To complete the experiment we had to use mathematics, physics, and we had to physically measure several variables in order to make needed calculations. We used a digital balance to measure the carts mass and two types of calipers to measure the diameters of the apparatus. Next, we used technology to measure the angular acceleration of the system because we needed it to calculate frictional torque of the apparatus since frictional torque is equal to the inertia of the system multiplied by the angular acceleration. Last we used physics to find the moment of inertia and frictional torque so we could use that data to calculate the time it would take for the cart to travel the previously mentioned destination and route.
For instance, we used a digital caliper and a non-digital caliper to measure the diameters of the apparatus that was composed of three pieces. As for the cart, we used a digital balance to measure the mass. Next, we calculated the volumes for each piece of the rotating apparatus in order to calculate the individual mass for each by dividing the individual volume of each by the total volume of the apparatus and then we multiplied by 100 %. Once we had the individual mass of each piece we calculated the total inertia of the apparatus in order to calculate the frictional torque that the system contained. But, for the angular acceleration of the system we used an iphone video of the apparatus spinning then we transferred that video to a computer application Logger-pro that has the capacity to calculated the angular acceleration. After that we used Newton's theorems of forces and torque to derive the acceleration of the system with a hypothetical cart of mass 0.500 kg to travel down a ramp angled at 40 degrees while connected to the apparatus. With all that data, we could know make a calculated predication for how much time it would take a cart of mass (m) to travel one meter on an angled ramp while connected to the apparatus we were using. Consequently, we used our formulas to predict how long it would take our experimental cart to travel the one meter while tied to our apparatus. Lastly, we completed three trials for the experiment and recorded the times per each trial in order to take the average value of those three quantities and divide it by our theoretical time then multiply that value by 100% to make sure we were within 4% of a deviation.
Image 1: These are the measurements and calculations for the dimensions of the apparatus. Note that mass 1 and mass 2 are equal in dimensions.
Image 2: These are the calculations for the moment of inertia and the frictional torque of the apparatus.
Image 3: This is the work to find the acceleration of the system. Note our actual cart mass and angle are illustrated on the upper right hand corner of the image. The angle was measured with an iphone compass application.
Image 4: These are the calculations to find the amount of time it take for a hypothetical cart of mass 0.500 kg to travel the one meter, and the calculations for the acceleration and time of our actual trial cart of mass 0.547 kg and chosen angle of decent to travel one meter as well.
Image 5: These are the trial times and the % error for the experiment with our actual cart.
Image 6: This is an image of the apparatus itself. Note, its mass is 4802 grams.
Image 7: This is a better look of the apparatus and the angled ramp.
Image 8: This is an image of one of the calipers that we used to measure the diameter of the small cylinder on the apparatus.
Image 9: This is an image of the apparatus set-up close to actual position that each component was in during the experiment. Note the larger caliper that we used to measure the diameter of the larger disk on the apparatus.
Image 10: This is an image during the stage when we were using Logger-pro to calculate the angular acceleration. Note the blue tape on the lower right side of the apparatus, we used the program to plot a dot on the blue tape's center mass while the video ran its length to plot a position vs. time graph that led to a graph of Omega vs. time graph to get angular acceleration.
Image 11: This is an image of the final graph that we used to calculate the angular acceleration of the system during our experimental massed car. Note to get this graph we first used a position vs. time graph, then we applied formulas to those values to get an omega vs. time graph that gave us a slope that is equal to the system's angular acceleration.
In conclusion, it seems like we did the experiment correctly because the accuracy was within 4%. More specifically it was within 1% as seen in image 5. Furthermore, the moment of inertia that we calculated (seen in image 2 ) was the same value that the professor told us it should of been so that makes me believe the calculations for the volume of the cylinders and main disk were correct. As for the forces calculations, I think they are correct because the trials that we did resulted in times very close to the calculated time of 6.67 seconds. Note, it was impressive to see that the car's decent lasted almost exactly the amount of time that we calculated. Lastly, the errors must have been few because all of the calculations resulted in correct values that matched the experimental expected outcomes like the moment of inertia value of 0.0208 kg*m^2. Although, I do believe that there was friction against the wheels of the cart during its decent but I don't know how it affected the experiment itself. Overall, I think the experiment resulted in an accepted moment of inertia value and an accepted frictional torque value because the resulting calculations that depended on those variables seemed to match what we saw in real life.
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