Newton's Second Law
for Rotations
The rotational version of
Newton's law is

(B.28) 
where
denotes the
angular
acceleration. As in the previous section,
is
torque
(tangential
force times a moment arm
), and
is the
mass moment of inertia. Thus, the net applied torque
equals the time derivative of
angular momentum , just as
force
equals the timederivative of
linear momentum :
To show that Eq.
(
B.28) results from
Newton's second law
,
consider again a mass
rotating at a distance
from an axis
of rotation, as in §
B.4.3 above, and
let
denote a
tangential force on the mass, and
the corresponding tangential acceleration. Then we have, by Newton's
second law,
Multiplying both sides by
gives
where we used the definitions
and
.
Furthermore, the lefthand side is the definition of
torque .
Thus, we have derived
from Newton's second law
applied to the tangential force
and acceleration
of the mass
.
In summary, force equals the timederivative of linear
momentum, and
torque equals the timederivative of angular momentum. By Newton's
laws, the timederivative of linear momentum is mass times
acceleration, and the timederivative of angular momentum is the mass
moment of inertia times angular acceleration:
Next Section: Equations of
Motion for Rigid BodiesPrevious Section: Torque