Interpretation of Data
The purpose of this lab was to determine the difference between starting heights of a ball when friction is not taken into account (our theoretical part), and in the experimental case when friction is accounted for in the ball's rotation. In the experimental case, we will use the law of conservation of energy with a kinetic energy of rotation term added to find what the calculated experimental starting height should be.
In the graphs of our data, we found that the actual experimental starting height needed for the ball to make it successfully around the loop was 0.425 m. This height is the actual height at which the ball was started, although it appears in negative time due to the -0.4 second time shift. This height, 0.425 m, was compared to the lower theoretical height of 0.206 m, which is the 5/2 R term derived from our theoretical calculation that was used in our Interactive Physics model. This difference can be explained by the fact that rotational inertia was not included in our theoretical calculation for starting height. With no rotational inertia due to friction, the ball has nothing to hinder its motion along the track. Therefore, it can start at a lower height and still make it around the loop without falling off.
To help determine the ball's location on the loop at important points on the position graphs here are some diagrams:
A is at the very top left of the loop-the-loop, from where the ball is released.
B is at the bottom of the loop where the ball enters the loop.
C is the rightmost side of the loop.
D is the top of the loop.
We calculated what the experimental initial height of the ball should be when rotational inertia is taken into account:
Substituting R = 0.137 m and r = 0.012 m, we obtain an initial height of 0.343 m. This is what our experimental initial height should be, if friction is accounted for and the ball does not slip during any part of its motion. We can see there is a considerable amount of difference between this calculated value and our experimental initial height value, 0.425 m.
This difference is most likely due to one of our major errors; we did not take enough videos around the one where the ball starts off at an initial height of 0.343 m to just make it around the loop. If we had done this, our experimental data would have been closer to the theoretical value of the initial height that we found.
Close to the end of the project, we decided to investigate more fully how this error influenced the experimental initial height we obtained. Using the velocity graphs generated in Excel, we decided to find the ball's velocity at the top of the loop in both the experimental and theoretical cases. We would then use these velocities and the law of conservation of energy to determine the starting height of the ball in each case.
To do this for each case, we found the top of the loop by locating the point on the spreadsheet where the y-position data was most positive. Moving over to the columns headed x and y velocity, we found the x and y-velocities at the top of the loop. Since these x and y-velocities were taken from a point that was close to, but was not, the top of the loop, we found the x and y-velocities of the points before and after the point where the y-position data was most positive. The x and y-velocities of these two points were then used to calculate the average x and y-velocities for both the experimental and theoretical cases. Substituting the average x and y-velocities into the equation
V = sqrt ( (avg x-v)^2 + (avg y-v)^2 ) ,
we determined the average velocities at the top of the loop for both the experimental and theoretical cases. These average velocities were 1.189 m/s and 1.243 m/s, respectively.
Using the law of conservation of energy and including the kinetic energy due to rotation, we found that the experimental initial height should have been 0.346 m, and the theoretical initial height, despite what we had calculated earlier, should have been 0.353 m. To see a table of how this process was carried out, click here.
From this, it appears that the ball started at a lower height than that required for the ball to make it around the loop without slipping. This is very surprising, since we had believed the ball started at a higher initial height than that required for the ball to complete the loop without slipping. The use of average velocity vs. instantaneous velocity likely is the source of this error, but we had to use average velocity since no function could be applied to our data from which we could obtain an instantaneous velocity.
Back to How We Analyzed Our Data
On To Conclusion
Table of Contents
Table of Contents
Introduction
Theory
How Interactive Physics Works
Our Materials List
The Experimental Setup
How Our Data Was Analyzed
Interpretation of Results
Conclusion