APR-01-2015 Centripetal Force with a Motor

Centripetal Force with a Motor

Purpose:  To determine the relationship between an angle theta with the vertical and omega on a motor-driven apparatus swinging a mass connected to the apparatus via a string.

Procedure:  The apparatus was set up previously by our instructor.  It is depicted below.

(Motor-driven apparatus swinging a mass connected by a string)

A stopwatch was used in order to determine the time required for the swinging mass to make 10 complete revolutions.  This is used to determine the period T of the circular motion.  The period can then be used to calculate what omega is, this gives us our measured value of omega.

We can also calculate what omega SHOULD be by taking additional measurements, including the radius of the motion, the height of the object from the floor, and the length of the string connecting the object and the apparatus.  We can use these values, along with the application of Newton's Laws, in order to come up with a theoretical value of omega.  Firstly, in order to accomplish this, we need to express omega in terms of an angle theta.  This derivation is depicted below (along with a diagram of the apparatus and relevant measurements).

(Derivation of omega in terms of an angle theta)

We took measurements in six different instances.  In each run, we recorded the time it took the object to make 10 revolutions about it's circular path, the objects total distance (radius) from the center of the circular path it's traveling, as well as the height at which the object was relative to the ground.  Having previously measured the total height of the apparatus to be 200 cm, we can utilize the object's height relative to the ground in order to determine the angle theta being made with the vertical.

Shown below is the gathered data; values in highlighted in green are our sources of uncertainty which propagate our uncertainty in our calculated value of omega, the values highlighted in orange should be nearly equal (except they are not, due to propagated uncertainty).  The measured omega value was attained using the formula omega = (2*PI) / T     where T is the period of circular motion.

(Excel spreadsheet organizing data)

If we plot our measured value of omega versus our calculated value of omega on an xy graph, the slope SHOULD be 1.  Displayed below is my attempt to use LoggerPro on my Windows machine...

(LoggerPro graph failed attempt at linear fit)

For some reason, after naming my axis and entering values, I'm unable to have the program proceed with a linear fit.  HOWEVER, I am able to input values on a plain xy graph and the linear fit works....  This is depicted below.

(LoggerPro graph displaying slope)

We came out with a correlation of 0.9994 and slope of 1.063.  These small discrepancies are due to the propagated uncertainty in our measurements of time (from stopwatch) and our height measurements (which was taken from a 2 - meter stick).