03-April-2017: Centripetal force with a motor

Lab 9: Centripetal force with a motor
03-April-2017
Idea: Model a relationship between angular velocity, and the angle at which an object is at, once a spinning motor rotates a hanging mass attached an R distance away from the center.
Summary:
1) In this lab, we had a set up wherein a hanging mass was attached to a meter stick a distance R away from the rotating center, powered by a motor.

2) By increasing the voltage of the machine, we could get a variety of increasing values of rotational velocity.

    - Before we even turned on the machine, we recorded the height of the machine, from its base to the meterstick of length R.

    - Then we measured the radius of the extended part of the m stick.

    - Once we got the machine spinning, we timed 10 rotations, with which we could calculate angular velocity. Next, by scooting a ring stand and marking the vertical height at which the angled string is now at, we could measure its location.

3) Afterwards, we tested 5 cases at increasing power levels.




Data Analysis:
As shown above, we observe that as the power increased, the time it took for the swinging mass to complete one full rotation naturally decreased. Conversely, when we measure the new heights of the ring stand, we observe that it goes higher and higher. We know that angular velocity is the quotient of a full rotation (in radians) with the time it takes for that full rotation.
Naturally, we discover as a result that angular velocity increases with each test case.
In order to find the angle theta of the swinging mass, we set up a triangle with the length of the string as the hypotenuse, and the height of the machine minus the height given by the ring stand, and use an inverse cosine to find the angle. This shows us again, that like angular velocity, the angle theta also increases as the power is increased as shown by our data below.

Afterwards, we had to accomplish the main purpose of the lab which was to create a model that has a relationship between angle Theta and angular velocity Omega.
To do this, we set up the net forces present in both the x and y direction. In the X direction, we observe that there is a horizontal component of tension in the string, and a force due to centripetal motion. In the y direction, there is a force of the string in the vertical component and force due to gravity. Although I do not know the tension in the string, it is present in both the x and y components of force: therefore I could substitute for it in the x direction, and plug it into my forces in the y direction. Doing so gave me a function for omega in terms of varying angles of theta, which let me see that as theta increased, omega also increased nearly proportionally with it.
Data Table/Calculations:

Data set based on the 5 trials. Note that in this table, Omega was calculated using 2*pi/t

How angle Theta was derived using the height of the stand, big H, and the height of the ring stand, little h.

How Angular Velocity was calculated in terms of angle Theta. We note that the radius of the system is not merely  just the radius of the extended meter stick, but combined with the horizontal component of length of the string!

Final Analysis:
By graphing the above function, I obtain a graph that looks like the following, where Omega is in the y axis, and Theta in the x axis. Looking at the graph we can visually recognize that, in general, as theta increases, so does angular velocity, with clear exceptions to the trend thanks to the asymptotes present in tangent theta.

Comparing this to our values of Omega obtained by using our intervals of Time was a matter of plugging in our values of theta, and before doing any percent error calculations, I observed that they are relatively very close to each other. The fact that my results appear to be within a 5-10% range of each other is a good sign that my measurements were relatively sound.

Conclusion:
Following my percent error calculations, I was pleased to find that none of my percent error exceeded 6%.  From this, I could determine that the time I recorded, AND the measurements of length I took, were both accurate at least to the tenths place, which was very good.

In determining the sources of uncertainty, there were a lot to consider.

1. For one, the spin of the machine was unreliable to a certain extent first due to the unknown fluctuations of power supply, which in turn would directly affect angular velocity, and the second, in a slight wobble in the meter stick holding the hanging mass. This would obviously affect the time it would normally take for a full rotation, and potentially the height recorded by the ring stand: measuring its height at a high point in its wobble would make the recorded height too large, and too short at a low point in a wobble. This would affect our angle Theta.
2. Finally, apart from that, the rest were from the standard limitations of our measuring tools. The first was uncertainty in our meter sticks, to a tenth of a centimeter, which again, would affect our derivation of angle Theta. The second was in the timers we used, primarily our phones. Because we didn't use a photogate, the times we record for a full interval were naturally going to be really inaccurate. We attempted to compensate for this by averaging the time for 10 full rotations to narrow this uncertainity as best as possible, but it would still affect our angular velocity.