Comparing these last two equations it is found that the time rate of change of
angular momentum is expressible in terms of the force F acting upon the particle.
In particular, one can write
dH
dt=r ×F =M .One of the many marvelous things introduced by the early Greek mathematicians
was that symbols represent ideas and concepts. The symbols in our last equation
tell us about a fundamental principal in Newtonian dynamics, that the time rate
of change of angular momentum equals the moment of the force acting on the
particle.
Angular Velocity
Arigid body is one where any two distinct points remain a constant distance
apart for all time. A rigid body in motion can be studied by considering both
translational and rotational motion of the points within the body. Assume there
is no translational motion but only rotational motion of the rigid body. A simple
rotation of every point in the rigid body, about a line through the body, can be
described by (a) an axis of rotation Land (b) an angular velocity vector ω.If the
axis of rotation remains fixed in space, then all points in the rigid body must move
in circular arcs about the line L. Consider a point Prevolving about Lin a circular
path of radius aas illustrated in figure 6-14.
The average angular speed of the point P is given by ∆∆φt,where ∆φis the angle
swept out by P in a time interval ∆t. The instantaneous angular speed is a scalar
quantity ωdetermined by
ω=dφ
dt= lim∆t→ 0 ∆φ
∆t.There is a direction associated with the angular motion of P about the line L and
thus an angular velocity vector ωis introduced and defined so that
(i) ωhas a magnitude or length equal to the angular speed ω,
(ii) ωis perpendicular to the plane of the circular path.(iii) The direction of ωis in the direction of advance of a right-hand screw when turned
in the direction of rotation.