10/3/16: Centripetal Force with Motor (Lab 9)

PURPOSE:
To create an expression that models the relationship between angular velocity, ω, and angle, θ, for a mass revolving around a central shaft.

INTRODUCTION:
As the motor spins at a higher angular speed ω, the mass attached to the central rod revolves at a larger radius and the angle θ increases. From this experiment, we can quantitatively define a relationship between angular speed and angle measure. The goal of this lab, therefore, is to derive an expression that defines ω in terms of θ. After this is done, we will test the accuracy of our model by comparing it to another method for calculating angular speed ω in terms of time, t.

APPARATUS:
Note: the apparatus was pre-assembled for this particular lab.

The real apparatus (top) and a diagram (bottom):

• An electric motor mounted on a surveying tripod 

• A long shaft going vertically up from the motor.

• A horizontal rod mounted on the vertical rod. 

• A long string tied to the end of the horizontal rod. 

• A rubber stopper at the end of the string. 

• A ring stand with a horizontal piece of paper or tape sticking out.

We utilized the apparatus to derive some key variables before proceeding with the experiment. For example, our group:

• Derived θ by looking at the right triangle -- from the diagram above -- with hypotenuse L and height H-h. Equation: θ = arccos((H-h)/L)
• Derived ω from timing the duration for the mass to make a number of revolutions around the shaft. Equation: ω = 20π/Δt (20π because we did ten rotations)
• Derived h by putting a horizontal piece of paper on a ring stand and slowly raising the piece of paper until the stopper just grazed the top of it as it passed by. Equation: None, just a ruler.

PROCEDURE:

1) Use a free body diagram (FBD) to create centripetal force equations of the system. Use these equations to create a mathematical model for ω in terms of θ. Here is our derivation of ω:

2) Use the apparatus to gather a sufficient amount of data to test your model. More specifically, you must collect values of h at a variety of values for ω. The professor adjusted ω by increasing the voltage to the motor driving the system.

3) Create an Excel Spreadsheet with the following variables: t, h, H-h, θ, ω (t), ω (h).

4) Create a graph of angular speed with respect to time t vs. angular speed with respect to angle θ.

DATA/GRAPHS:
Uncertainty in  r, L, and H: +/- 0.1 cm
Uncertainty in t: +/- 0.25 s (due to reaction time)
Uncertainty in h, H-h: +/- 0.1 cm

ω(t) = angular speed with respect to time.
ω(h) = angular speed with respect to θ.

***In chart "r" = "R" on the apparatus diagram.

Data Table with Relevant Variables and Constants

ANALYSIS: 

There are two forces acting on the stopper as it is spinning: tension and gravity. The horizontal component of tension is providing a net centripetal force on the rubber stopper, helping it accelerate towards the center and rotate at a constant speed. Since the stopper is not moving in the y-direction, Newton's first law dictates that the net force must be equal to zero. Therefore, we now know that the vertical component of the tension force is equal in magnitude but opposite in direction to the weight of the rubber stopper. Using these principles, we formulated our force equations, manipulated them, and solved for angular speed ω. The slope of the previous graph, therefore, represents the accuracy of our derived value of ω with respect to θ. The slope is in the form: 1 + uncertainty in ω(θ), (where ω(θ) is angular speed with respect to theta). Additionally, R² represents the correlation value of our graph. In other words, it quantifies how close our linear fit of the data came to matching all of our data points. According to our graph, we accumulated an uncertainty of about 4.82%.

CONCLUSION:

Overall, our calculation of angular speed with respect to an angle was very accurate. This is because our expression for ω(θ) calculated a value for angular speed that was within 5% of the angular speed we calculated with respect to time. While some 4A students would be satisfied with this minute margin of error, I for one find it much more satisfying to explore the ways in which our group could have mitigated a multitude of these myriad mistakes. The first step in minimizing this uncertainty is identifying the root causes. One of the main causes of uncertainty in our lab was human error. For example, it is possible that our group mate inaccurately measured the period of rotation due to our inherent lag known as "reaction time." Moreover, it is also possible that we could have inaccurately measured distances on the apparatus by a small margin as well. This is because the tick marks on the meter sticks can be difficult to distinguish at times and different perspectives can lead to inconsistencies over the most precise measurement. Another source of error in our lab were variables we omitted from our calculations. For instance, our class decided that air resistance due to drag was negligible. Thusly, we did not take drag force into consideration when creating our FBD and force equations. Another source of uncertainty in our calculations arose from the reverberation of the ruler atop the central shaft. As the motor spun the central shaft, the ruler would slightly oscillate, causing the radius of rotation to change as well. For the simplicity of the lab, however, our class decided that this effect was negligible as well and we treated the radius r as a constant for a given angular speed ω. In an ideal world, our group would redo this lab with top notch equipment and extreme attention to detail, but in recognition of the technical limitations and time constraints of a 4A class, our group accepts that our current experimental results will suffice. 

GROUP MEMBERS: Matthew I., Xavier L., Billy J.