place a meter stick between the sonic ranger, which is attached to the rod and clamp setup, and the perpendicular cardboard surface, which is attached to the glider. Then run the Science Workshop (sonic ranger package) software to calibrate the distance to be about 1.00 meters.
6. Attach the springs to the glider, as well as to the two ends of the air track as required.
7. Allow the glider to settle into it's equilibrium position, and using a meter stick note the distance from the glider to the sonic ranger, which we will call B.
8. Use the Position vs. Time application in Science Workshop and prepare to record data using a graph as well as a table.
9. Attach the magnets in various positions on the angled base of the glider.
10. Using the tape measure that is on the air track, pull the glider out of its equilibrium position and let go of the glider. Click record in Science Workshop. *Note: Be sure to record the distance you pulled the glider, which we will call A and do not let the glider have any initial velocity.
11. As the glider is oscillating Science Workshop records the distance of the glider every 1/40th of a second.
12. Allow Science Workshop to record several periods, you should take note of how the periods look. If a linear damping force does not occur, adjust the magnets until a desired result is achieved. *Note: The linear damping force is created when the magnets move along the metallic track. The magnets produce tiny electrical currents in the track due to electromagnetic induction.
13. The resulting data is the position vs. time as shown in the Data portion of this report.
14. Select roughly 200 points from the acquired data and Copy and Paste these points over to an Excel spreadsheet.
15. Use these 200 points to create a graph which should give us a nice graphical representation of the linear damping force.
16. Select three peaks from your graph and use these three points to calculate R1, R2, and R3, using the formula: -R/2msys*t=ln(x-B)/A. R is the strength of the damping. *Note: Be sure to change the values of x(position) and t(time) with respect to R.
17. Use your 200 points to select roughly 50 points that best represent the resultant curve. Select a couple points in-between each peak and valley, and about ten points in each peak and valley.
18. Circle these points on the graph, and highlight them on the spreadsheet.
19. Use MathCad to fit your position vs. time data into standard form.
20. Simply open the Damped Harmonic Motion application and plug in your 200 points, value for R (either an average or R2), keff, A, B, C (which is zero, unless the sonic ranger was not in line with the direction of motion of the glider), msys, and F (which is 0 if your starting point is a peak, and 3.14 if your starting point is not a peak).
21. Compare the experimental values of A, msys, and keff with the best-fit values obtained by the computer.
Analysis
K1 = M (delta) g / x5 - x4
K1 = M (delta) g / x5 - x4
Keff = K1avg + K2avg
Msys = mg + 1/3 (mk1 + mk2)
X = Ae^(-Rt/2m) * cos(w't + Phi ) + B
X = Ae^(-Rt/2m) + B
ln(x1-B)/A = ln(e^(-Rt1/2m) = Rt1/(-2Msys))
A: this is just the initial amplitude of the oscillation.
k: use the effective spring constant of the two-spring system
B: this is an overall constant which reflects the distance between the glider and the sonic ranger when the glider is at the equilibrium position. This can be measured with a ruler.
C: this is zero unless the axis of the sonic ranger is not in line with the direction of motion of the glider.
R: this parameter controls the strength of the dampening.
m: use the mass of the glider, plus 1/3 of the mass of both springs.
y(R,m,A,K,B,C,t) = Ae^(-Rt/2m) * cos( (k/m - (R/2m)^2 * t)^(1/2) + Phi ) + B + Ct
Discussion
