We already showed that the torque about a point
can also be thought of as a decomposition.
We take the vector from the point P to the center of mass
and apply all the forces acting on the particle
at the center of mass.
And we can calculate the torque about the center
of mass due to the action of some forces
where we're having forces acting about the center of mass.
Now if we choose the point P to equal the center of mass,
then we know that the vector r center of mass
to the center of mass is 0.
So the torque about the center of mass
is just equal to the forces about that point.
We know that torque is always just Lcm/dt.
Now, again, how could we justify that statement
that because we're only calculating
the torque about the center of mass,
it's only the rotational angular momentum about the center
of mass that's changing.
We saw before that if we thought of Lp, again,
as a translational and a rotational angular momentum--
I'm sorry, rotational angular momentum, omega--
and the point p was equal to the center of mass,
then this first piece would be 0.
And L about p is only Icm omega and dLp/dt
is equal to Icm alpha for a fixed axis.
And that's exactly equal to dLcm/dt.
And so the point here is that when
we're applying problems involving rotation
and translation, we can just analyze the torque
about the center of mass and only consider
how the angular momentum about the center of mass is changing.
And then for the center of mass motion--
so this gives us our rotational dynamics.
And for our linear dynamics, we will still
apply F equals the total mass times Acm.
So that's our linear dynamics.
And this is our overall decomposition
of rotational motion.
To analyze it, we study the rotational dynamics
and the linear dynamics.