Re: Friction causes increase in speed?
- From: "Greg Neill" <gneillRE@xxxxxxxxxxxxxxxx>
- Date: Fri, 3 Apr 2009 13:17:58 -0400
jbriggs444@xxxxxxxxx wrote:
On Apr 3, 12:19 pm, "Greg Neill" <gneil...@xxxxxxxxxxxxxxxx> wrote:
Ray Vickson wrote:
On Apr 3, 7:30 am, "Greg Neill" <gneil...@xxxxxxxxxxxxxxxx> wrote:
jmorr...@xxxxxxxxxxx wrote:
I was thinking about a satellite in a circular obit around the
earth...an orbit high enough that orbital decay from air friction is a
very minor concern.
However, there is <some> air friction, so the satellite is gradually
slowing and dropping into lower and lower, still almost perfectly
circular, orbits.
But those lower circular orbits have a <higher> orbital velocity. So,
is the air friction causing an increase or decrease in velocity?
The friction is "stealing" mechanical energy from the
orbital system. By lowering the orbit, energy is being
extracted from the gravitational potential energy, some
of which is also moved into kinetic energy in order to
keep the system's angular momentum constant.
Why should the angular momentum be conserved in this case? The
friction force has a non-radial component, so theorems about angular-
momentum conservation do not apply. For a more earth-bound example,
consider a spinning turntable mounted on a spindle that has friction,
or a spinning top with some friction between the tip and the table.
Angular momentum is always conserved. If you look
closely at your examples you'll see that frictional
forces are allowing angular momentum to be transferred
to the table --> planet.
True, but irrelevant.
Your original claim was that kinetic energy had to increase in order
to keep the system's angular momentum constant.
That claim is manifestly false. As you point out, the system's
angular momentum will be conserved in any case. If, for instance, you
replaced the atmosphere with molasses then the satellite would be
reduced to negligible kinetic energy in spite of the fact that angular
momentum would still be conserved.
That's a good example, and it certainly points out
the mistake I made. Mea culpa. Thanks for pointing
this out.
What is true is that there is a balance between orbital velocity,
orbital radius and gravity that must be maintained in order to have a
[nearly] circular orbit. That balance requires that orbital velocity
increase as orbital radius decreases. That balance has next to
nothing to do with angular momentum conservation.
As you say, the balance is between potential and kinetic energy,
thus the specific mechanical energy for a body in orbit:
E = v^2/2 - U/r U being the planet's gravitational parameter, GM.
A circular orbit further constrains things so that
v = sqrt(U/r)
.
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