01-April-2015: Centripetal Force with a Motor

The purpose of this lab was to come up with a relationship between the angle at which the mass is rotating and the angular speed.

Setup and Demonstration:
For this lab, there was only one setup that the entire class was to observe. The setup consisted of a long vertical rod. Attached to this vertical rod was a horizontal rod. At the end of this horizontal rod was a string to which a mass was connected. The horizontal rod was driven by a motor which set it into constant circular motion. The magnitude of this velocity was controlled by this motor. The entire apparatus can be seen in Fig. 1.

Fig. 1

We measured the the height of the vertical rod to be 2.00 meters, the radius at which the horizontal rod to be 0.97 meters, and the length of the string at which it rotated from to be 1.65 meters. We considered the angle the string made with the vertical to be θ, and the combined radius at which the mass rotated to be (0.97+1.65sinθ) meters. Since we could make a right triangle with the string, we could express a segment of the radius to be a trigonometric function of θ (Fig. 2).

Fig. 2
We needed to model our apparatus in order to find our expression for omega. 

Fig. 3-2
We used a FBD to find our expression for
angular speed.

Once we had a general idea of how the system would work, it was time to put some math to our ideas. We used Newton's Second Law to find an expression for the angular speed, omega. This omega would depend on the angle θ, created by the string and vertical. To find θ, we broke down the diagram into a right triangle from which we could solve for θ. (Fig. 3-1 and Fig. 3-2). Once we had this, it was time to begin recording data.

Fig. 3-1
Our expression for omega needed a way for theta to be measured.

Collecting Data:
The only thing we needed to (and could) measure from this experiment was the height at which the mass was rotating and the period at which it rotated. With the height, we would be able to find θ, which would in turn allow us to find the angular speed omega. To do this, we set the object into circular motion using our setup. Once it had reached some consistent circular motion, we placed an adjustable rod near the edge at which the object was rotating. attached to this rod was an extended piece of note card paper. As the object rotated, we slowly began to raise the note card. Once the note card hit the rotating mass, all we then had to do was measure the height of the note card relative to the ground. This height was the height at which the mass was rotating (Fig. 4). The period we mesured would allow us to determine whether our model for omega using Newton's Second Law was sufficient or not.

Fig. 4
As the rod spun, we adjusted the height of the note card to find the height of the mass. 

We observed the mass rotate for six separate trials, recording the height and period of the rotating mass for each trial. Our data table can be seen in Fig. 5. 

Fig. 5

Analyzing Data and Verifying Our Expression:

Once we had all of our data, it was time to begin using it to test the relationship we found between the angle at which the mass rotated and angular speed. As a reminder, we found the angular speed to equal the square root of (gtanθ)/(0.97+1.65sinθ). We plugged this formula into logger pro to produce our experimental value for omega. Along with this, we plugged our periods for each trial into LoggerPro to produce our "real" value for omega. We used the formula omega=2π/"period". (see Fig. 6)

Fig. 6
We used LoggerPro to calculate our experimental values for omega.

Once we had produced our two different angular speeds, we graphed them versus each other. The slope produced was 0.9947, which indicated that our model for angular speed was good. (see Fig. 7)

Fig. 7
Since the slope was practically 1, our values for omega were extremely close and similar, indicating that we
had found a good expression for omega for our apparatus.

There are two main areas of uncertainty that yielded us a slope that was not exactly one. First, our measurements of height of the apparatus, length of the string, etc. Using rulers, we could only be so close to the actual value of the dimensions of our apparatus. Also, our measured period was dependent on our reaction time for pressing our stopwatches. Since we did not know the exact spot where a rotation ended, we had to estimate. Overall, however, even with our uncertainty, we were able to yield pleasing results.