The measurements that we recorded for this experiment were done so in hopes of finding the relationship mentioned. More specific, we wanted to measure how long it takes a point on a disk to make a number of rotations in a known time interval and the Centripetal acceleration it would experience during each trial. Physics says, that any force equals mass times acceleration so then the centripetal force must equal mass times acceleration. Thus we measured the mass of our object with a digital scale and the acceleration by using the rotational angular acceleration physics formula that says radius times omega squared is equal to angular acceleration. Accordingly, we measured the force a mass would experience during radius of our object for each trial with a ruler. As for the omega, we had to record the time interval per trial and the number of revolution per that same trial to convert that information into rads per second by using mathematical knowledge. Lastly, we varied omega by changing the time intervals for several trials during the experiment to calculate the force per each trial.
This experiment began with us gathering around an apparatus that was set-up by the professor. The apparatus was a wooden wheel that spins in a circular motion supported by makeshift desk wheels and powered by an electrical scooter motor to create the spinning action. Also, a photogate sensor (which was connected to logger pro) was set-up to measure how fast a piece of tape that was attached to end of the wheel rotated in a timed interval. Once that was in place, a mass was attached to a string and a wireless force sensor (which was attached to the center of the spinning disk) and the wheel was allowed to spin. Then, the power supply was adjusted and the disk spun until it had a fairly constant speed before the force was measured for an interval of ten rotations with the photogate sensor. Since we had to vary variables independently, we recorded three good trials per each variable change. For example, the mass was simply varied be changing the object that was used and the radius was varied by changing the length of the string per its respective trial. As for omega, it was varied by adjusting the dial to the power supply so that the time intervals of the ten rotations would be faster with increased power or slower with decreased power to the electrical scooter motor. The force and time interval for each respective trial was recorded by logger pro with the help of the professor who counted the rotations with a graph that logger pro illustrated per ten rotations. Lastly, we entered the information into logger pro to create graphs and linear trend lines that illustrated the slope per each graph that helped us conclude our findings when two variables were left constant. For instance, we made one graph of force vs. mass that gave us a slope almost equal to the radius times omega squared, another for force vs. radius that gave us a slope value almost equal to the mass times omega squared, another for force vs. omega squared that gave us a slope almost equal to the mass times the radius, and two more that gave us values of slopes almost equal to radius and mass respectively.
Table #1.
| Data when only the mass was changing after 10 rotations | |||
| Variables | trial #1 | trial #2 | trial #3 |
| mass (kg) | 0.10 | 0.05 | 0.30 |
| radius (m) | 0.47 | 0.47 | 0.47 |
| power supply (P) | 3.00 | 3.00 | 3.00 |
| start time(s) | 2.68 | 1.54 | 2.35 |
| end time (s) | 17.00 | 14.97 | 16.46 |
| force(N) | 1.05 | 0.57 | 2.80 |
| time interval (s) | 14.32 | 13.43 | 14.11 |
Table #2
This is a graph of force vs. radius(omega squared) that was taken from the values of data in table #3 above that has the power source value changing only. The slope is supposed to be equal to the mass of the object that was calculated as .3 kg. The value of this slope is 0.2292.
| Data when only the radius was changing after 10 rotations | |||
| Variables | trial #1 | trial #2 | trial #3 |
| mass (kg) | 0.20 | 0.20 | 0.20 |
| radius (m) | 0.23 | 0.29 | 0.47 |
| power supply (P) | 3.00 | 3.00 | 3.00 |
| start time(s) | 1.43 | 1.73 | 1.57 |
| end time (s) | 14.77 | 14.88 | 15.60 |
| force(N) | 1.20 | 1.47 | 1.48 |
| time interval (s) | 13.34 | 13.15 | 14.03 |
Table #3
| Data when only Omega was changing after 10 rotations | |||
| Variables | trial #1 | trial #2 | trial #3 |
| mass (kg) | 0.3 | 0.3 | 0.3 |
| radius (m) | 0.4699 | 0.4699 | 0.4699 |
| power supply (P) | 3.5 | 3.8 | 4.1 |
| start time(s) | 1.24 | 1.13 | 1.46 |
| end time (s) | 11.48 | 10.74 | 10.6 |
| force(N) | 5.39 | 5.94 | 6.43 |
| time interval (s) | 10.24 | 9.61 | 9.14 |
Table #4
| Calculations of 10 rotations into Omega (rad/sec) | ||
10 rotations = 10 revolutions: 2(3.14159) rad = 1 revolution: 10 revolutions = 20(3.14159) rad
| ||
Table #5
| calculations for Omega and Omega squared taken from the Data chart for when omega was changing | |||
| trial #1 (rad/sec) | trial #2 (rad/sec) | trial #3 (rad/sec) | omega squared (rad/sec) |
| 2(3.14159)/10.24=6.14 | 2(3.14159)/9.61=6.54 | 2(3.14159)/9.14=6.87 | trail #1 = 6.14x6.14=37.7 |
| Trial #3 omega squared= 47.2 | trail #2 = 42.8 | ||
Formula one.
Force toward the center = ma = r(m)(omega squared)
Graph #1
This is a graph of force vs. radius(omega squared) that was taken from the values of data in table #3 above that has the power source value changing only. The slope is supposed to be equal to the mass of the object that was calculated as .3 kg. The value of this slope is 0.2292.
Graph #2
This is a graph of force vs. mass(omega squared). Note that the image says mass(radius squared) but that is wrong. These values were also taken from table #3. Its slope is supposed to give us a value of the radius of the object from the center of the spinning disk. The illustrated value of the slope in this image is 0.3667, and our measured radius was .4699 meters.
Graph #3
This graph illustrates force vs. omega squared and its slope is supposed to be the value of the mass times the radius. The values were taken from table #3. The slope value is 0.1078, and from the table values it should be 0.1410 kg(meters squared).
graph #4
This is the graph for when the mass of the object changed. The slope is supposed to be the value equal to the radius(omega squared), and the illustrated value is 8.869. These values were taken from table #1 and the calculated value should of been 9.072.
In conclusion, we tried to prove that the centripetal acceleration is equal to the mass*(radius)*(omega squared) by completing several experimental trials while changing one variable at a time. The graphs that were created with the respective data displayed slope values that were close to the expected values, but the slope values were never equal to the intended and calculated values. For instance, graph #1 has a slope that is supposed to be equal to the mass of the object tied to the end of the string when radius was supposed to be the only variable changing. Yet, the absolute difference was .07 kg. Also, graph #2 was supposed to have a slope representing the value of the radius of our tied object but it had an absolute difference .10 meters. Respectively, graph #3 and #4 had an absolute difference of .03 kg*m and .2 m*rad/sec^2. So the experiment did not give us exact values for the intended variables, but they were close. The inaccuracy can probably be due to several factors. One may be the that the length of the string could have been calculated with some error because it was measured with a simple ruler with a simple glance. That simple error may have carried throughout the calculations that we did for this lab and resulted in an even larger numerical value error. Another factor is that the power source that supplied energy to the scooter motor was not calibrated in a manner that we knew it was accurate in its gauge reading of energy output, so that could of resulted in the calculations that we did for omega to be wrong. For example, graph #2 represents force vs. mass*omega squared with a slope that is supposed to be equal to the radius (but is not) with an absolute difference of only two tenths of a second. Lastly, the apparatus that spun around had very unbalanced motion due to the makeshift engineering that most likely was the main cause of the faulty conclusions. It had wheels the supplied the spinning possibilities to the apparatus that seemed close to breaking that must of supplied resistance to the system, in return affecting the calculations significantly very likely by affecting the omega calculations just like the uncertain power supply connected to the system. To summarize the experimental values were not exactly the correct values that they should have been, but I believe that the experiment did prove that the centripetal acceleration is a value that can be calculated with the above mentioned formula of radius*mass*omega squared.
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