October 3, 2016: Centripetal Force with a motor

Anthony Betancourt
Lab Partner: Josh Fofrich
Professor Wolf

Lab #9: Centripetal Acceleration with a Motor

Purpose:

The purpose of this lab is to familiarize the student with the acceleration forces involved with circular motion.  The apparatus used in this lab can replicate a centripetal acceleration acting upon an object suspended by a length of string (L) above the ground of a height (H) at a calculated angular velocity (omega) that is derived from counting rotations over a timed interval.  From the derived data, an angle (theta) can be derived along with a radius from the axis of rotation (R+X).

Procedure: 

1. Check to make sure the apparatus is setup for the lab.  Take measurements of the height of the swinging arm down to the floor (H).
2. Take the measurement of the length of string the rubber stopper is hanging from (L).  Take the measurement from the center of rotation of the swinging arm to the attached string (R).  
3. Turn on the motor to the apparatus and wait for it to reach a constant velocity. Once the velocity stabilizes, use a piece of paper attached horizontally to a lab stand and slowly move the paper up until the rubber stopper make contact, then move slightly back down. This measurement will be h.  
4. As the rubber stopper spins around, start timing how long it takes to complete 10 revolutions.  Take the number of revolutions times 2pi and divide by the time taken for all 10 revolutions and this will give you omega.  
5. Repeat steps 1-4 at different voltages to increase the speed of the motor, thus giving different height (h) measurements, different theta angles (angle created by string from attached point on swing arm), and different omega values.  Complete 6 different trials.  

Theory:  

As the rubber stopper moves in a circular motion, the voltage on the motor will dictate how fast the rubber stopper spins.  As the angular velocity increases, so does the angle (theta) at the point where the string is attached changes.  To establish the relationship between omega (⍵) and theta, a free body diagram can show the forces acting in the y and x direction.  The equations for sum of forces can be re-arranged to show the value of omega.  

Measured Data:

(Apparatus diagram with measurements)

(Derivation for omega)

(Spreadsheet table with measured data and 𝜔 calculations)

(𝜔 calculations from timing 10 rotations of the apparatus; Trial #1 is calculated)

(Correlation between timed omega and derived omega)

Analysis: 

Once the timed rotations are calculated for omega (10 rotations multiplied by 2𝞹 and divided by the time it took) those values are then compared to the values from the derived equation of 𝜔.  An Excel spreadsheet can easily correlate the two data sets into a graph with a linear trend line.  Overall, both data sets were very close in values but tend to differ as the speed of the motor was turned up.  The faster the rubber stopper went around, greater the angle that was created from the the attached string to the swinging arm.  The derived values for omega did not increase as rapidly as the timed values for omega.  Once the data values were compared, a correlation of .90589 resulted between the Derived vs. Timed omega values.  This correlation shows the relative accuracy for the timed portion of the experiment but with some added uncertainty in the end calculation. 

Conclusion:

In order to calculate a relatively accurate value for an objects angular speed, these two methods can be utilized.  The timed is accurate up to a certain extent.  As the object increases in angular speed, the method of counting the revolutions and timing them becomes less accurate as the speed increases.  This is due to the fact of having to watch the object rotating and count how many revolutions while simultaneously using a stop watch to timed the accurately. One way to increase the accuracy of this method for higher speeds, would be to allow the object to complete more revolutions while allowing more elapsed time and result in greater accuracy for average angular speed.  The derived equation for omega also has certain uncertainties associated with it.  One being the forces acting on the spinning object, such as air resistance, preventing the omega value to be as accurate as the calculated result.  Other uncertainties in this experiment could be from the swinging arm (the meter stick used) was not completely level from the start of the experiment causing the height of the actual swinging arm to be slightly off.  With the inherent air resistance acting on the swinging rubber stopper, the string might not be  exactly in line with the swinging arm as the apparatus is moving in a circular motion.