Using Newton's Second Law and knowledge of centripetal acceleration, an equation for the sum of the forces as it relates to centripetal motion can be derived.
The above diagram models the experiment and shows the free body diagrams associated with the two masses. A free body diagram shows the forces acting on an object, and the direction of the forces. m1 represents the mass of the unknown object, while m2 is the known mass of the weight being revolved. The equations to the left go with m1. It demonstrates that T (tension) is equal to the mass of the object times the acceleration due to gravity (9.8 m/s^2).
For the m2 object, calculations are done in the x-direction because that is the direction in which centripetal motion occurs (in this experiment other forces prevent the string from being perfectly horizontal but that is considered negligible in calculations). Forces pointing inwards are positive because centripetal acceleration is center-seeking.
Substituting m1g in for T in the equation for the x-direction and solving for velocity gives and equation that relates velocity to the value of m2. r, m1, and g are all constants.
We chose to conduct our experiment by timing how long it takes to complete 30 revolutions, rather than counting how many revolutions are completed in a certain amount of time. This ensures that we did not have to deal with fractions of revolutions.
The math to the left shows how to derive an equation as a function of the time it takes to complete 30 revolutions, rather than velocity.
We chose to use two different radiuses, one for larger masses and one for lighter masses (so that we could still count the number of revolutions with a higher mass). This meant that we had two equations with different r values, as seen below.
Substituting in the constants gives two equations to calculate the theoretical mass of the unknown object. The equation used depends on the radius used during the experiment. We used a radius of 0.5m to test objects in the 100g-400g range and a radius of 0.75 to test objects in the 400-700 range.
Mass of Object vs. Time for 30 Rev. at Radius of 0.5m
Mass of Object vs. Time for 30 Rev. at Radius of 0.75m