Lab 9: Centripetal Force with a Motor

Physics Lab #9 started on 3/25/15
Centripetal Force with a Motor
Annemarie Branks
Professor Wolf

Objective: Determine the relationship between the the angle created by a revolving mass tied to a string, and the mass's angular speed.

The Apparatus:

A motor will spin the central shaft which in turn spins the meter stick. Attached to the meter stick is the string with the mass tied to the end. The string will make an angle, as shown in the picture.

Procedure:
     1.  First, measure the length, L, of the string, the internal radius, R, the apparatus makes with the end of the meter stick, and how high, H, the meter stick is above the floor from where the string is attached. For our experiment, the apparatus was 2.00 +/- 0.001 meters tall, the radius was  0.870 +/- 0.001 meters, and the string was 1.664 +/- 0.001 meters long. 
     2.  The first trial will require a certain number of voltage to give the system some angular speed. After making a mark on one point of the circle, time the system for 10 rotations.
     3.  Near the edge of the circle use a ring stand and a horizontal piece of paper that you will slowing raise up so you can find where the mass hits the paper. Once the mass hits the paper, find the height, h, the paper was at when the mass hit. 

     4.  Repeat steps 2 and 3 using more voltages for each trial than the last, until you have a total of 6 trials.
     5. In each trial you found a height of the mass from the floor, h, and an angular speed which you find from multiplying the number of revolutions you timed for by 2*pi, and divide that value by how much time it took to complete those 10 revolutions.
     6. Draw a free body diagram with all the forces on the mass. Calculate the sum of forces and find the relationship between angular speed and the angle of the string. Here are our calculations: 

     7.  We don't know theta, but we know other values that will help us find theta as shown in the picture above. The only unknown variable leftover is h. We want to be able to compare the values we got for angular speed in step 5 to the angular speed we will get from the equation we just calculated. If they are similar and we plot them on a graph then the graph should be directly proportional and the slope should be about 1 radians/second^2.
     For my group's experiment, we graphed the angular speed that we got from the equation above by the angular speed that we calculated in step 5. Our slope, A, was 1.006 +/- 0.006427. Just for fun, we flipped the axes and got a slope of 0.9935 +/- 0.006345.

Errors:

Obviously, there will always be a certain amount of uncertainty in ever measurement we take and even tough we produced nearly perfect results we must account for the uncertainty. 

• For the Length of string, the internal radius, and the height of the apparatus, we said they each had an uncertainty of +/- 0.001 meters. 
• For the height of the spinning mass above the ground we said there was an uncertainty of +/- 0.005 meters for each trial except for the last one, because it was a spinning so fast it was had to see where it hit. For trial 6 we said the uncertainty was +/- 0.01 meters. 
• There was also uncertainty in the reaction time of whoever was timing the spins. In our group we had two people timing the 10 revolutions and some times the people timed it more than once. For every trial the times were always within about 200 milliseconds, or less, of each other. Some online sites say that this average human reaction is slightly above 200 milliseconds, but since were young people with better reaction time than people of other ages, 200 ms is a good uncertainty for time.

Changes in the graph:

The uncertainties mean that the "real" value of the points could be up/down and left/right a certain amount. I wanted to focus on one point to show what all the possible values could be. I am keeping all calculations with the proper amount of significant figures. 

• The very first trial gave us 10 revolutions in 37.67 seconds which is 1.668 radians/second. If  we include the uncertainty of 200 milliseconds, the values are 1.677 radians/second and 1.659 radians/second. If these values for angular speed were on the y-axis, the values are within 0.009 radians/second up or down.
• Using the height that we got from the first trial as well as the height of the apparatus and the length of the string with all of their uncertainties, we are going to find a range for the x-axis. Without the uncertainties, the angular speed  is 1.67 radians/second. 
• Rather than doing the propagated uncertainty method that we have been doing in previous blogs, I used the small H, big h, big L, and small R values to find the largest the angular speed could be. I did the opposite to find the smallest angular speed. These ended up being 1.68 radians/second and 1.65 radians/second. If these values for angular speed were on the x-axis, the values are within 0.02 radians/second up or down. Reminder, I kept this in the proper amount of significant figures so 0.02 is not an exact number.

One could repeat this process for all the points but the general understanding is that the value we calculated has a range around it in the x and y directions for what the real value is. The real values should line up to give a slope of 1.