Results Data
Looking at the results, I found the angular velocity to be of a decreasing exponential curve. (See Below). This is what I expected for the velocity would be fast early, and then gradually slow down to a stop. The trend I had was about 30 seconds from the time I started spinning the bike to the time it came to a stop. The graph below shows the bikes angular velocity as a position of time or data point number; each data point is equal to .1 seconds.
Now in this graph, you can see the trend of the points. There were approximately 350 points in this graph and most fit into the decreasing exponential graph, however some do not. There is a margin or error and many of these points would fit into that. However we can defer from the graph that the angular velocity decreases exponentially.
We calculated the angular velocity by taking the difference in the positions (revolutions) and divided it by the difference in time. For example my first two points were 0sec, .106rev and .1sec, 228rev. I then take the difference of the revolutions (.122) and divide it by the time difference to get an angular velocity of 1.2 radians per second. I did that for all of the data points to get the graph above.
The next step was to figure if the force of air friction was in fact proportional to the velocity of the object. To do this I had to derive a formula, which I got from the Newton’s law of cooling formula. That formula has a Differential equation of Y= K (Y-t) and a solution of Y’= T + Ae^kt. For me I used the solution of this to solve for K. Using my starting velocity of Wo of 1.25 radians per second, and e is a constant, there is no A for the starting velocity and the velocity at time of 0 seconds is the same. I then picked a random point on the graph and solved for it using the information below.
K= (ln
(abs W-Wo))/T
I then used this formula to solve for K with all my data points to get the chart below:
In theory the solutions for the Constant (K) should be all the same, though this is actually an exponentially increasing graph. There is a large margin of error on this or perhaps there is something else affecting this constant. However, the constant seems to be the same or very close to each other for the most part after about point 35. So I took the average of the last 300 data points to figure out a constant constant. This number came out to be around .03. Once I had a constant I decided to figure out if in fact force of friction depended on the angular velocity by a constant K. To this I used my formula with the constant in it and solved for the velocity at different times, which it should be the same as the actual velocity at that time. I used the formula:
W=Wo + e^-.03 * T
I then plugged in different times to see what the velocity would have been using this constant and graphed the results below:
I then compare this graph to the graph of the actual velocity and compare it to my graph here of the predicted velocity to see if they are in fact the same.
