Re: kinetic energy paradox?

["Carl E. Mungan" <mungan@USNA.EDU>]

> Consider a ball bearing rolling across a frictionless (yes, I know
> there must be a minimal friction force or the thing would never
> roll...) tabletop or

nitpicker's note: Consider a wheel on your car rolling along pavement
at constant speed on a flat road. Suddenly you hit an icy patch. But
you don't panic. You keep the wheel straight and you keep your feet
off the accelerator and brakes. Doesn't the wheel keep rolling
merrily along?

>  desktop parallel to the edge of the table or desk. It's rolling
> with a constant speed, and therefore a constant kinetic energy.
> Suppose at a given instant, you give the ball bearing an impulse
> directed perpendicular to it's initial velocity at that instant. The
> ball bearing's parallel velocity component will not change, but
> there is now a non-zero prependicular velocity component. The
> magnitude of the total velocity is now different, and therefore the
> ball bearing's kinetic is now also different; it's increased.
>
> As a result of the new non-zero perpendicular velocity, the total
> velocity is now pointing in a slightly different direction (the
> smaller the duration of application of the force the smaller the
> change in direction, right?).
>
> Now, if you look at the gravitational force on Earth by the Sun, and
> assume Earth's orbit is circular, the Sun's gravitational force does
> no work on Earth and Earth's kinetic energy is constant. The
> gravitational force only serves to change the direction of Earth's
> velocity.
>
> My quandry is this: why is Earth's kinetic energy conserved and the
> ball bearing's kinetic energy NOT conserved? In both cases, an
> impulse is imparted perpendicular to the velocity.
>
> Is the answer just that Earth receives MANY such impulses
> continuously, one right after the other, and each impulse is
> *instantaneously perpendicular to the displacement (and velocity)
> vector at that same instant*? And it's the *tangential* speed that
> determines the orbital kinetic energy, and this component isn't
> changed.

Just to add to what Joel already said, the magic is in the
*constantly changing direction* of the force while it is being
exerted, rather than in the number of impulses or how nearly
instantaneous they may or may not be.

I could perhaps imagine making gravity turn on and off periodically
in short bursts. (A bunch of equally spaced gravity shields
surrounding the sun with only small open slits between them?) Then
earth would go around the sun in a nearly equilateral figure. (With a
smaller average radius, assuming we start with the same total
mechanical energy, owing to the decreased duty cycle.) Yet its KE
would still be constant.

It is amusing to push this idea to its limits and consider earth
moving around the sun in an equilateral triangle (with very slightly
rounded corners).
--
Carl E. Mungan, Asst. Prof. of Physics  410-293-6680 (O) -3729 (F)
      U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
mungan@usna.edu    http://physics.usna.edu/physics/faculty/mungan/