Partners: Matthew Ibarra, Billy Justin
Date of Lab: 3 October 2016
Centripetal Force
Mission Statement:To determine the relationship between the angle theta and the angular velocity of the system.
Theory:When an object is spun around a fixed axis, the object is subject to a centripetal acceleration which propels the object farther and farther from the axis of rotation until the opposing forces supplied by gravity and the string balance the outward motion.
Experimental Procedure:
In order to determine the relationship between the angle theta and the angular velocity of the system , we utilized a system involving a tripod which, when powered via electric motor, spun a ball around its axis at different angular velocities, which in turn yielded two quantities, namely the angle theta and the height h, depicted below.
From the above diagram, it is evident that the angle theta and the
height h varied depending on the angular velocity of the ball being
spun around the tripod. We first collected the data which we could
obtain directly before beginning the experiment, namely the fixed height
H, the horizontal distance R, and the length of the string L. With
these measurements, we could then proceed to symbolically solve for the angular velocity, shown below.
The above equation, w = sqrt((g*tan(theta))/(r+L*sin(theta)), does not depend on time, but rather the difference in heights (H-h) and the specific angle theta for that particular instant, shown below.
While the above equation we derived does link the angular velocity with the angle theta, thereby satisfying our mission to establish a relationship between the aforementioned quantities, we had yet to confirm if our derived equation could accurately model reality.
In order to do so, we needed to solve for omega using a different method first, then determine if the resulting value approximately corresponded with or ideally even matched the value calculated via our derived equation. This method involved counting the time transpired for the ball to complete ten revolutions around the axis and then multiplying this ratio by the unit definition of angular velocity, shown below.
This equation provided an alternative way of determining the angular velocity. Next we proceeded to watch the professor perform trial after trial, shown below.
During each new trial, the professor increased the power being provided to the spinning mechanism, thereby outstretching the ball further and changing the magnitudes of theta and h. We recorded all relevant information from each trial and then proceeded to type in this data into Excel, with two distinct columns for the omega calculated using our derived equation with respect to the angle theta and for the omega calculated using the known temporal relationship between angular velocity and the number of revolutions per seconds transpired. Below is the completed table.
The two rightmost columns, w(t) and w(h), show the calculated values using the temporal relationship and the theta relationship, respectively. We then proceeded to plot these two columns against each other and determine if the slope looked linear. If it did indeed seem linear, then this would imply that there is an approximately one-to-one correspondence between the two equations, thereby providing confirmation that our derived equation does indeed model reality effectively. Below is the graph.
From this graph, we can plainly see that the slope of the w(h) verses w(t) curve is within 5% of one. This implies that the w(h) equation, our derived equation, yields a result which is consistently 5% higher than the result we would find using w(t), the temporal relationship equation.
Experimental Procedure:
In order to determine the relationship between the angle theta and the angular velocity of the system , we utilized a system involving a tripod which, when powered via electric motor, spun a ball around its axis at different angular velocities, which in turn yielded two quantities, namely the angle theta and the height h, depicted below.
In order to do so, we needed to solve for omega using a different method first, then determine if the resulting value approximately corresponded with or ideally even matched the value calculated via our derived equation. This method involved counting the time transpired for the ball to complete ten revolutions around the axis and then multiplying this ratio by the unit definition of angular velocity, shown below.
During each new trial, the professor increased the power being provided to the spinning mechanism, thereby outstretching the ball further and changing the magnitudes of theta and h. We recorded all relevant information from each trial and then proceeded to type in this data into Excel, with two distinct columns for the omega calculated using our derived equation with respect to the angle theta and for the omega calculated using the known temporal relationship between angular velocity and the number of revolutions per seconds transpired. Below is the completed table.
Conclusion- The 5% difference between our w(h) and w(t) values can best be attributed to a number of systemic deficiencies and environmental forces which were neglected from our derivation.In terms of systemic deficiencies, the most obvious involve the oscillating, swaying motion experienced by the ruler of protruding length R when the spinning motion becomes more intense. Under such circumstances, the height H would tend to be lower than previously presumed.
Under the opposite circumstances, when the axis is being spun very slowly, the counterweight on the opposite end of the ruler of protruding length R might have actually caused the height H to be higher than previously presumed.
Furthermore, with regard to the environmental forces in play, the most obvious force we deliberately neglected is air resistance. How significantly air resistance affected the motion of the ball is difficult to discern, but it can be inferred that for the higher angular velocity trials, the air resistance mattered more.
Lastly, human error and poor timing coordination during the stop-watch phase of the lab likely contributed to the discrepancy.
Under the opposite circumstances, when the axis is being spun very slowly, the counterweight on the opposite end of the ruler of protruding length R might have actually caused the height H to be higher than previously presumed.
Furthermore, with regard to the environmental forces in play, the most obvious force we deliberately neglected is air resistance. How significantly air resistance affected the motion of the ball is difficult to discern, but it can be inferred that for the higher angular velocity trials, the air resistance mattered more.
Lastly, human error and poor timing coordination during the stop-watch phase of the lab likely contributed to the discrepancy.
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