Finding a relationship between mass and period for an inertial balance
Purpose:
This lab aims to find a mathematical relationship between the mass and period of an object when placed on an inertial balance which would allow an individual to predict an object's mass based on measured period.
Theory:
Typically, mass is measured using a scale dependent on gravity. This means that in low or zero-g environments, conventional gravity-based scales cannot be used. An inertial balance does not require the presence of gravitational pull and instead allows us to measure the period of an object as it oscillates on a fixed spring. Since gravity does not affect the motion of the system, the object's mass is the only variable that would influence its period. The goal of this lab is to produce a model for predicting mass based on measurements of period of an object oscillating on the spring and compare it to the actual mass of an object. The equation to be used is T = A(m+Mtray)^n where T is the period in seconds, A is an experimentally determined constant, m is the mass of the object, Mtray is the mass of the tray on the inertial pendulum, and n is an experimentally determined constant.
Apparatus:
- Inertial pendulum (1)
- 500g mass (1)
- 200g mass (1)
- 100g mass (2)
- Object of unknown mass (2)
- LabPro photogate (1)
Procedure:
Attach a small piece of tape to the end of the inertial pendulum so that it passes through the photogate when oscillating. Starting from 0g of mass on the pendulum and increasing in increments of 100g up to 800g, measure the period of oscillation. Using the data collected in Logger Pro, create a plot of lnT vs ln (m+Mtray). Power fit the graph and adjust the value of Mtray manually to produce a correlation coefficient of at least 0.9997. Set A equal to the y-intercept and n equal to value of the slope. Substitute these values into the original equation and solve for m. Determine the accuracy of the model by predicting mass of the two unknown objects through measurements of period and comparing the predicted mass to measured mass.
Data and Graphs:
Analysis:
The graph of lnT vs ln (m+Mtray) produced a slope of 0.6558 and a y-intercept of -4.953. When assigned to the appropriate experimental variables in the equation, solving for m produces the equation seen near the center of the photo of data and calculations. Equations of m were also gathered from two other groups for comparison. Using these equations, the masses of the two objects (a wallet and a hole puncher) were predicted using measurements of period and found to be consistently higher than actual masses. The wallet had an average predicted mass of 86.6g, a difference of 6.6g or 8.25% from actual mass while the hole puncher had an average predicted mass of 1076g, a difference of 39g or 3.62% from the actual mass.
Conclusions:
Based on the results, our model for predicting mass is only adequate for non-precision measurements as both tests using unknown masses produced values much greater than actual measured masses. The consistently higher measurements are most likely due to inaccurate values of Mtray as these values were only estimates. Because all predictions were consistently high across all 3 models of m, it may be assumed that the instruments did not contribute significantly to error in calculations.
Data and Graphs:
Analysis:
The graph of lnT vs ln (m+Mtray) produced a slope of 0.6558 and a y-intercept of -4.953. When assigned to the appropriate experimental variables in the equation, solving for m produces the equation seen near the center of the photo of data and calculations. Equations of m were also gathered from two other groups for comparison. Using these equations, the masses of the two objects (a wallet and a hole puncher) were predicted using measurements of period and found to be consistently higher than actual masses. The wallet had an average predicted mass of 86.6g, a difference of 6.6g or 8.25% from actual mass while the hole puncher had an average predicted mass of 1076g, a difference of 39g or 3.62% from the actual mass.
Conclusions:
Based on the results, our model for predicting mass is only adequate for non-precision measurements as both tests using unknown masses produced values much greater than actual measured masses. The consistently higher measurements are most likely due to inaccurate values of Mtray as these values were only estimates. Because all predictions were consistently high across all 3 models of m, it may be assumed that the instruments did not contribute significantly to error in calculations.
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