Uncertainty in the Expected Value: A Ballistic Pendulum

In considering error analysis for an experiment, we often treat the "experimental" values--the measurements which we make--as having some uncertainty (e.g. a standard deviation) and the theoretical value as being certain. This is not, however, always the case. For example, the "nominal" value of a thing--resistor, mass, pull strength, etc--often has associated with it some error. The resistance in a resistor is specified by its colored bands, with an uncertainty value* explicitly specified by the fourth colored stripe. Thus, the "nominal" value is not without some uncertainty.

Furthermore, the "theoretical" value--calculated from some curve--may often have some uncertainty of its own. After all, the theoretical curve from which the theory value may be obtained is itself computed using a set of measured values. As an example, consider the maximum change in height for a ballistic pendulum as a function of the projectile's mass. If the pendulum is shot at using a spring loaded gun to launch the projectile, then this maximum height should be calculatable using the following parameters: effective mass of the pendulum, mass of the projectile, spring constant of the spring, initial and final compression of the spring, mass of the driver used to propel the ball, frictional forces within the system.

The maximum displacement height for a ballistic pendulum as a function of the mass of the projectile. This particular pendulum had an effective mass of 75 grams, a driver assembly mass of 42 grams, and a spring of spring constant 2730 N/m and a measured compression of about 2.3 cm (solid curve).This is plotted with data (averaged 10 shots, with error bars being standard deviations of the means) as well as two enveloping lines (dashed lines) representing 0.5 mm more (gray) or 0.5 mm less (black) for the spring compression.

Now consider a spring-loaded projectile gun with a very stiff spring and a very short compression distance. Any difference in the measured compression distance will easily result in a noticeable shifting of the theoretical curve computed using this distance (see the image above). An error of only 0.5 mm in measuring the spring's compression--and this is a decent guess as the the uncertainty of the measurement, which was made using a ruler--could result in the maximum displacement height's being "off" by 1 cm. Thus, the uncertainty in the "theoretical" curve may be great enough that the entire curve could overestimate or underestimate most values of the displacement. Instead, the curve is a sort of "theoretical envelope" within which the experimental values should fall.

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*Technically, this is the tolerance, and the uncertainty is much lower if the resistance is actually measured via an Ohmmeter.