through the angle φis given by s=aφ. The magnitude of the linear speed v, of the
point P, is given by
v=ds
dt
=adφ
dt
=aω =|v |.
Figure 6-14. Rotation of a rigid body about a line.
The geometry in figure 6-14, is investigated and indicates that a=|r |sin θ, and
hence the magnitude of the velocity can be represented as
ds
dt =|v |=|ω||r|sin θ.
The velocity vector is always normal to the plane containing the position vector and
the angular velocity vector. Therefore the velocity vector can be expressed as
dr
dt =v =ω×r =|ω||r |sin θˆen,
where ˆenis a unit vector perpendicular to the plane containing the vectors ωand r.
The above arguments demonstrate that the expression for the velocity of a rotating
vector is independent of the orientation of the cartesian x-,y-,z-axes as long as the
origin of the coordinate system lies on the axis of rotation. To prove this result let
O′denote the origin of some new x′, y ′, z ′cartesian reference frame with its origin on
the axis of rotation. If r 1 is the position vector from this origin to the same point P
considered earlier, one finds that
dr 1
dt
=v 1 =ω×r 1.
