| Most small mechanical oscillations can be
approximated as simple harmonic oscillations and can be
modeled as a spring. You remember that for a spring, the
restoring force is F = kx, where k is known as the "restoring
constant". The force is directly proportional to
how far you stretch the spring. That lets it be simple
harmonic.
Simple harmonic oscillations are described by a differential
equation. For a spring and mass, neglecting friction
and gravity,
d2x/dt2 - (k/m) x = 0
and the solution for x is a sinusoid
x = A cos(ωt +
φ), where A is the maximum displacement or the Amplitude.
The total energy for an oscillating mass is simply
the sum of its potential energy at any time (from compressing
and extending the spring) and its kinetic energy at
any time (from the motion of the mass itself).
E = 1/2 m v2
+ 1/2 k x2
Taking x and
dx/dt = v and substituting, you find that the
total energy has a very simple form:
E = 1/2 k A2
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So, if I take a pendulum, like a bowling ball on the
end of a cable, and bring it to the nose of my TA, Paul,
and then let it go, the total energy of the bowling
ball only depends on how high Paul's nose is. Unless
Paul gives it a push, the bowling ball will only return
to Pauls' nose and no further
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| (The angle that the bowling ball pendulum
makes with the vertical is way too big to let it be simple
harmonic motion. But the total energy is still represented
by how high you start the ball.) |
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