The purpose of this experiment was to find the moment of inertia of a right uniform triangular plate about the center of mass in the following orientations:
Apparatus:
Figure 1: Apparatus used in this experiment
The apparatus consisted of the same rotational stand used in the angular acceleration lab. However, in this experiment only one disk was rotated (by allowing air to run through the end), and the same disk was used throughout. The same torque pulley (large) was also used throughout the experiment. The counts per revolution was also set at 200 as in the angular acceleration lab. In this lab, however, the triangle holder was used instead of a drop pin. The holder was used to hold the uniform triangle used in this experiment from its center of mass, as shown in the figure above. Lastly, the hanging mass was also used to accelerate the disk, as in the angular acceleration lab.
A computer with LoggerPro was also used to collect the data recorded by the rotational sensor on the apparatus for analysis. Lastly, analytical balances and calipers were used to measure the masses of the hanging mass, the uniform triangle, the disk, and the pulley / triangle holder system as well as to measure the dimensions of the pulley, the triangle, and the disk.
Procedure:
First, the moment of inertia of a uniform triangle about its center of mass was derived. This derivation was used to determine the accuracy of the moments of inertia obtained experimentally in this lab. However, the parallel axis theorem was used in this derivation, because it turns out that the calculus involved in finding the moment of inertia directly at the center of mass is complex. Therefore, it was easier to find the moment of inertia of the triangle about its edge and then use the parallel axis theorem to find it about the center of mass.
Figure 2: Parallel-Axis Theorem for a uniform right triangle
Using the parallel axis theorem, the moment of inertia of the triangle at the edge can be found and then subtracted by the mass of the triangle multiplied by the distance from the edge to the center of mass squared. That distance d, based off of center of mass in the x direction, is b/3, where b is the length of the base of the triangle.
Figure 3: Derivation of the moment of inertia of the right triangle about its center of mass
The moment of inertia about the edge was found to be (1/6)(m)(b^2), and the moment of inertia about the center was found to be a third of that, or (1/18)(m)(b^2).
Next, the masses and dimensions of the parts of the apparatus were measured. The mass of the triangle was recorded as 456 g. The shorter leg was found to be 9.80 cm, and the length of the longer leg was found to be 14.85 cm. The mass of the large torque pulley used to wind and unwind the string attached to the hanging mass was found to be 36.3 g, and it diameter is 5.36 cm. The mass of the rotating disk used in the apparatus (the large steel disk) was found to be 1.360 kg, and it diameter was 12.630 cm. Lastly, the mass of the triangle holder was found to be 26 g.
Then, once the equipment was all set up, the air was turned on to the apparatus with the hose clamp open so that the bottom and top disk rotated independently of each other. Then, once the string was wrapped around the torque pulley and the hanging mass was at its highest point, the data measurements were started and the mass and disks were released. Using the angular velocity graphs obtained from the experiments, the angular accelerations of the rotating disks as the mass was going upward and as the mass was going downward were found by looking at the slopes of the angular velocity graphs. The angular velocity graphs obtained as well as data obtained for each experiment is found below. In the first experiment, the triangle was not attached; the only attached object was the triangle holder. In the second experiment, the triangle was attached and oriented such that the shorter leg was parallel to the floor. In the third experiment, the triangle was also attached and oriented such that the longer leg is parallel to the floor.
Figure 4: Angular velocity graph of the first experiment
Figure 5: Angular velocity graph of the second experiment
Figure 6: Angular velocity graph of the third experiment
As can be seen, the angular acceleration as the hanging mass was moving up is not the same as when it was going down, due to the fact that there is frictional torque. To lower the effect of the frictional torque onto the results, the same approach will be taken as in the angular acceleration lab, which is to take the average of the magnitudes of the angular accelerations.
Discussion:
The angular acceleration of the system without the triangle attached was found so that the moment of inertia of the triangle in each orientation can be found. Using the angular accelerations, the moment of inertia of all three systems can be found. Because of this, the moments of inertia of the triangle in the two orientations alone can be found by subtracting the moments of inertia of the systems with the triangles by the moment of inertia of the system with no triangle.
The theoretical moments of inertia of the triangle in each orientation can be found using the derived equation and the dimensions measured. However, to determine the accuracy of this value, the moments of inertia of the triangle in those orientation are found experimentally using the procedure explained above. To find the experimental moments of inertia, the same derived equation was used as in the angular acceleration lab, displayed below,:
1.
where r is the radius of the torque pulley, and m is the mass of the hanging weight. Using the experimental angular acceleration data and the measurements of the dimensions. the actual moment of inertia of the system with no triangle attached was 0.00294 kg m^2, obtained as follows:
2.
To find the moment of inertia of the triangle when the base was parallel to ground, the moment of inertia of the system with the triangle attached was found.Then, the experimental moment of inertia of the triangle was obtained by subtracting the moment of inertia of the system with the triangle and the moment of inertia with the triangle. Using the same method used above from the moment of inertia without the triangle, the moment of inertia of the systems in this experiment are tabulated below.
Figure 7: Experimental moments of inertia of the three systems
Subtracting the moments of inertia of trial 2 and 3 by trial 1, the experimental moments of inertia of the triangle in the two orientations was found:
Figure 8: Experimental moments of inertia of the triangle in the two orientations about its center of mass
It was expected to find that the moment of inertia of the triangle when the longer leg is parallel to the ground (its base) is higher than the moment of inertia when the shorter leg is parallel to the ground. This is because the moment of inertia of the triangle depends on its mass (which stays constant) and its base; the triangle with the longer base, by the equation, should have a larger moment of inertia than the triangle with the shorter base. a sample calculation for finding the experimental moment of inertia of the triangle is found below:
3.
To measure the accuracy of the experimental moments of inertia of the triangle, the theoretical moments of inertia were computed using the derived equation for the moment of inertia of a triangle about its center of mass, and were compared to the experimental values. The theoretical values are displayed below, along with a sample calculation for finding the theoretical moment of inertia.
Figure 8: Theoretical moments of inertia of the triangle in the two orientations about its center of mass
4.
Conclusion:
The theoretical and experimental moments of inertia of the triangle about its center of mass when its shorter leg is parallel to the ground are very similar; there is only a 4.12% percent error of the experimental value, showing that the method used to find the moment of inertia of the triangle in that orientation was accurate and a useful method. This also proves that the moment of inertia derived for a triangle rotating about its center of mass is the correct formula.
On the other hand, the difference between the experimental and theoretical moments of inertia of the triangle in the orientation where the longer leg is parallel to the ground is fairly large (10.2% percent error). This is mainly due to error in the angular acceleration values obtained during the experiment. In addition, because the values used in this experiment are small, any small change in the angular acceleration would have a large effect on the accuracy of the results and would cause the difference to be larger. The uncertainty in the measurement of the length of the longer leg is very small, having little effect on the error in the experimental moment of inertia.
Because the experimental value for the moment of inertia of the triangle with its longer side parallel to the ground is larger than the theoretical value, the angular acceleration obtained was smaller than actual. This error could have resulted from friction in the disks due to not having the air turned on as much as needed. Another source of error could be that the triangle was not perfectly oriented such that the longer leg was not parallel to the ground, resulting in a different value range obtained for the moment of inertia. In addition, as seen in the angular acceleration lab, the rotational sensor and the LoggerPro software already contain uncertainties in their measurements of the angular displacement and angular velocity of the rotating disks.
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