# Circular Motion

An object moving in a circular path, such as a person on a merry-go-round or a satellite orbiting the Earth, has an interesting motion where the speed is constant but the direction is continually changing. This is accelerated motion because in any chosen direction, the speed is changing. At the high school level (without calculus), it is interesting and rewarding to find the central formula for circular motion through an experiment that can be done with no special equipment.

## Circular Motion Experiment

A force is required to provide the acceleration of an object in circular motion. This force is always toward the center of the circle and is called the centripetal force. Its formula can be found in this little experiment. It requires no special equipment, but has a fairly complex analysis.
Procedure
Swing a rubber ball attached to a string in circular motion at various speeds and string lengths. Record the time for 10 complete circles, the length of the string (to the center of the ball) and the angle of the string from vertical. For larger radii, swing it around your head. For smaller radii, swing in front of you. A good way to observe the angle is to use a light in an otherwise darkened room to cast a shadow of the swinging ball and hand holding the string onto a wall or chalkboard where a partner can measure it. The angle is the largest source of error in the experiment and needs to be measured as accurately as you can.
Analysis
Two forces act on the ball (middle diagram), gravity down and the Tension force of the string. In the right diagram, the tension force has been replaced by its horizontal and vertical components. The total vertical force must be zero because the stopper does not accelerate vertically, so T*cos(A) = mg. (1) The horizontal force is the centripetal force accelerating the stopper toward the center of the circle: Fc = T*sin(A). From equation (1), T = mg/cos(A) so we have Fc = mg*sin(A)/cos(A) = mg*tan(A)
The radius is R = L*sin(A). The velocity is distance over time: v = 2πR/t
Finding the Formula
Graph Fc versus various expressions like vR, v$^2$R, vR$^2$ until you get a graph that appears to be a straight line. Identify the slope (use its units as well as its magnitude) as some quantity in the experiment. A straight line graph of y vs x has the formula y = mx + b where m is the slope and b the y-intercept. For the graph of Fc, you will have Fc in place of y and some quantity such as v$^2$/R in place of x. Hopefully the y-intercept will be zero.
This part may not be essential, but it makes the experiment much more convincing. Repeat one of the trials and estimate the accuracy of the angle measurement - perhaps plus or minus 3 degrees. With one line in the table, do the calculations with 3 degrees added and again with 3 degrees subtracted to see the plus or minus variation in the Fc and in the expression that results in the best straight line. Use these variations to make error bars on the graph. Is the graph a straight line to within the accuracy of the experiment?
Conclusion
Done carefully, this experiment shows that Fc = mv$^2$/r to within experimental error. Here m is the mass of the object in circular motion.