The purpose of this experiment was to use our torque equation to ultimately derive an acceleration for an object and compare it to its experimental value.
PROCEDURES
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| Figure 1 |
Before setting up our experiment, we made some calculations to find the moment of inertia of an inertia wheel. We did this by imagining that the inertia wheel was composed of two cylinders and a disk. We found the moment of inertia of each component and added them up to find the total moment of inertia of the entire inertia wheel. However, in order to do, we had to first find the mass of each component. We did this by measuring their diameters and thicknesses and calculating their volumes. We then used the ratios of their volumes with respect to the total volume of the inertia wheel and set them equal to the ratio of their masses over the inertia wheel's total mass. With these masses, we were able to find the moment of inertia of each component by assuming that they were all disks and using the equation I = ½MR². When we added these individual moments of inertia, we got a value of 0.020439 kg*m². This whole process is shown in Figure 1.
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| Figure 2 |
We began the procedures of this experiment by giving the inertia wheel a spin and allowing it to come to a stop by itself. We captured the motion of the wheel with a camera (Figure 2) and used video analysis to measure the negative angular acceleration caused by the axle.
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| Figure 3 |
When we analyzed the video, we used the piece of tape on the wheel as a reference point and placed a data point on the top right corner of it every time the wheel completed a rotation (Figure 3). We set the origin at the axle and rotated the xy-plane until the y-axis was somewhat lined up with the data points, as shown in Figure 3.
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| Figure 4: Angular displacement vs time |
Next, we created a "Manual Column" in Logger Pro called "theta" and plugged in 0, 2π, 4π, and 6π, which corresponded to each time the inertia wheel made a rotation. Then, we plotted "theta" versus time as shown in Figure 4. We applied a "Quadratic Fit" to the graph. We noticed the resulting equation was one of the constant angular acceleration equations that we learned in class i.e. Δθ = ωot + ½αt². From this relationship, we were able to deduce that A was equal to ½α. Therefore, we found α by multiplying A by 2, which was -1.0322 rad/s². From this value, we were able to find the negative torque caused by the wheel's axle by multiplying it by the wheel's moment of inertia (I = 0.020439 kg*m²). We found this value to be -0.02114 N*m.
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| Figure 5 |
In order to predict the time it would take the cart to travel 1 m down the track, we applied Newton's second law of motion on the cart in the x-direction and the torque equation on the wheel as shown in Figure 5. We assumed that the only two forces acting on the cart in the x-direction were the tension (T) in the string and the horizontal component of the cart's weight. We set up the force equation according to this assumption and set everything equal to T. Then, we set up the torque equation on the wheel. Besides the negative torque caused by the axle, the only other force contributing to the torque on the wheel was T. In addition, we related the angular acceleration (α) to the translational acceleration of the cart (a) with the equation a = rα. Then, we set the resulting torque equation equal to T as well and set the force and torque equations equal to each other. This got rid of T and we were left with only a as an unknown. This allowed us to solve for a, which we found to be 0.02101 m/s². We used this value in the kinematic equation to solve for t (the time taken for the cart to travel 1 m down the track), which was 9.756 s.
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| Figure 6 |
After completing our calculations, we set up the track as shown in Figure 6. We wound up the string on one of the smaller "cylinders" and released the cart to allow it to accelerate. We used our phones to measure the time it took for the cart to travel down 1 m. We measured this time to be 10.58 s. Compared to the theoretical value found earlier, the percent error was 8.445 percent. This was a bit higher than what we would have liked (~5 percent). However, we concluded that the results were acceptable because there were several factors that probably slowed down the cart that we did not consider in our calculations.
CONCLUSION
As mentioned before, the percent error was slightly higher than we would have liked. This was a result of several factors. One of the biggest sources of error probably came from the way we measured the time it took for the cart to move down 1 m. First of all, the response time of starting and stopping the "stopwatch" added to the overall time. In addition, it was difficult to determine exactly when the cart passed the 1 m mark since the cart was moving at a relatively fast speed by the time it reached that point. Another factor that contributed to the error came was the friction between the string and the cylinder as the cylinder was assumed to be a frictionless "pulley" in our calculations.
Despite these sources of error, our experiment was a moderate success as our percent error was not extremely high. Furthermore, we gained valuable experience in implementing what we learned about moment of inertia's and torque in a real-life experiment. It was interesting to see that our calculated values were close to the experimental values.
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