PURPOSE: To determine the moment of inertia of a right triangular thin plate around its center of mass for two perpendicular orientations of the triangle using the parallel axis theorem.
For this lab, we first calculated the moment of inertia of a thin triangular plate about its edge, and used the parallel axis theorem to verify the moment of inertia around its center of mass.
PROCEDURE:
We used a device that rotates a disk on a cushion of air, which has a string attached to a pulley on its center. The string hangs over a frictionless pulley to a hanging mass. By measuring the angular acceleration of this system (using logger pro), we were able to find the moment of inertia of this system.
We measured angular acceleration in both the ascending and descending positions of the mass to get the average of the two, because of the frictional torque in the system. This was found by taking the slopes of the velocity vs. time graph in both directions.
We then repeated this experiment, this time adding the triangular plate first with the tall side vertical (we will call this vertical position), and then with the short side vertical (horizontal position). Both times the plate was mounted around its center of mass.
From these 3 experiments, we derived the following data:
For the system in which the triangle was mounted on its center of mass in the vertical position, moment of inertia was 0.001179 kg m^2:
For the system in which the triangle was mounted on its center of mass in the vertical position, moment of inertia was 0.001448 kg m^2:
Our experimental moment of inertia for the plate in the horizontal position was 5.402 x 10^-4 kg m^2. This gives us an error of uncertainty of 10.71%.
CONCLUSION:
By finding the difference between the moment of inertia of a rotating system and then the moment of inertia of a triangular plate mounted about its center in two positions, we were able to find the moments of inertia of the triangular plate itself. Then, we calculated the moment of inertia about the triangular plate's edge using calculus, and plugged all of this into the equation for the parallel axis theorem to compare our mathematical results for moment of inertia about the triangle's center to our experimental results for moment of inertia about the triangle's center.
In the vertical position, our experimental value was 2.712x10^-4 kg m^2 and our theoretical value was 2.44x10^-4 kg m^2, giving us an error of uncertainty of 10.02%. In the horizontal position, our experimental value was 5.402 x 10^-4 kg m^2 and our theoretical value was 6.05x10^-4 kg m^2, giving us an error of uncertainty of 10.71%.
The value of these numbers represent the quantity that it takes for the triangle to spin around a fixed axis. Our results show that it takes a higher value for the triangle to spin around its center of mass in the horizontal position, which makes sense because the mass is distributed further out away from the axis. Our errors in theoretical and experimental values could have been made from a number of factors: the mass of the triangular plate may not have been completely in uniform density; the air being released to spin the rotating disks may have fluctuated, causing acceleration to fluctuate; there also could have been error in our measurements.
For this lab, we first calculated the moment of inertia of a thin triangular plate about its edge, and used the parallel axis theorem to verify the moment of inertia around its center of mass.
PROCEDURE:
We used a device that rotates a disk on a cushion of air, which has a string attached to a pulley on its center. The string hangs over a frictionless pulley to a hanging mass. By measuring the angular acceleration of this system (using logger pro), we were able to find the moment of inertia of this system.
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| Setup of system without triangular plate added. |
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| System with triangular plate mounted about its center of mass, in vertical position. |
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| Initial system without the triangular plate. The slope of the velocity vs. time graph gives us angular acceleration in both the up and down directions of the mass. |
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| Velocity vs. time graph for the triangular plate in its vertical position. |
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| Velocity vs. time graph for the triangular plate in its horizontal position. |
From these 3 experiments, we derived the following data:
We used an equation derived from the previous lab to find moments of inertia in each system. This equation uses hanging mass, radius of pulley, and average acceleration, which canceled out the small frictional torque in the system.
We measured the hanging mass to be 0.025 kg and radius of the pulley to be 0.025 m. For the system with no triangle, moment of inertia was 0.0009078 kg m^2:
For the system in which the triangle was mounted on its center of mass in the vertical position, moment of inertia was 0.001179 kg m^2:
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We were then able find the difference of moment of inertia of the system with no triangle with the triangle in each position. Moment of inertia of the triangle in the vertical position minus moment of inertia of the system with no triangle was 2.712 x 10^-4 kg m^2, giving us the moment of inertia of the vertical triangle by itself:
Moment of inertia around the edge of the triangular plate: 1/6Mb^2.
After finding this, we used the parallel axis theorem to calculate what the moment of inertia is about its center of mass. Since we found the moment of inertia around its edge, we can subtract the mass of the triangle times the distance squared from the edge to the center of mass (Mh^2). We measured the mass of the triangle to be 0.462 kg.
Doing the same for the triangular plate in the horizontal position, we find that its moment of inertia is 5.402 x 10^-4 kg m^2:
After experimentally finding the moments of inertia of the triangle in 2 perpendicular orientations around the center of mass, we used our calculations to verify the results. Using calculus, we derived the moment of inertia for the triangle, and evaluated it with the axis at its edge because the limits of integration were simpler.
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| Calculations for determining the moment of inertia for triangular plate. |
After finding this, we used the parallel axis theorem to calculate what the moment of inertia is about its center of mass. Since we found the moment of inertia around its edge, we can subtract the mass of the triangle times the distance squared from the edge to the center of mass (Mh^2). We measured the mass of the triangle to be 0.462 kg.
In the triangle's vertical position, the base was 0.0987 meters and the distance from the center of mass to the tall vertical edge was 0.0331 meters. Plugging this in, we found the moment of inertia around the center of mass to be 2.44x10^-4 kg m^2.
Our experimental moment of inertia for the triangular plate in vertical position was 2.712 x 10^-4 kg m^2. This gives us an error of uncertainty of 10.02%.
In the triangle's horizontal position, the base was 0.149 meters and the distance from the center of mass to the short horizontal edge was 0.0489 meters. Plugging this in, we found the moment of inertia around the center of mass to be 6.05x10^-4 kg m^2.
CONCLUSION:
By finding the difference between the moment of inertia of a rotating system and then the moment of inertia of a triangular plate mounted about its center in two positions, we were able to find the moments of inertia of the triangular plate itself. Then, we calculated the moment of inertia about the triangular plate's edge using calculus, and plugged all of this into the equation for the parallel axis theorem to compare our mathematical results for moment of inertia about the triangle's center to our experimental results for moment of inertia about the triangle's center.
In the vertical position, our experimental value was 2.712x10^-4 kg m^2 and our theoretical value was 2.44x10^-4 kg m^2, giving us an error of uncertainty of 10.02%. In the horizontal position, our experimental value was 5.402 x 10^-4 kg m^2 and our theoretical value was 6.05x10^-4 kg m^2, giving us an error of uncertainty of 10.71%.
The value of these numbers represent the quantity that it takes for the triangle to spin around a fixed axis. Our results show that it takes a higher value for the triangle to spin around its center of mass in the horizontal position, which makes sense because the mass is distributed further out away from the axis. Our errors in theoretical and experimental values could have been made from a number of factors: the mass of the triangular plate may not have been completely in uniform density; the air being released to spin the rotating disks may have fluctuated, causing acceleration to fluctuate; there also could have been error in our measurements.
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