LAB 1 - Deriving a Power Law for an Inertial Pendulum
Tony Wu, Leslie Zhao, Isaiah Hernandez
August 29, 2016
Lab Goal: To derive an equation that relates the period (T) and mass (g) using an inertial balance device. The inertial balance device measures the inertial mass of an object by comparing its resistance to changes in motion.
Theory/Introduction: Mass is a quantity that is solely dependent upon the inertia of an object. A more massive object has a greater tendency to resist changes in its state of motion. In this lab we measured the period of oscillations for a range of masses. We modeled the behavior in a power law equation:
T: Period
A: Constant
: Mass Added
: Mass of the Tray
n: Power constant
In this equation we have three unknowns n, A, and. In order to solve for these unknowns, we use a little bit of Calculus. By taking natural logs of both sides, we can get the power equation into slope-intercept form:
Apparatus: We secured the inertial balance to a tabletop using a C-clamp. At the end of the inertial balance is taped a thin piece of masking tape. A photogate is set up to record the oscillations of the inertial balance each time the tape passes through the photogate beam.
Experimental Procedure: Using the slope-intercept equation and the data collected, we are able to calculate the unknowns. We plugged the data into Logger Pro to create a graph of 
vs. 
. At this point the value of 
is still unknown. We arbitrarily adjust the value of the parameter until the graph gives a straight line. The correlation coefficient is close to .9998 or .9999 for a straight line. There is an upper estimate and a lower estimate for , creating a range of values for that generate a correlation coefficient of .9999. The following graphs are line plots of the measured periods for each mass.
Upper Limit Estimate for": 440g
Using the graphs above, we are able to find the remaining unknowns in our slope-intercept power law. We now have values for, ln A (Y-intercept), n (slope).
With the information that we have now gathered, we return to the original power law equation:. Using some basic math skills, we can rearrange the equation and isolate the variable .
We measured the gravitational mass of each item using a balance. Below is a table containing the period of oscillations of each mass, as well as the two unknown masses.
Theory/Introduction: Mass is a quantity that is solely dependent upon the inertia of an object. A more massive object has a greater tendency to resist changes in its state of motion. In this lab we measured the period of oscillations for a range of masses. We modeled the behavior in a power law equation:
A: Constant
n: Power constant
In this equation we have three unknowns n, A, and
Apparatus: We secured the inertial balance to a tabletop using a C-clamp. At the end of the inertial balance is taped a thin piece of masking tape. A photogate is set up to record the oscillations of the inertial balance each time the tape passes through the photogate beam.
Lower Limit Estimate for : 295 g
Upper Limit Estimate for
Using the graphs above, we are able to find the remaining unknowns in our slope-intercept power law. We now have values for
With the information that we have now gathered, we return to the original power law equation:
Data and Calculations: We have now derived a power law for an inertial pendulum. Given any item of unknown mass, we can measure the period of oscillations and calculate the mass. We picked two items of unknown mass in order to test our power law equation. Item #1 was a snack bar and Item #2 was a stapler. We placed each item on the inertial balance and recorded the period of each. Below are the calculations for estimating the mass of each item. Take note that we must use both the upper limit and lower limit estimates for the mass of the tray, each resulting in a different estimated mass.
We measured the gravitational mass of each item using a balance. Below is a table containing the period of oscillations of each mass, as well as the two unknown masses.
Conclusions: The range of the calculated masses of both unknown items is an accurate estimate of the actual gravitational mass. However, a possible source of error for this lab is the variation in the period of each mass if the masses were not placed directly in the center of the inertial balance.
In conclusion, the power law equation we derived is an accurate approximation for the relationship between mass and period for an inertial balance.
