The purpose of this experiment was to find the period of physical pendulums of different shapes that oscillate at small angles.
Apparatus:
Figure 1: Apparatus of the first portion of the experiment
The apparatus of the first part of the experiment involves two ring stands, with a knife edge attached horizontally. The knife edge is used to hold up the ring that is concaved to half its thickness at a part of the inner side. There is another ring stand, with a photogate attached. Tape is attached to the ring so that the photogate can detect the position of the ring. The photogate measures how much time it takes for the physical pendulum to pass the photogate twice. This relates to the period of the pendulum, where it reaches the amplitude in the positive and negative direction from equilibrium and reaches equilibrium. A computer with LoggerPro was also needed to collect the data from the photogate and provide the period of the system.
Figure 2: Apparatus of the second portion of the experiment
The second apparatus of this experiment involves the same setup, except that different physical pendulums are attached. A semicircle by the center of its diameter was pivoted, and it was also pivoted for a second trial by the highest tip of the curve.
Procedure (Part 1):
First, the moment of inertia of the ring was obtained using the following calculation:
Figure 3: Finding moment of inertia of the ring
The ring was treated as a cylindrical shell, which has a moment of inertia of 1/2MR^2 rotating about its center. Then, the parallel axis theorem was used to find the moment of inertia halfway through its thickness. Using the calculated moment of inertia, torque was used to find the angular acceleration of the pendulum while oscillating at small angles. The form of alpha = omega^2 * theta was found, and the angular velocity of the system of obtained using this form. Then, the period was found by dividing 2pi by the calculated angular velocity; the theoretical period was found to be 0.717 seconds.
Next, the actual experiment was performed where the ring was oscillated at small angles, and the experimental period was measured by the photogate. Once the equipment was set up as shown in the apparatus, the experiment was performed, and the result is shown below:
Figure 4: Experimental period of the physical pendulum involving the ring
Discussion (Part 1):
The experimental period of the physical pendulum with the ring was found to be 0.720 seconds. The percent error of the period of this physical pendulum was calculated below:
1.
As can be seen, the percent error in this part of the experiment is very small, indicating that the theoretical period is the true period. It also shows that the assumption that when the angle is small, sin theta is almost equal to theta is also a good assumption. Lastly, it also shows that the method used to obtain the period is also accurate and correct.
Conclusion (Part 1):
One reason that the experimental period value is slightly larger than the theoretical is due to uncertainties in the measurements of the dimensions of the ring. In addition, there is also uncertainty in the measurement of the period by the photogate. Also, some friction is present in the pivot where the ring is in contact with the knife edge, lowering the angular frequency by a little and resulting in a higher frequency. Lastly, air resistance as the pendulum is oscillating can result in a slower angular velocity and a larger period. As can be seen though, because the percent error is very small, these errors and uncertainties have little effect on the system.
Procedure (Part 2):
First, the center of mass of the semicircle pendulum was obtained, using the procedure shown below:
Figure 5: Deriving the center of mass of a semicircle
Picking dm as a strip of the semicircle, the center of mass in the x direction was zero, based on an origin in the center of the diameter. The center of mass in the y direction was found to be 4R/3pi.
Next, the moments of inertia of the semicircle about the center of the diameter, the center of mass, and the tip of the curve were obtained, using the same procedure as in part 1.
Figure 6: Deriving the moment of inertia of a semicircle by the middle of the diameter
Using semicircular shells, the moment of inertia of the semicircle was found to be 1/2MR^2. Next, the moment of inertia of the semicircle at a pivot at the center of mass was found using the parallel axis theorem, and its derivation is found below:
Figure 7: Deriving the moment of inertia of the semicircle about its center of mass
Lastly, the moment of inertia at the tip of the curve of the semicircle was found using the parallel axis theorem again.
Figure 8: Deriving the moment of inertia of a semicircle about the highest tip of the rim
Then, using the derived moments of inertia, the periods for the semicircle pendulum oscillating about the center of the diameter and the tip of the rim were calculated, as shown below:
Figure 9: Deriving period of pendulum oscillating about the center of diameter
The same was performed for the semicircle oscillating about the highest tip of the rim.
Figure 10: Deriving period of pendulum oscillating about the tip of the rim
Next, the shapes were measured and cut out from foam, and the actual experiment was performed, as in part 1, and the actual periods were obtained. The period found for the pendulum about its center of diameter is shown below:
Figure 11: Actual period of oscillation about center of diameter
Likewise, the period is shown below for the pendulum about the tip of the rim:
Figure 12: Actual period of oscillation about tip of rim
Discussion (Part 2):
The percent error between the actual and theoretical values for the period of oscillation about the center of diameter was found to be 4.8%. Likewise, the percent error for the the period of oscillation about the tip of the rim was calculated as -4.7%. The percent error is high for a system such as a physical pendulum. These high percent errors, along with the percent error of the first part, show that the errors in this experiment are resulting from human error, and not error from the path taken to calculate the theoretical values and the procedure used.
Conclusion (Part 2):
The reason for such a high percent error is that, due to the lack of time, the physical pendulums were not pivoted correctly, resulting in the wobbling of the pendulums and movement back and forth, instead of only side to side movement like a regular pendulum. In addition, there was some noticeable friction at the pivots, resulting in the physical pendulums slowing down over time during the experimental gathering of data. To make this experiment better, better pivots should be used for the semicircle pendulums that does not result in the wobbling of the pendulums, and more time would be needed to perfect the experiment. Lastly, more dense material should be used for the pendulums instead of foam), giving the pendulums more control during the oscillations.
