and so it is placed at the end of the position vector, as illustrated in the figure 6-13
to show that the velocity is tangent to the circle. The magnitude of the velocity v
is the speed vgiven by
|v |=v=√
r^2 ω^2 sin^2 ωt +r^2 ω^2 cos^2 ωt =rωOne can define an angular velocity vector ωas follows. Use the right-hand rule
and point the fingers of your right-hand in the direction of the position vector r
and then rotate your fingers in the direction of motion of the particle. Your thumb
then points in the direction of the angular velocity vector. For circular motion
counterclockwise in the x, y -plane, one can define the angular velocity vector ω=ωˆe 3.
By defining an angular velocity vector one can express the velocity vector of a
rotating particle by
v =ω×r =∣∣
∣∣
∣∣ˆe 1 ˆe 2 ˆe 3
0 0 ω
rcos θ r sin θ 0∣∣
∣∣
∣∣=ˆe^1 (−ωr sin θ)−ˆe^2 (−ωr cos θ), θ =ωt (6 .57)and this equation can be compared with equation (6.56).
The acceleration of the rotating particle is given by
a =dv
dt=−rω^2 cos ω t ˆe 1 −rω^2 sin ω t ˆe 2 =−ω^2 rThis shows the acceleration is directed toward the origin. It is therefore called a
centripetal acceleration.^8 The magnitude of the centripetal acceleration is
|a |=ω^2 r=v2
r =vωThe acceleration can also be obtained by differentiating the vector velocity given by
equation (6.57) to obtain
a =dv
dt=ω×dr
dt+dω
dt×rand since ωis a constant, then ddtω= 0 so that the above reduces to
a =ω×dr
dt=ω×v =ω×(ω×r ) = −ω^2 r(^8) Centripetal means “center-seeking”.