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I timed 50 periods of vibration with a stopwatch. Each mass was timed 3 times. I noticed that for masses of 500 grams or more, there was a noticable buckling of the balance at the endpoints of its vibration. The data table is reproduced below.
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The graph of the mean time for 50 periods (in seconds) vs. the added mass (in grams) is shown below. (It was constructed using the Graphical Analysis program by Vernier.) The graph appears to be pretty-much linear between mass = 50 grams and mass = 450 grams. The curve shown is approximately:
T50 = 16.23 + 0.07287m - (8.988 x 10-5)m2 + (1.144 x 10-7)m3
This equation was derived by using the automatic curve fitting option (polynomial) in the Graphical Analysis program. (The program reports a mean square error of 0.007208 for this equation.)
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The graph below is a plot of the data from mass = 50 gm to mass = 450 grams, which represents the linear part of the graph shown above. The line shown is the linear regression line drawn by the Graphical Analysis program, which gives the equation:
with a regression coefficient of 0.9997
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This inertial balance could be used to accurately determine the mass of an object in the range 0 - 500 grams. It seems to me (although I need to investigate further) that the behavior of the balance for mass = 0 differs from its "loaded" behavior because its mass is distributed more uniformly when no mass is added. When mass is added to the balance, almost all of the mass is concentrated at the same distance from the support.
In the same way, the behavior of the balance for masses above 450 grams was probably due to the flexing of the balance lengthening its period.
The behavior of the balance for masses between 50 gm and 450 gm is also mysterious. I expected that the period of the balance would be proportional to the square root of the mass (SHM) but it turned out to be proportional to the mass. I really don't know why, but I hope I can find some time to investigate it.
