01-April-2015: Centripetal force with a motor

     The purpose of this experiment was to come up with a relationship between angular speed and the angle with the vertical at which an object spins. The setup for this experiment used the apparatus shown below.
 

     The apparatus had an electric motor mounted on a tripod. A vertical shaft coming out of the motor. A horizontal rod attached to the top of the vertical rod. A string tied to the end of the horizontal rod and a rubber stopper at the end of the string.
     As the motor spins at higher speeds, the mass (rubber stopper) revolves around the vertical shaft at a larger radius and the angle it makes with the vertical increases. Below is the diagram for measurements made on the apparatus.

     The height from the ground to the horizontal rod (H) was measured with a meter stick. As was the distance from the vertical rod to the end of the horizontal rod (R), and the length of the string (L). To find the height from the ground to the mass (h), a piece of paper was put in place under the revolving mass. As the mass revolved, the paper was slowly risen up until the mass hits the top of the paper. The height from the ground to the top of the paper was then measured with a meter stick for each trial. The distance from the edge of the horizontal rod to the mass (r) was found by Lsin(theta), and the angle was found by arccos((H-h)/L).

     Finding a relationship between the vertical angle, theta, and angular speed, omega,  called for equations to solve for the variables. The equation for the angle was found with simple trigonometry to be arccos((H-h)/L). To get an equation for angular speed, omega, a free body diagram for the mass was used. Below is the free body diagram for the mass in motion.

Finding the sum of forces in the x and y directions gives

     Now that the sum of forces were found, the variable omega (angular speed) can be solved for. After some manipulations, an equation for omega gave

and since r=R+Lsin(theta), the end equation is

     So, the relationship between the angle and angular speed was found.

     After the equations were solved for, the trials began. The motor was turned on and the mass began to revolve. The period for the revolution was found by using the stopwatch feature on a smartphone. Each time the mass passed by a point of reference, the time was recorded. That was done for a total of 10 times for each trial. Below is the data recorded and the subsequent calculations for the angle and angular speed. Note that calculating the angular speed required the angle to be converted to radians.

     Next, the actual measured angular speed needed to be found. This was found by the formula 

     The period, T, for each trial is located in table above, in the column "T for one revolution."

The measured angular speed and the predicted angular speed were put into a table in Logger Pro (shown below).

The data was then graphed and a linear fit was done.

     This graph shows something very important. It shows that there is indeed a relationship between angular speed and the angle at which an object revolves, and that relationship was shown in the equation from earlier:

 

     If there were no relationship between them, the graph would not produce a straight line. Below is the data for the linear fit of the above graph.

     The slope is 1.042, very close to 1. Of course, since there were uncertainties in all measurements made with the meter stick, the slope was never going to be exactly one. There were also some factors that could have negatively affected the data. When the mass was revolving, the entire apparatus was rocking back and forth. Also, there was human error in trying to record the period of the revolution. It would be extremely difficult to accurately record the period with the human eye and a stopwatch.