Re: [Phys-l] Queston: simple pendulum lengthen suspension.

["Jeffrey Schnick" <JSchnick@Anselm.Edu>]

> -----Original Message-----
> From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
> bounces@carnot.physics.buffalo.edu] On Behalf Of Bernard Cleyet
> Sent: Saturday, March 17, 2007 5:42 PM
> To: PHYS-L Maillist
> Subject: [Phys-l] Queston: simple pendulum lengthen suspension.
>
>
> What happens to the amplitude if one suddenly lengthens the suspension
> at equilibrium position or at max. displacement?
>
> bc, puzzled

Treating the simple pendulum as a point mass m on the end of a rigid
massless rod of length L which is suddenly extended to length L', and
defining the amplitude of the oscillations to be the angle that the
pendulum makes with the vertical when it is at maximum displacement from
its equilibrium position:

If the length is increased at maximum displacement, when the angular
momentum is zero, then the amplitude stays the same.  The pendulum is at
its maximum angular displacement from equilibrium both before and after
the extension.  In requiring a sudden extension of the rod at maximum
displacement, we are stipulating that the amplitude is the same.  The
potential energy difference that matters, the gravitational potential
energy of the bob/earth system at maximum bob displacement from its
equilibrium position minus the potential energy of the bob/earth system
at the equilibrium position of the bob, actually increases as a result
of the extension.
U=mgL(1-cosA) is proportional to L, with A being constant.
(This seems counterintuitive--by decreasing the potential energy by
lowering the bob, we have increased the potential energy relative to the
potential energy at the equilibrium position because we have lowered the
equilibrium position more than we lowered the bob.)

If the length of the extensible rod increases at the equilibrium
position, conservation of angular momentum, L'mv'=Lmv leads to
v'=(L/L')v The speed of the bob at equilibrium decreases as a result of
the extension.  A decrease in speed means a decrease in the kinetic
energy of the bob at its equilibrium position which means a decrease in
U (the potential energy difference that matters) which means that A'<A
(in order for
U'=mgL'(1-cosA') to be less than U=mgL(1-cosA) with L'>L).


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