Kepler's laws
Torque



Torque.

Essential to proving Kepler's second law (and further laws)
is the concept of torque. A torque is a tendency to change something's
state of rotation; it is the rotational analogue of force.
For instance, if I apply torque to a wheel, I'm providing a tendency
to rotate that wheel. Torque is in rotational mechanics what force is
in linear mechanics.
Torque, tau can be employed as

tau = r cross F

equation 1

where F is the impressed force and r is the lever arm over which
it acting; that is, the vector that begins at the axis of rotation and
ends at the point where the impressed force is acting. Note that
torque is a vector quantity; this means that it has a direction. The
direction of the torque indicates in which direction the body tends to
rotate.
That doesn't seem very directly related to celestial mechanics,
does it? But while torque is usually applied to rigid bodies, such as
wheels and levers, it does not have to be. The concept of torque can
be applied to any body with respect to a fixed point in space. The
vector between this fixed point and the body then becomes the lever
arm, although it is by no means a solid one.
We shall apply this notion of torque to a planet orbiting the Sun.
Here, however, the impressed force will be gravity. Our fixed
reference point will be the Sun itself. We know that r = r r
and F = (G m M/r^{2}) r so we can get

tau = r cross F

equation 2


tau = (r r) cross [(G m M/r^{2}) r]

equation 3


tau = (G m M/r) (r cross r)

equation 4


tau = o.

equation 5

We know that any vector crossed with itself is the zero vector, o, so
the Sun never impresses a torque on a planet. This makes perfect
sense: if you can only pull radially on bucket (as the Sun can only
pull radially on a planet), you won't be giving the bucket rotation
about an axis a tendency to speed up or slow down in its rotation.

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