AP Physics 1 Unit 9, Uniform Circular Motion and Gravitation

Have you ever wondered why water doesn't fall out of a bucket spun around in a vertical circle?

Paradigm:  We will use a rubber stopper whirling around in a horizontal circle on the end of a string in the classroom, or a "blinky buggy" secured to a post, but the paradigm relates to any object moving in a circle at a constant speed, like a bucket on a string, or a car rounding a corner of constant radius.  Since we are only considering UNIFORM circular motion (for simplicity's sake), the speed will remain constant.  

In the paradigm lab we whirled a stopper in a (sort-of) horizontal circle.  Some groups varied the velocity, keeping the radius constant, and found that force (applied by hanging masses) had to vary, too.  So, we graphed velocity versus force for a "whirling stopper" to obtain a side opening parabola like the one below:

If we square the velocity data, we get a graph that looks like this:

Does the slope have any meaning?  What are the units of v^2/F?  Work it out, remembering that Newtons are kg*m/s^2.

We also tried to keep the force (the hanging masses) AND the radius constant while varying the mass of the whirling stopper.  We obtained a graph that looks like an inverse graph:

It was a little more complex than an inverse graph, however, since it took two operations to make it look like a line (squaring the velocity and taking the inverse of the mass):

Again, it's a good idea to think about the units of the slope.  What would be the units of v^2/(1/m)?

Lastly, we kept the hanging mass (approximately proportional to the net force) constant, and varied the radius of the swing.  Something had to change, and it was the period, "T".  From period, we calculated the velocity (circular motion, remember?).  And we obtained:

Another "side-opening" parabola.  Linearize this one, and decide what, if anything, the slope is units of.