c/The amplitude would usu-
ally be defined as the distance
from equilibrium to one extreme
of the motion, i.e., half the total
travel.
we want to analyze small oscillations, it doesn’t even matter which
function we assume. For simplicity, we’ll just useU(x) = (1/2)kx^2
from now on.
Now we’re ready to analyze the mass-on-a-spring system, while
keeping in mind that it’s really only a representative example of
a whole class of similar oscillating systems. We expect that the
motion is going to repeat itself over and over again, and since we’re
not going to include frictional heating in our model, that repetition
should go on forever without dying out. The most interesting thing
to know about the motion would be the period,T, which is the
amount of time required for one complete cycle of the motion. We
might expect that the period would depend on the spring constant,
k, the mass,m, and and the amplitude,A, defined in figure c.^11
In examples like the brachistochrone and the Apollo 11 mission,
it was generally necessary to use numerical techniques to determine
the amount of time required for a certain motion. Once again, let’s
dust off the time3function from page 93 and modify it for our
purposes. For flexibility, we’ll define the functionU(x) as a separate
Python function. We really want to calculate the time required for
the mass to come back to its starting point, but that would be
awkward to set up, since our function works by dividing up the
distance to be traveled into tiny segments. By symmetry, the time
required to go from one end to the other equals the time required
to come back to the start, so we’ll just calculate the time for half a
cycle and then double it when we return the result at the end of the
function. The test at lines 16-19 is necessary because otherwise at
the very end of the motion we can end up trying to take the square
root of a negative number due to rounding errors.(^11) Many kinds of oscillations are possible, so there is no standard definition of
the amplitude. For a pendulum, the natural definition would be in terms of an
angle. For a radio transmitter, we’d use some kind of electrical units.
116 Chapter 2 Conservation of Energy