Lab 9 - April 7, 2015 - Centripetal Force with a Motor

Lab 9:
Brandon Elder

        Pic. 1 The apparatus set up in class by the instructor.     

4/7/2015

Centripetal Force with a Motor

Purpose: 

The purpose of this lab is to observe an apparatus swing a weight around the central shaft at a known radius and to create a model that properly relates theta and omega. We will be comparing the data we record from the experiment and comparing it to the values that we measure for omega and see how they compare.

Set-Up: 

The set-up for this experiment was put together by the instructor in the class (see Fig. 1) for us to observe and record data as he performed the six different experiments. The apparatus was spun around at six different speeds. The time for the weight hanging from the string to swing around for ten revolutions was recorded each time. See Fig. 2 for all the important elements that need to be measured in order to compute for theta. 

Fig. 2 The elements H, R, h, can all be measured by the
apparatus. The x, y, and theta will be calculated after the
experiments.

Measurements: 

The H was measured to be 2 meters. The R was the distance from the center shaft to the string, this distance was measured to be 98 cm or 0.98m. It is important that this distance is not confused with the radius later on during the calculations. The lower case h varied between all the different experiments, so there was six different recordings. The time is also important to record. At each speed we recorded the amount of time it took to make 10 revolutions. This time was then divided by ten to determine the amount of time for one revolution. From the picture you can tell that the hand-drawn in x and y can be calculated with trig. Once all those elements are measured and calculated, the theta can be calculated. It is important not to lose track at this point of the purpose. We needed to determine a relationship between the angular speed and the angle theta. The model that can be derived for theta is in terms of theta, gravity, and the radius. This model was determined from the free-body diagram drawn of the mass hanging from the string (see Fig. 3). The sum of forces equations were also written down after observing the free body diagrams. The sum of forces in the y direction gave us a formula for mass. This can be seen in the top right corner of Fig. 3. This was plugged into the equation for the x direction and then rearranged to give us our final model for omega, the angular speed of our model. This model is seen in Fig. 4. It is again important to note here

that the r in this model is not the same as the R from earlier. The r in this model represents the radius from the central shaft to the mass being rotated. This r would be the sum of the R and the x that were measured earlier. 

Fig. 3 Free body diagrams for the hanging mass. The
sum of forces equations are also written here.

Fig. 4 The model for omega in terms of theta. 

Calculations: 

The calculations for all the above elements were performed during class and recorded onto a sheet of paper (see Fig. 5). The equation from above is re-written down at the top of the picture. For each experiment, the time was recorded first. Then the h value was recorded. The h value was the height off the ground that the weight was for each experiment. This was then plugged into the triangle above it to help solve for the y value. The value of y is measured by taking h and subtracting it from the overall height, H, of 2 meters (200 cm) from earlier. This value was then plugged into the quadratic formula to determine the third side of the triangle, since we measured the length of the string in the beginning. The square root of the length of string squared minus the y value squared was the x value. Theta was then found by taking the inverse sine function of the x value over the length of string value. Finally, at the far right is the omega values. That would be our calculated values of omega based on the angle theta that we calculated from the experiment. The theta value that was just determined was plugged into our model from Fig. 4 and we wrote down our data for omega values.

Fig. 5 All recorded calculations for all the elements discussed above. Take note of the highlighted figures for the first
experiment. The same logic was followed for the next 5 experiments, I just highlighted the first one to help with clarity.

Fig. 6 The values of the measured omega using the time we      
measured in class for one revolution and the values that
were calculated for omega from the experimental data used
to determine theta. 

Finally, we must measure what the actual omega value from the experiment was. This is done by taking the formula highlighted in the top left of the picture in Fig. 5. The measured value was derived by dividing 2*pi and T (time in seconds for one revolution). The experimental values of omega are all listed under the calculated values for omega in the right side of the picture. A comparison chart of these values is shown in Fig. 6. The numbers for the experiments were cut off in the picture, but they are listed in chronological order, all six experiments.

Comparison:

Now that all six values of theta and omega have been calculated, we can compare these values to each other and determine how close our data was experimentally vs calculated. It appears that the data is very close, but one way of determining how close to accurate these values are, would be to graph the data points and then to apply a best fit line to the data. The slope of the graph of the data points should be very close to one. This would tell the user that the data sets were equal to each other and that the experiment was a valid way of determining the angular speed of a system. If you take a look at Fig. 7, you will determine that our calculations/measurements have quite a high degree of error in them, as the result (slope of the line) should have been much closer to one.

Fig. 7 The slope of the best fit line was 0.78, this is not as close to 1.0 as we were hoping for. This means that there was error in our measurements.

Uncertainties in Lab, Assumptions We Made, and What Was Ignored:

It is evident that some uncertainty in this lab became apparent when reviewing the graph and noticing how far off the slope of the best fit line was from 1.0. This was most likely due to some error in the recording of the data. Two of our data points appeared to be off much more than the other points. I used the strike-through data cells in Logger Pro to remove these two points from the best fit line and noticed that the slope of the line became much more closer to 1.0 (see Fig. 8).

Fig. 8 With two of the points being taken out of the factoring of the best fit line, the slope moves much more closer to 1.0.

Other uncertainties came from the measurements in our data. The time had an uncertainty of plus or minus .01 second and the height measurements had an uncertainty of plus or minus .5 cm. These amounts of uncertainty would have caused the data to have some error introduced into our measurements and calculations. The exact moment at which the weight traveled around for 10 revolutions would have varied between each experiment as the recording was subjective based on the person hitting the start and stop button on the watch.

Overall, we learned that we can derive an equation for angular speed from observing a system in class that allows us to measure the speed of an object throughout several revolutions. This equation was then compared to our known equation of 2pi/time. The end result was that our model was an accurate method to determine angular speed and did NOT depend upon mass, but only on the radius, gravity, and the angle.