Just as we did for the angular velocity in equations (4 3) and (4 4) (but using a vector notation), we can define an average angular acceleration over a time
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- 62- Chapter 4. Rotation and Conservation of Angular Momentum Notes: • Most of the material in this chapter is taken from Young and Freedman, Chaps. 9 and 10. 4.1 Angular Velocity and Acceleration We have already briefly discussed rotational motion in Chapter 1 when we sought to derive an expression for the centripetal acceleration in cases involving circular motion (see Section 1.4 and equation (1.46)). We will re visit these notions here but with a somewhat broader scope. We reintroduce some basic relations between an angle of rotation θ
about some fixed axis, the radius, and the arc traced by the radius over the angle θ. Figure 1 shows these relationships. First, the natural angular unit is the radian, not the degree as one might have expected. The definition of the radian is such that it is the angle for which the radius r
and the arc s have the same length (see Figure 1a). The circumference of a circle equals 2π times the radius; it therefore follows that1 rad=
3602π =57.3 (4.1) Second, as we previously saw in Chap. 1, the angle is expressed with θ= s r
(4.2) and can be seen from Figure 1b. We can define an average angular velocity as the ratio of an angular change Δθ
over a Figure 1 - The relations between an angle of rotation about some fixed axis, the radius , and the arc traced by the radius over the angle .
- 63- time interval Δt ave,z z Δt(4.3) For example, an object that accomplishes one complete rotation in one second has an average angular velocity (also sometimes called average angular frequency) of 2π
rad/s. If we make these intervals infinitesimal, then we can define the instantaneous angular velocity (or frequency) with ω
z =limΔt→0
z Δt dθ z dt(4.4) That is, the instantaneous angular velocity is the ti me derivat ive of the angul ar displacement. The reason for the presence of the subscript "z
" in equations (4.3) and (4.4) will soon be made clearer. It should be noted that an angular displacement Δθ
can either be positive or negative; it is a matter of convention how the sign is defined. We will define an angular displacement as positive when it is effected in a counter-clockwise direction, as seen from an observer, when the fixed about which the rotation is done is pointing in the direction of the observer. This is perhaps more easily visualized with Figure 2. 4.1.1 Vector Notation Since a rotation is defined in relation to some fixed axis, it should perhaps not be too surprising that we can use a vector notation for angular displacements. That is, just as we can define a vector Δr
composed of linear displacements along the three independent Figure 2 - Convention for the sign of an angle.
- 64- axes in Cartesian coordinates with Δr=Δxe x +Δye y +Δze z (4.5) we can do the same for an angular displacement vector Δθ with Δθ=Δθ x e x y e y z e z (4.6) It is understood that in equation (4.6) Δθ x is an angular displacement about the fixed x-axis, etc. The notation used in equations (4.3) and (4.4) is now understood as meaning that the angular displacement and velocity are about the fixed z-axis
. An example is shown in Figure 3, along with the so-called right-hand rule, for the angular velocity vector ω=
dθ dt(4.7) The introduction of a vector notation has many benefits and simplifies the form of several relations that we will encounter. A first example is that of the infinitesimal arc vector dr
that results for an infinitesimal rotation vector dθ of a rigid body (please note that we have intentionally replaced s for the finite arc in equation (4.2) with dr and not ds ). Let us consider the special case shown in Figure 4 where an infinitesimal rotation dθ=dθ z e z about the z-axis is effected on a vector r=re x aligned along the x-axis . As can be seen from the figure , the result ing infinitesima l arc dr=dre y will be oriented along the y-axis . We know from equation (4.2) that dr=rdθ,(4.8) Figure 3 - Shown is the vector representation for an angular velocity about the , along with the so-called right-hand rule.
- 65- but how can we mathematically determine the orientation of the infinitesimal arc from that of the rotation and radius? To do so, we must introduce the cross product between two vectors. Let a
and b two vectors such that a=a x e x +a y e y +a z e z b=b x e x +b y e y +b z e z (4.9) Then we define the cross product a×b=a y b z -a z b y e x +a z b x -a x b z e y +a x b y -a y b x e z (4.10) It is important to note that a×b=-b×a. (4.11) It is then straightforward to establish the following e x ×e y =e z e y ×e z =e x e z ×e x =e y (4.12) and e i ×e i =0, (4.13) where i=x,y, or z. Coming back to our simple example of Figure 4, and considering equations (4.8) and (4.12) we find that dre
y =dθ z e z×re
x(4.14) Although equation (4.14) results from a special case where the orientation of the different vectors was specified a priori, this relation can be generalized with z
d d r r x y (+z)Figure 4 - Infinitesimal rotation of a rigid body about the . - 66- dr=dθ×r, (4.15) as could readi ly verified by changing the orientation of r and dθ in Figure 4. We therefore realize that the infinitesimal arc dr represents the change in the radius vector r under a rot ation dθ; hence the chosen notati on. Moreover, we can fi nd a vector generalization of equation (1.44) in Chapter 1 that established the relationship between the linear and angular velocities by dividing by an infinitesimal time interval dt
on both sides of equation (4.15). We then find v=ω×r, (4.16) where v=drdt and ω=dθdt. Just as we did for the angular velocity in equations (4.3) and (4.4) (but using a vector notation), we can define an average angular acceleration over a time interval Δt
with α ave Δt (4.17) and an instantaneous angular acceleration with α=limΔt→0
Δt dω dt(4.18) Combining equations (4.7) and (4.18), we can also express the instantaneous angular acceleration as the second order time derivative of the angular displacement α=
dω dt d dt dθ dt d 2 dt 2(4.19) 4.1.2 Constant Angular Acceleration We have so far observed a perfect correspondence between the angular displacement dθ
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