One way to look at the mass hanging from the bottom
of a spring, the top of which is simusoidally driven,
is begin with a Lagrangian.
In this diagram, upward is to the left. The position
of the driven top of the spring is x(t). The position
of mass m is y. Assume that the spring has negligible
mass and, for the moment, that there is no damping.
OOOOOOOOOOOOOOOOOOOO---[m ]
<---|---> | <---- upward
x(t)=x_0 cosWt y
Lagrangian L = (1/2)m[dy/dt]^2 - (1/2)k[x(t)-y]^2
Then the equation of motion is
m[d^2y/dt^2] +ky = kx_0 cosWt
Linear damping can be added, as usual, by adding a term
b[dy/dt]. Other damping can be added similarly.
That is, this is a straightforward forced oscillator for
which the applied force is kx_0 cosWt.
For many years we have used such a forced-oscillator in
our introductory course. Driving angular frequency is
varied by moving a wheel to different radii on an old-
fashioned record turntable. A string attached eccen-
trically to the wheel then, via pulleys, drives the top
of the spring. Damping is provided by vanes attached to
the bottom of the mass that move in a dashpot (a long tube)
of water. The amplitude of the motion of m is read
directly from a meter stick attached to the device.
A spark device enables determination of the phase
of the displacement of the mass relative to the driving
force.
In our experiment, we find the spring constant from the free-
motion period, then look at damped unforced motion, and
finally look at the motion as W is varied from well below
to well above resonance.
+++++++++++++++++++++++++++++++++++++++++
+ Herschel Neumann, Professor +
+ Department of Physics and Astronomy +
+ Physics Building, Room 214 +
+ University of Denver +
+ Denver, CO 80208-2238 +
+ Telephone: 303-871-3544 +
+ Department Telephone: 303-871-2238 +
+ Fax: 303-871-4405 +
+ E-Mail: hneumann@du.edu +
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