The purpose of this experiment was to find a model that relates the angle a hanging mass tied to string makes with the vertical and the angular speed of the hanging mass as it rotates around a central axis.
Apparatus:
The apparatus for this experiment consisted of an electric motor that spins at a certain speed depending on the voltage that runs through it. The motor is mounted on a surveying tripod. It also consists of a vertical rod attached to the motor that also spins with the motor, with a horizontal rod that is attached to the vertical one. A long string is attached to the end of the horizontal rod, with a rubber stopper attached to it. This apparatus was used to find the relationship between the angle and angular speed by performing different trials with running increasing voltages through the motor, resulting in a faster angular speed. The angular speed was measured by measuring the time it took for the hanging mass to go through 10 revolutions, then finding the period by dividing the time by 10. Then, angular speed is found by dividing 2pi by the period. The experimental angular speed was used to measure the accuracy of the model.
In order to find the angle, the height from the top of the horizontal rod to the mass when it was rotating had to be measured. In order to be possible to find that height, the height of the whole apparatus was measured and it was subtracted by the height of mass from the ground as it was rotating. The height of the hanging mass was found by using a ring stand with a horizontal piece of paper; the height of the paper was modified until the hanging mass barely scraped the paper, and the height of the paper from the floor was measured.
Procedure:
First, the dimensions of the apparatus were measured. This includes the height of the apparatus (200.0 cm +/- 0.1 cm), the length of the string (166.4 cm +/- 0.1 cm), and the distance from the center of the horizontal rod to the edge where the string is (87.0 cm +/- 0.1 cm).
Using the apparatus explained above, six trials were performed, with the voltage running through the motor increased between each trial. Each trial, the time it took for 10 revolutions was measured and the height of the mass from the ground as it was in motion was measured. The period and angular speed of the mass for each trial was calculated. As expected, the period was smaller and the angular speed larger every increasing trial. The data is found tabulated below:
Figure 1: Data table for experimental values of w and height
The uncertainty in the height h of the first 5 trials is 0.5 cm, and the uncertainty in last trial is 1 cm. The uncertainty in the time for 10 revolutions is about 0.2 s (average reaction time), so the uncertainty in the period is 0.02 s.
Once the experimental angular speed for each trial was found, the relationship between the angle the mass makes with the vertical and the angular speed was calculated. The derivation is shown below in the results section.
Results:
Figure 2: Free body diagram of hanging mass
The relationship between theta and omega was found through analyzing the forces acting on the hanging mass in motion through the use of the free body diagram above. The two forces acting on the mass are its weight and tension of the string, with the centripetal acceleration pointing inward to the center of rotation parallel to the horizontal direction. Summing forces in the x and y direction, we get the equations shown in the figure below:
Figure 3: Derivation of the model that relates theta and omega
In this experiment, centripetal acceleration was substituted with r times omega squared to get theta and omega in the same formula. Theta can be found by looking at the triangle formed when the mass is spinning, seen in Figure 2. The vertical leg is the height of the whole apparatus (H) subtracted by the height of the mass relative to the ground (h). The horizontal leg is the distance r from the mass to the end of the horizontal rod. The hypotenuse is the length of the string (L). Since r cannot be measured, using cos(theta) is the best alternative. Then, theta = acos((H-h)/L). Using the sum of froces in the x direction, T can be solved for in terms of theta, which is what is needed for this model. The distance r, since it cannot be measured, can be substituted for Tsin(theta), where T is mg/cos(theta). Plugging in the expressions for T, r, and theta, an expression for angular speed that can be used with the limitations of this experiment is found. The model derived that relates both theta and angular speed is:
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The model that uses the dimensions measured instead of T, theta, and r is:
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Using the dimensions of the apparatus and the heights of each trial, the expected angular speed for each trial was calculated, and are tabulated below.
Figure 4: Calculated angular speed for each trial using the derived model
To compare the calculated angular speed and the experimental one, a graph was made that plots the calculated angular speed on the y axis and the experimental on the x axis. Perfect proportionality between the two (that is, all of the values for all six trials are exactly equal to each other) results in a slope of exactly 1. The closer the slope is to 1, the better the derived model is. The graph is shown below:
Figure 5: Graph of calculated w vs experimental w
The slope of the above graph is 0.9943 +/- 0.003639. As expected, the slope of the graph cannot possibly equal to 1 because of all the uncertainties in the measurements in this experiment. However, the slope, along with its uncertainty, is fairly close to 1, indicating that the model derived in this experiment is good enough to relate theta and omega.
Conclusion:
To see if the error/propagated uncertainty in this experiment is only based on the uncertainties in the measurements, dw could be found by taking partial derivatives with respect to T, H, h, L, and R. However, taking partial derivatives of such a model proves to be difficult and is not the best alternative to use since most of the uncertainties in the measurements are not large enough to have a great impact on the uncertainty/error in this experiment. The measurements that have the highest uncertainty is the height h of the mass relative to the floor and T, making them have the biggest impact on the uncertainty/error.
The best alternative to finding if the error is caused all by the uncertainty in this experiment, one of the values of h from one of the trials can be used. Using the uncertainty of that value of h, the change in the value of w for that trial can be seen upon addition and subtraction of the uncertainty from the value of h. Then, the change in the value of w upon changing the value of h with respect to its uncertainty can be compared with the error of the slope. If the error in the slope of the graph is less than that of the error from the uncertainty in w upon changing the value of h, then the propagated uncertainty in this experiment is all caused by the uncertainties in the measurements of the dimensions of the apparatus and time.
Using the value of h from trial 4 (118.5 +/- 0.5 cm), the original value of angular speed is 2.742 rad/s. Using the value of h with the uncertainty added (119.0 cm), the value of omega obtained is:
The difference between that value and the actual value of omega (2.742) is + 0.009. Doing the same with the value of h obtained by subtracting the uncertainty in h (118.0 cm), the value of omega obtained is:
The difference is -0.010. Therefore, the average uncertainty in omega is +/- 0.0095, which is 0.35% of the actual value of angular speed for that trial (2.742 rad/s). Doing the same with T for trial 4, the average uncertainty in omega is +/- 0.0237, which is 0.87% of the experimental value of angular speed for trial 4 (2.731 rad/s). Subtracting the two uncertainties, since w - the derived equation (sqrt(...)) should equal zero, the uncertainty obtained is 0.52%.
The uncertainty in the slope of the graph is +/- 0.003639, which is 0.37% of the slope (0.9943). Since the uncertainty in the graph is smaller than the uncertainty in the angular speed, it is safe to say that the uncertainty in the slope is caused by the uncertainty in the measurements of the dimensions and period.
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