A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.3 Is rotation a vector?

velocities are conventionally measured in radians per second, whereas angular

accelerations are measured in radians per second squared.

For a body rotating with constant angular velocity, ω, the angular acceleration

is zero, and the rotation angle φ increases linearly with time:

φ(t) = φ 0 + ω t, (8.4)

where φ 0 = φ(t = 0). Likewise, for a body rotating with constant angular accel-
eration, α, the angular velocity increases linearly with time, so that

ω(t) = ω 0 + α t, (8.5)

and the rotation angle satisfies

φ(t) = φ 0

+ ω 0

t +

1

α t^2. (8.6)

2

Here, ω 0 = ω(t = 0). Note that there is a clear analogy between the above equa-

tions, and the equations of rectilinear motion at constant acceleration introduced

in Sect. 2.6—rotation angle plays the role of displacement, angular velocity plays

the role of (regular) velocity, and angular acceleration plays the role of (regular)

acceleration.

8.3 Is rotation a vector?

Consider a rigid body which rotates through an angle φ about a given axis. It

is tempting to try to define a rotation “vector” φ which describes this motion.
For example, suppose that φ is defined as the “vector” whose magnitude is the

angle of rotation, φ, and whose direction runs parallel to the axis of rotation.

Unfortunately, this definition is ambiguous, since there are two possible directions

which run parallel to the rotation axis. However, we can resolve this problem by

adopting the following convention—the rotation “vector” runs parallel to the axis

of rotation in the sense indicated by the thumb of the right-hand, when the fingers

of this hand circulate around the axis in the direction of rotation. This convention

is known as the right-hand grip rule. The right-hand grip rule is illustrated in

Fig. 68.