CLASSICAL MECHANICS & THERMODYNAMICS - Chapter 6, Exercise 3 Solution
==========================================================
Conceptual: Kepler's Second Law via Conservation of Angular Momentum

PROBLEM
-------
Explain, using this chapter's own explanation of Kepler's Second Law,
why a planet moves fastest when closest to the Sun (perihelion) and
slowest when farthest away (aphelion) - connecting the explanation
directly to conservation of angular momentum from Chapter 5.

SOLUTION
--------
Chapter 5 established that angular momentum, L = I omega, is
conserved in any system with no external torque acting on it. For a
planet orbiting the Sun, gravity is the only significant force
involved, and this chapter's own explanation of Kepler's Second Law
points out that gravity always pulls directly along the line between
the planet and the Sun - meaning it exerts zero torque about the Sun
(torque requires a force with a component PERPENDICULAR to the
radius, per Chapter 5's own tau = rF sin(theta) formula; here theta =
0 degrees between the force and the radius line, so sin(theta) = 0
and torque is exactly zero).

With zero torque acting on it throughout its orbit, the planet's
angular momentum about the Sun stays exactly constant everywhere
along its path - at perihelion, at aphelion, and everywhere in
between.

For a planet in an elliptical orbit, the distance r to the Sun changes
constantly. Angular momentum for orbital motion can be written
L = m v r (mass x speed x distance from the Sun, for motion
perpendicular to that radius). Since L, and m, must both stay
constant:

    v r = constant

This means v and r are inversely related: when r is small (perihelion,
the planet's closest approach), v must be large to keep v times r
constant. When r is large (aphelion, the planet's farthest point), v
must be correspondingly smaller.

ANSWER: A planet moves fastest at perihelion and slowest at aphelion
because its angular momentum (L = mvr) is conserved throughout its
orbit - gravity's force always points directly at the Sun and so
produces zero torque - meaning that as the planet's distance from the
Sun shrinks, its speed must increase to compensate, and vice versa.

----
WHY THIS WORKS AS AN ANSWER

This is precisely the same real physics behind Chapter 5's own figure
skater example, applied to an orbit instead of a spin: pulling in
(reducing r, like the skater pulling in her arms) forces speed to
increase to keep angular momentum constant, while moving farther out
(increasing r) forces speed to decrease. Kepler observed this pattern
purely empirically in 1609, without knowing why it happened; this
chapter's own explanation - that it follows directly and necessarily
from conservation of angular momentum under a purely radial force -
is exactly the kind of "why" that Newton's approach added on top of
Kepler's original "what."