1-Apr-2015: Centripetal force with a motor

PURPOSE

The purpose of this experiment was to use the relationship between centripetal acceleration and angular speed to find the theoretical value of angular speed and compare it to the value that we found experimentally. 

PROCEDURES

Figure 1: Spinning apparatus

This experiment was conducted with the set-up shown above (Figure 1). The apparatus consisted of a motor (circled in blue) that spun around a hanging mass (circled in red). In addition, a piece of paper was attached to a ring stand (Figure 2) whose height was adjusted until the hanging mass just barely grazed the piece of paper. The purpose of this step was to determine the distance between the mass and the ground (h), which we would use later to find θ.

Figure 2: 2D drawing of the set-up, where H = 2.0 m, L = 0.9 m and R = 1.645 m

We ran the experiment with the motor spinning at six different speeds and recorded the time it took for the mass to make ten revolutions. As we increased the mass' speed, h increased as well. From h, we were able to find θ by taking the inverse cosine of L/(H-h). The data that we gathered is displayed below in Figure 3.

Figure 3: Data table

From this data, we were able to find ω by applying Newton's second law of motion in the x- and y-directions and combining the equations together. This process is illustrated in Figure 4 below. One thing to note about this equation is that r was not equal to R = 2.0 m. Instead, it was equal to R + Lsinθ since r is the distance between the axis of rotation and the rotating mass.

Figure 4: Mathematical process used to find ω

Rather than solving for ω by hand six different times, we decided to carry out the task in excel. Next, we used a second approach to find ω. We first found the time it took for the mass to complete one revolution by dividing t_10 by ten. Then, we divided 2π rad by this value to find ω. The results are shown below in Figures 5 and 6.

Figure 5: Resulting ω values from first approach

Figure 6: Resulting ω values from second approach

Finally, we plotted the resulting ω values against each other to see how accurate our measurements were. We did this by looking at the slope of the graph. If we were a hundred percent accurate with our measurements, the slope of the graph should have been equal to one. As it can be seen from the graph in Figure 7, the slope was 1.0399 and was actually very close to our expected value.

Figure 7: Graph of values found from second approach vs values found from first approach

CONCLUSION

In this experiment, we applied our knowledge on centripetal acceleration and how it relates to Newton's second law of motion in a real-life situation. Then, we compared our theoretical values to our experimental values and found that we were relatively successful in properly carrying out this experiment. In fact, our percent error was only 3.99 percent. Given the circumstances, this result was as accurate as it could have been. There are a few reasons why the result was not and could not have been a hundred percent accurate. First, the measurements that we made for the different lengths could not have been completely accurate because we were limited by the tools that we used to make the measurements. Another reason why our result could not have been fully accurate was that we ignore the effects of air resistance in our calculations. Although air resistance may not have played a significant role on such a small object, it could have been enough to affect the results noticeably since the object was moving at high speeds.