*Procedure: Using a inertial balance tray, secured to a table with a clamp, tape, some masses, and the LabPro to measure periods. We will make the tray sway with a variety of different masses, measure the periods between sways, and derive and test a power law with two separate unknown masses (Object A and Object B)
*Theory: With the data gathered from the LabPro we can derive a mathematical formula that from the oscillations of a inertial balance (periods between swaying) tray the mass of objects can be determined. Period=some scalar (total mass)^some power (T=A(M_tray +M_object)^n)
The Apparatus
The data of periods known masses
|
Mass in Balance, g
|
Period, sec
|
|
0
|
0.284
|
|
100
|
0.35
|
|
200
|
0.406
|
|
300
|
0.457
|
|
400
|
0.504
|
|
500
|
0.551
|
|
600
|
0.596
|
|
700
|
0.644
|
|
800
|
0.691
|
Graphing Data on Mass/Period
*This graph isn't very informative, but if we take the log of our theorized power law (ln T= n ln (M_tray+M_object)+ln A) we create a formula with a resemblance to a standard line (y=mx+b).
*adjusted graph for T(period) vs ln (M_tray+M_object)
Graph with T(period) vs ln (M_tray+M_object)
*Using a line fit and "fiddling" with the inputed mass of the tray we created a line graph correlation coefficient of .9999
*We determined the A in our power law was e^y-intercept and n was the slope of the line.
*The current formula T=e^y-intercept (total mass)^slope of line
Graph of fitted line
*After adjusting the mass of the tray further (higher and lower) until the correlation coefficient deviates from .9999 we create a maximum and minimum mass of the tray (280 g and 260 g respectively).
*(low)260 grams slope= .6245 and y-intercept=-4.733
*(high)280 grams slope=.6495 and y-intercept=-4.915
* Object A (a small unmarked mass) is then placed in the tray, and the period of oscillations is measured to be .474 sec.
*Plugging .474 into the formula for low gives us a mass for the Object A as 330 grams.
*Plugging .474 into the formula for high gives us a mass for the Object A as 332.7 grams.
*Actual mass of Object A is measured to be 332 grams.(falls within the two formulas)
* Object B (a filled water bottle) is then placed in the tray, and the period of oscillations is measured to be .608 sec.
*Plugging .608 into the formula for low gives us a mass for the Object B as 619 grams.
*Plugging .608 into the formula for high gives us a mass for the Object B as 618.9 grams.
*Actual mass of Object B is measured to be 593 grams. (does not fall within two formulas)
Conclusion
* Our model gave us a power formula that was very predictive for one object, but not the other.
*I believe this discrepancy has more to do with the selected object however.
*I believe our model has given us a power law for an inertial pendulum.
Uncertainties
*Air resistance
*Object B was a water bottle placed vertically, increasing air resistance.
*Object B was filled with water that would "slosh" around during oscillations, shifting the mass, which contributed to the discrepancy.
