01-April-2015: Centripetal Force with a Motor

Lab 9: Centripetal Force with a Motor Lab

Purpose: To determine a relationship between Angle of Rotation (theta) and Angular Speed (w).

Apparatus: 

Here we see, an electric motor mounted on a surveying tripod. There is a long shaft going vertically up from the motor. A horizontal rod is mounted on the vertical rod. A long string is tied to the end of this horizontal rod. A rubber stopper is tied to the end of the string. Essentially, this is a swing ride at an amusement park. As the motor picks up speed, the angle of rotation will increase.

 

Above, you see a ring stand with a horizontal piece of paper. This was used to measure, the vertical distance from the ground to the rubber stopper. The piece of paper was raised until the stopper just grazed it as it passed by.

 

 

Required Measurements:

• H = 2 m +/- .001
• L = 1.664 m +/- .001
• R = .87 m +/- .001
• theta: varies depending on angular speed
• h: varies depending on angular speed

To find theta, we used:

• theta = arccos((H-h)/L)

In order to determine a relationship between theta and omega, we collected values of h at a variety of values of w. So, for each run we calculated theta  and omega. These values will then be used in two expressions found  below. In which, we compared a theoretical value of omega to the experimented value of omega. Here, the theoretical value of omega depends upon theta. Now, by graphing these values we will be able to determine a relationship between omega and theta based on the slope of the graph.

Collecting Data:
 In all, we had six runs. We collected theta, omega, h, and period (T) for each run.

• h (m): measured height when stopper hit paper
• period (s): Time for 10 rotations/ 10
• theta (degrees): arccos((H-h)/L)
• omega (rad/sec): (2*pi)/T

Run 1:

• 37.67 sec for 10 rotations, T = 3.767 sec
• w = 1.67 rad/sec
• h= .473 +/- .005 m
• theta = 23.4

Run 2:

• 32.75 sec for 10 rotations, T = 3.275 sec
• w = 1.92 rad/sec
• h= .624 +/- .005 m
• theta = 34.2

Run 3:

• 28.04 sec for 10 rotations, T = 2.804 sec
• w = 2.24 rad/sec
• h= .850 +/- .005 m
• theta = 46.3

Run 4:

• 23.01 sec for 10 rotations, T = 2.301 sec
• w = 2.73 rad/sec
• h= 1.185 +/- .005 m
• theta = 60.7

Run 5:

• 19.30 sec for 10 rotations, T = 1.930 sec
• w = 3.26 rad/sec
• h= 1.408 +/- .005 m
• theta = 69.2

Run 6:

• 15.30 sec for 10 rotations, T = 1.530 sec
• w = 4.11 rad/sec
• h= 1.610 +/- .01 m
• theta = 76.4

To recap:

Analyzing Data:

Next, we setup a FBD for the rubber stopper.

We solved for forces in the y and x-direction.
R=.87 m +/- .001
Theta: Calculated
w:Calculated

• Fx: Tsin(theta)=m*R*w^2
• Fy: Tcos(theta)=mg

Then we solved the Fy equation for w and the Fx equation for T. We combined them by plugging in T and simplified the equation.

• w = sqrt((g*tan(theta))/(R+ Lsin(theta)))

Now, we can setup a relationship between w and Theta by graphing w vs. w.

• Theoretical Values = y-axis: w (theta) = sqrt((g*tan(theta))/(R+ Lsin(theta)))
• Experimented Values = x-axis: w =  (2*pi)/T

By graphing this setup, we are implying that w and theta are proportional to one another. This means that the slope of the graph should equal 1 or something very close to it.
These are the points for our graph:

 

We plotted the points and took a proportional fit of the graph to determine the slope.

• Slope: .9935 +/- .006345

 As you can see the slope of our graph is .9935. This shows us that we are about 1.3% off from theoretical values because they are not exactly proportional. This is due to the amount of uncertainty from measuring the height of the stopper as well as measuring the periods. Overall, we determined that as angular speed increases, the angle of rotation will also increase.