Kaxa2000 said:
If you were to measure the area of a sector that a planet would sweep out in one week around the sun. It would be the same no matter what time of the year it was. What conservation principle is this example demonstrating? Linear, angular or both? and why?
Angular momentum.
From the perspective of a Sun centered system, linear momentum is *not* conserved. You have to look at things from the perspective of the center of mass and consider both the motion of the planet and the Sun in order to find that linear momentum is conserved. Even then, conservation of linear momentum is trivially conserved.
Over a sufficiently small period of time, the angular rate of the planet's motion with respect to a non-rotating, Sun-centered frame and the distance between the Sun and the planet are more or less constant. The area swept out by the planet during this short period of time is thus
\Delta A = \frac 1 2 r^2 \Delta \theta
Dividing by the length of the time interval \Delta t and taking the limit \Delta t \to 0 yields
\dot A = \frac 1 2 r^2 \dot \theta
Kepler's second law says this is zero. Now look at the specific angular momentum of the planet. This is
\vec l = \vec r \times \vec v
Representing the position vector as \vec r = r \hat r, the velocity vector is
\vec v = \dot r \hat r + r\dot \theta \hat{\theta}
Thus
\vec l = r^2\dot \theta \hat z
The magnitude of this vector is twice the areal velocity, the quantity that Kepler's second law says is constant.
