Physics of Music Physics 121
Griffioen Fall 1994
A Mass On A Spring
Here are the results from the measurement in class of the period of
a mass on a spring. We suspended the foot-long, 1-inch diameter spring
from a lab stand and hung masses of 50, 200, 500 and 1000 grams on the
end. In each case, we determined that the period was independent of
the amplitude and we were justified in measuring the time for 10 periods.
For each mass, we took 3-4 measurements to reduce the chances of random
error. The average measured time divided by 10 periods is listed
in the table below. The accuracy of any one
measurement was about 0.2
seconds, which we determined from the spread in the measured points.
The error on the period, then, is 1/10 of this number, because our
uncertainty is spread over 10 periods. Thus, the uncertainty of the
measured period is 0.02 seconds. For 1000 g, the period is longer
and our accuracy is somewhat greater. The
graph of these data show that the relationship between mass
and period is not linear. In fact, the fit shows that the
period is roughly proportional to the square root of the mass.
In fact, a simple theory of the spring
which assumes Newton's Laws and linear restoring forces, predicts the
square-root dependence on the mass. In this way, we can see how our
experience (the measurement) corresponds to our intellectual framework
(the mathematical, theoretical description). The validity of the latter
is always driven by the accuracy and reproducibility of the former.
It's truly astounding that a mathematical description of our
experience works so well!
Table
mass (g) period (s) uncertainty (s)
50 0.64 0.02
200 0.98 0.02
500 1.46 0.02
1000 2.03 0.01