Re: [Phys-l] Force and potential

[brian whatcott <betwys1@sbcglobal.net>]

On 4/4/2012 2:11 PM, LaMontagne, Bob wrote:
> Deep in the sugar shock of eating too many Easter "Peeps", I came up with the following:
>
> 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).
>
> :-(
>
> Bob at PC
I thought this was a splendid puzzle from Bob.
He led us to a category error in fact. Bravo!

If I check the form of the expected horizontal  force on a ball in a 
bowl, by inspection I evidently expect  zero initial horizontal  force, 
with a gradual increase  to some intermediate position, followed by a 
decrease to zero at the base of the bowl. Why does Bob lead us in 
another direction?
He cleverly uses the equation of the locus of a point on a semicircle 
which you might describe as a static, kinetic or geometric construct, 
and starts applying dynamic operations.
We all know perfectly well the shape of the tension on a string and bob, 
a similar physical setup,  and can easily decompose it to x and y 
components. This would then be  a dynamic equation - one that deals with 
forces and accelerations - a fish of quite another stripe. (a force 
whose y maximum is greater than mg of course, when the ball passes its 
nadir so to speak.)

Brian W

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