Re: oscillations test question

[John Mallinckrodt <ajm@CSUPOMONA.EDU>]

> When the mass is hung on the spring, the spring stretches by an
> amount d = mg/k. (Non-conservative work was done by your hand taking
> the mass to the equilibrium point - otherwise the mass would
> oscillate!)
>
> If we call this position x=0 and now pull the mass downward by an
> additional amount, x, the potential energy term is 1/2 k (d + x )^2
> - m g x.   Because of the cross term in the square, 1/2kx^2 does not
> seem to take on any easily identifiable meaning (at least not for an
> introductory physics class).
>
> Perhaps someone out there has thought about this enough to have come
> up with an insightful meaning to the 1/2 k x^2 - please post it if
> you have.

As Robert Cohen said in his post "If the [mass] is ... stretched a
distance x from the new equilibrium point, the potential energy
relative to the new equilibrium point is equal to 1/2 kx^2."

This is easily shown:

Potential energy relative to the new equilibrium point
   = (1/2 k (d+x)^2 - 1/2 k d^2) - mgx
   = kdx + 1/2 kx^2 - mgx
   = 1/2 kx^2

I recall as a student being mystified by the fact that my textbook
and my professors seemed to take it for granted that the oscillations
of a mass hung from a spring should not be affected by the fact that
the spring is stretched in the equilibrium position.  It seemed
obvious to me that a stretched spring would be "stiffer" than an
unstretched spring and that the oscillation frequency would,
therefore, be higher than it would have been if we had attached the
same mass to the same spring in a gravity-free (or horizontal and
frictionless) configuration.  It took me some effort even to make one
of my professors understand what I was worried about.  It still seems
very odd to me that textbooks don't generally address this point, but
I have to admit that few if any students ever seem troubled by it in
the way that I was!

--
John Mallinckrodt        mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.