Daniel Guzman
Lab#17 moment of inertia of a uniform triangle)
Physics 4A.
The purpose of this particular experiment is to determine the moment of inertia of a right triangle about its center of mass , for two different orientations of the right triangular thin plate.
The theory in this experiment is very interesting due that it allows one to determine the moment of inertia of an object about a point, which is not necessarily the center of mass of the object, for this experiment we used the parallel axis shift theorem, to find the moment of inertia of a right triangular plate about a center of mass; nevertheless, we found the moment of inertia of the triangle about other point and then used the parallel axis shift theorem to find the moment of inertia of the triangle about the center of mass.
The parallel axis theorem is I parallel axis= I around the center of mass + M(d parallel displacement)^2.
From this theorem one is able to determine the moment of inertia about the center of mass, if one finds the moment of inertia about the other point, so in this case the parallel axis theorem would be
I around center of mass = I around one vertical end- M(d parallel axis displacement)^2. This theorem is used in this particular experiment to find the moment of inertia of a right triangle about its center of mass when the triangle has two different orientations.
Apparatus description: The apparatus for this experiment consisted of two rotating discs a thin triangular plate, a triangular thin plate, a pulley, a hanging mass, a lab pro and a laptop. The set up for this experiment was quite simple because it only consisted on wrapping the sting around the pulley which rotates between the two rotating steel discs, the apparatus for this experiment uses air which minimizes the friction in the apparatus and allows the pulley to rotate freely, Once the string is wrapped around the pulley and the hanging mass is attached to the string, one can proceed and connect the lab pro to the apparatus and computer which is going to collect the data for the experiment, once this has been done the set up of the experiment is pretty much done, and one can proceed and collect the necessary data for the experiment.
Experimental procedure: The experimental procedure for this experiment was not very difficult due that the variations for the experiment were very simple and minimum. The only two variations for this lab were changing the orientation of the thin triangular plate which made the the procedure quite simple. The first thing we did in the experiment was to measure the mass, this is very important because the equation derived for the moments of inertia depends on this particular value. Once this measurement was made we proceeded and open the valve that lets air flow so the friction of the rotating pulley would be decreased and the friction between the two discs would be decreased as well. Once this was done and when the pulley was rotating already back and forth we started to collect the data for the apparatus itself without the the triangular plate on it. Once the angular accelerations : acceleration up and acceleration down are determined for the apparatus itself, we proceeded and put the triangle with one orientation (small base) on the apparatus and repeated the same process as we did for the apparatus itself, we determined the acceleration up and acceleration down of the apparatus, after this was determined and collected we proceeded and changed the orientation of the triangle again, and repeated the same process , we determined the angular acceleration down and up when the triangular thin plate is on the apparatus.
using the lab pro one can record the position and velocity over time graphs, and from the velocity and and time graph one can take the slope of two different segments, one that is going up and one that is going down and from this one can determine the angular acceleration up and angular acceleration down. ( This graph is just a sample graph of how the angular acceleration was determined).
Data for the experiment:
The data collected in the experiment was the angular acceleration up and down for three different variations: the first variation was for the apparatus itself, the second variation was for the triangular plate over the small base, and the third variation was for the triangular plate over the big base.
The dimension of the triangular plate also were collected and recorded: Base, Height, and the mass of the triangular plate
To calculate the moment of inertia for the three different variations we used the equation derived in class by the professor, equation that was also used for the angular acceleration lab. This equation uses the mass of the hanging mass, the radius of the pulley, and the average between the acceleration up and down, which were collected using the lap pro.
After the moments of inertia are calculated for the three different systems: 1. the apparatus by itself, 2. the apparatus with the triangle over the small base, 3. the apparatus with the triangle over the big base. I proceeded and subtracted the moment of inertia of the apparatus by itself from the variation 2 and variations 3. and in this way one is able to determine the moment of inertia of the triangle: over the small and big base.
Derivation for the moment of inertia of a triangle about its center of mass.
This is very important due that it allows one to compare the moments of inertia obtained experimentally to the moments of inertia obtained theoretically for the experiment. When comparing these two values the percent error would be expected to be very small because the pulley was assumed to be rotating without friction.
When comparing the moments of inertia found experimentally and the moments of inertia found theoretically I found that they were really close to each other, when comparing them and calculating the percent error I obtained the percent error to be less than 5%, which lets me infer that the experiment was carried out correctly.
Conclusion
The moments of inertia about the center of mass of the triangles found theoretically were really close to the experimental values for the moment of inertia, due that when comparing them i obtained percent errors that were relatively small; for the first case the percent error found was 1.67% and for the second moment of inertia the percent error found was 2.91%. These two percent errors are relatively small which lets me infer that the experiment was carried out correctly, the sources of error for this experiment are minimum and might come from making a wrong measurement or from making a wrong assumption such as assuming that the system is friction-less; even though the friction in the system is reduced by the air flow which creates an air cushion on which the disk rotate and the pulley, friction still exists; nevertheless is minimum.
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