Purpose:
The purpose of this experiment was to derive expressions for the period of various physical pendulums and to verify the predicted periods by experiment.
Part 1:
Apparatus:
The apparatus of the first part of the experiment consisted of two stands, with a knife edge attached horizontally. The edge was used to hold the ring in place as it oscillated. A photogate was attached to one of the stands. A tape marker was attached to the ring so that the photogate would track its movement. In this case, the photogate measured the period of the oscillating ring. In addition, we also used a computer with LoggerPro to collect and interpret the data from the photogate.
Procedure:
First, we calculated the moment of inertia of the ring. We treated the ring as a hollow cylinder rotating about its center. Then, we used the parallel-axis theorem to find the moment of inertia from its new pivot point.
The dimensions of the ring and equations used can be seen below.
- I(ring)= 0.00328 kg*m^2
Next, using the calculated moment of inertia and torque, we were able to determine the period of the ring, while oscillating at small angles.
First, we drew and RBD of the system and setup its respective equations. In this case, we used Torque=I*alpha.
Next, we solved for alpha and noticed that it took the form : acceleration=-constant*displacement
From here, we can interpret it as: acceleration=-Omega^2*displacement
Finally, we solved for omega and inputted into T=(2*pi)/Omega.
As you can see, our theoretical value for the period of the oscillating ring is 0.7162 seconds.
Next, we verified the predicted period by running the experiment. During the experiment, the ring was oscillating at small angles and the photogate was used to measure the period. The result can be seen below:
To recap:
- Theoretical Period:0.7162 sec
- Experimental Period:0.7211 sec
Conclusion:
The percentage error for this experiment was 0.7%. This shows us that the theoretical period is accurate. Furthermore, it also proves that the method used to obtain the period is accurate. The discrepancy between the two values can be the result of uncertainty in the dimensions of the ring. In addition, there can also be an uncertainty from the data recorded by the photogate. Fortunately, these errors are small enough that they are negligible for this experiment.
Part 2:
Apparatus: The same setup was used for the second part of the experiment. However, for this part of the experiment, we used the following physical pendulums:
- Isosceles triangle, base B, height H, oscillating about its apex (Trial 1)
- Isosceles triangle, base B, height H, oscillating about the midpoint of its base (Trial 2)
- Semicircular plate of radius R, oscillating about the midpoint of its base (Trial 3)
- Semicircular plate of radius R, oscillating about a point on its edge, directly above the midpoint of the base (Trial 4)
 |
| Trial 4 |
 |
| Trial 3 |
 |
| Trial 1 |
 |
| Trial 2 |
Procedure:
First, we measured the appropriate dimension of each shape. Next, we used electrical connectors and a pin to setup the pivot points for each object. This allowed the object to oscillate freely. Like in Part1, we attached a tape marker to each object so that the photogate would pickup its movement.
Next, we derived the center of mass of each shape and used this information to derive the moment of inertia of both the triangle and semicircle from each orientation.
 |
| Center of Mass of Semicirlce |
- Semicircle Center of Mass: 4R/(3*pi)
- I(cm): 1.20*10^-5 kg*m^2
- I (trial 3) = 1.88*10^-5 kg*m^2
- I(trial 4)= 2.45*10^-5 kg*m^2
Next, we determined the periods of the oscillating semicircle. First, we drew a RBD of the system and set up the sum of torques equation. Then, we followed the same steps as in Part 1 and setup the equations in the following format:
acceleration=-constant*displacement & acceleration=-Omega^2*displacement
From here, we solved for omega and inputted into T = (2*pi)/omega
 |
| Trial 3 |
 |
| Trial 4 |
As you can see, we solved for the period of the semicircle in both orientations.
Next, we conducted the experiment as shown above and recorded the period of the oscillating semicircle. The results can be seen below:
 |
| Trial 4 |
 |
| Trial 3 |
To recap:
Next, we found the periods of the oscillating isosceles triangle. The same steps were used as before. First, we found the center of mass of the triangle. Then, we found the moment of inertia of the triangle about its center of mass. Next, we used the parallel axis theorem to find the moment of inertia about its new pivot points.
- Triangle Center of Mass: (1/3)H
Above you can see how we derived the moment of inertia of the triangle about its apex.
Next, we used the parallel axis theorem to find the moment of inertia about its center of mass. Then, we used the parallel axis theorem once more to determine the moment of inertia about its base. These derivations can be seen below:
Next, we determined the periods of the oscillating triangle. First, we drew a RBD of the system and set up the sum of torques equation. Then, we followed the same steps as in Part 1 and setup the equations in the following format:
acceleration=-constant*displacement & acceleration=-Omega^2*displacement
From here, we solved for omega and inputted into T = (2*pi)/omega
 |
| Trial 1 |
 |
| Trial 2 |
Finally, we ran the experiment as shown above and recorded the period of the oscillating triangle. The results can be seen below:
 |
| Trial 1 |
 |
| Trial 2 |
To recap:
Trial 1:
- Theoretical Period: 0.748 sec
- Experimental Period: 0.784 sec
- Percentage Error: 4.8%
Trial 2:
- Theoretical Period: 0.640 sec
- Experimental Period: 0.661 sec
- Percentage Error: 3.2%
Conclusion:
The margin of error for the second part of the experiment is low. This shows us that the theoretical periods are accurate. Furthermore, it also proves that the method used to obtain the periods is correct. The difference in the values can be the result of uncertainties present during the experiment. In this experiment there is uncertainty in the mass and dimensions of both the semicircle and triangle. Lastly, there could also be a source of error if the physical pendulums were not pivoted correctly. Since the margin of error is low, we can assume that these factors are negligible in this experiment.
No comments:
Post a Comment