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Re: [Phys-l] Force and potential



On 04/04/2012 12:11 PM, LaMontagne, Bob wrote:

Consider a bowl formed from the lower half of a sphere. In 2D I can
write the equation for the bowl as x^2 + (y-1)^2 = 1.

This gives y = 1 - sqrt(1-x^2)

If I place a marble at the inside rim (x=1, y=1) of the bowl and
release it, the marble will oscillate back and forth, repeatedly
coming back to the release point.

The potential energy is given by U=mgh=mgy = mg[1 - sqrt(1-x^2)]

Using F_x = - dU/dx , then we have F_x = -x/sqrt(1-x^2) which implies
that F_x is infinitely strong when x = 1 (the release point).

Well, mathematically speaking, the derivative does not exist at
all at x=1. The function is not differentiable there.

Physically, it is _half true_ that the force in the x direction is
infinite under the stated conditions. In particular, if you start
at (x,y) = (1,1) and push in the +x direction, you are resisted by
an infinite force, i.e. the force of constraint. If we imagine that
the particle is constrained to move along the surface of the bowl
(as opposed to the interior), then the force is infinite in the -x
direction as well.

Even more physically, the force in the x direction has no physical
significance at this point. It contributes nothing to the energy,
since the particle is constrained to move perpendicular to the x
direction.
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