The period of oscillation T of a simple pendulum for small amplitude is given by:
where g is the standard acceleration of gravity and L is the length of the string of the pendulum (or, more precisely, the distance from the point of suspension to the center of the pendulum bob).
A student constructs a pendulum to determine a value for g. The student measures L to be 153.2 centimeters (cm). It is decided that the uncertainty in this measurement is primarily due to the difficulty in measuring L to the center of the bob, and that this uncertainty is about plus or minus 2 millimeters (mm). (In other words, it is estimated that the measurement is probably within 2 millimeters of the correct value.) Thus, L is taken to be 153.2 +/- 0.2 centimeters. In standard units:
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| (6.1.1) |
Now, the student sets the pendulum swinging with a small amplitude (a few centimeters). Using a stopwatch, the student measures the time for 30 oscillations to be 75.3 seconds (s). The uncertainty in this measurement is due to the human reaction time in operating the stopwatch, and so the time measurement is taken to be 75.3 +/- 0.5 seconds.
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| (6.1.2) |
Because the number of oscillations (30) is exact, the period of a single oscillation is:
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| (6.1.3) |
The previous formula rearranges to:
and so, the student calculates the experimentally determined value of g to be:
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| (6.1.4) |
The accepted value of g is:
Assuming the error in the experimental value represents a standard deviation, the accepted value thus lies farther than one standard deviation from the experimental central value. The student then must decide whether to repeat the experiment, or consider possibly overlooked sources of measurement error.