Lab 9 - Centripetal Force with a Motor

Purpose: The purpose of this lab was to create a relationship between angular speed ω and the angle Θ that the string in the apparatus below made with the vertical.
Apparatus: The apparatus that we used was a motor that was attached to a surveying tripod, positioned with the axis of rotation pointing upward and an extension rod. The top of the rod had a meter stick taped to it in a horizontal fashion. The far end of the meter stick had a string tied to the hole of length 187.1cm, with a rubber stopper attached at the loose end. The opposite side of the meter stick had some counterweights taped on.
Theory: The theory behind this lab is intuitive; if you spin something in a lateral circle, the faster you spin it the higher it will move. It is easy to say what will happen, however the challenge that was posed here was to quantify the angular speed based on what angle the string makes with the vertical.
Procedure: When the motor was on and the apparatus was spinning, we had a ring stand on the floor with a piece of paper taped on it. The height of the paper was adjusted until it was at the lowest height required to make contact with the swinging stopper. The height off the ground was measured, and we were able to calculate the angle made by the string by using the dimensions of the apparatus. The relationship we came up with was:
ω=[(g*tanΘ)/(R+L*sinΘ)]^.5
g=gravity
R=length from string to vertical shaft on meter stick
L=length of string

 

This is a table of all the data that we collected and calculated. The column on the left was measured by using a stopwatch. 10 revolutions were timed in order to get an average value, and these average values were compared to our calculated angular speeds (which are shown on the far right column in the table). (Note: for some reason I was having a hard time figuring out how to rotate the table properly).

 

This was a graph with calculated angular speeds, and the measured average values, each on their own axis. Based on the line, there was approximately 15% error in the two different sets of data.
Conclusion: I think that this lab was pretty cool because we were able to calculate the angular speed of the apparatus just by using a free-body diagram and an angle. The amount of error is appropriate for the setup that we had. The angular speed that we measured with a stopwatch was on average 15% lower than what we calculated. This is due to many reasons, such as: air resistance in the string and the rubber stopper as it went around; friction in the apparatus; and the fact that a meter stick was not the best choice for a revolving beam. The air resistance had the most impact on the error because it is very apparent that the differences between the measured and the calculated angular speeds gets larger as the speeds increase.