Re: Question about Kepler's second law
Re: Question about Kepler's second law
Source: http://sci.tech−archive.net/Archive/sci.physics/2007−04/msg01180.html
• From: "PD"
• Date: 11 Apr 2007 10:08:36 −0700
On Apr 11, 11:11 am, "Peter" wrote:
On Apr 11, 11:16 am, "PD" wrote:
On Apr 11, 8:08 am, "Peter" wrote:
On Apr 10, 8:58 pm, "PD"
wrote:
One more thing.
It crosses my mind you may be confused
about the term "tangential".
Someone else mentioned this to you already.
In a *circular* orbit, the direction along the
planet's motion is
purely tangential.
But in an elliptical orbit, there are two
possible meanings of
"tangential" and I think this is causing you
trouble.
One meaning is "touching the curve at one
point". In your drawing of
the planet crossing the minor axis, this is the
horizontal line that
just touches the ellipse where the planet is.
The other meaning is "perpendicular to the
radius", where the radius
is the line from the sun to the planet.
Obviously, the horizontal line you've drawn
in your picture is not
Re: Question about Kepler's second law 1
Re: Question about Kepler's second law
perpendicular to the line from the sun to the
planet.
Your memory tells you that tangential speed
of the planet increases if
there is a net torque on it. This is correct, but
that pertains
strictly to the *second* meaning of
"tangential" and NOT the first
meaning.
I apologize on behalf of the community of
physics teachers for this
ludicrous source of confusion, and I hope
that you can forgive and get
around it.
PD− Hide quoted text −
− Show quoted text −
Do you think it is wrong to say that only a net force in the
direction
of the velocity of a planet can change its speed?
Let's answer this carefully. Consider a bullet that is fired
horizontally and let's do it somewhere where air friction is not a
significant concern or where the curvature of the Earth is not big
enough to matter. Gravity acts on the bullet directly vertically. So
as the bullet falls, does it speed up, slow down, or stay the same
speed? The answer is that it *speeds up*. The reason is that the
horizontal component of the velocity is unchanged by gravity, and so
it stays constant. But the vertical component of the velocity
increases in magnitude, due to the influence of gravity. Since you
have a constant horizontal component and a growing vertical component,
the magnitude of the total velocity is increasing.
Re: Question about Kepler's second law 2
Re: Question about Kepler's second law
You can look at the planet in a similar way. Draw a line from the sun
to the planet (let's call this the V line). Now draw a line at the
planet that is perpendicular to the first line (let's call this the H
line). The velocity vector of the planet (as it crosses the minor
axis) will be an arrow that lies somewhere between those two
perpendicular lines. Therefore this velocity vector has components
along both the H line and the V line. The force of gravity toward the
sun is strictly along the V line. The component of the planet's
velocity along the H line will not change, but the component along the
V line will change. Just like in the bullet case, the total velocity's
magnitude will then change.
Do you think linear
and angular momentum are two different kinds of
momentum?
They are conserved separately, in that thelawof conservation of
linear momentum and thelawof the conservation of angular momentum
both apply independently. It is certainly possible for there to be a
force that changes the linear momentum of an object without changing
its angular momentum and vice versa. I've already given two examples
of that.
If you'd like to see some math that supports that, then let's take the
simplest possible case: a small object with mass m and velocity v,
passing by an axis at distance r. The angular momentum of that object
is m*v*r. Now is it possible to change the linear momentum m*v without
changing m*v*r? Certainly. Simply arrange it such that r changes as
much in the opposite direction as m*v changes. (This is essentially
what happens with a comet orbiting the Sun. As r decreases, m*v
increases, keeping m*v*r constant. This, by the way, is the conceptual
essence of one ofKepler'slaws −− the equal swept areaslaw. If you
look at two different wedges in an elliptical orbit swept by equal
time spans, then the wedge with the smaller radius will have the
greater arc length covered in the same time −− smaller radius, greater
linear speed.)
PD− Hide quoted text −
− Show quoted text −
Re: Question about Kepler's second law 3
Re: Question about Kepler's second law
If the speed of a planet increases because it is falling due to the
attraction by the Sun, since it is always falling, why
doesn't its speed increase continuously? Or, Is this a dumb question?
Peter
It's not dumb, you just have to look at the other side of the orbit,
where the planet crosses the minor axis again. Now you see there is a
component of the force of gravity that lies along the line of motion
of the planet, but it's pointed in the opposite direction. So here the
planet slows down for *exactly* the same reason that it sped up on the
other side.
You may want to Google "Kepler applet" to see animations.
PD
.
Re: Question about Kepler's second law 4