University Physics I - Classical Mechanics, 2019

174 CHAPTER 8. MOTION IN TWO DIMENSIONS
where ω is positive for counterclockwise motion, and negative for clockwise. (There is asensein
whichitisusefultothinkofω as a vector, but, since it is not immediately obvious how or why, I
will postpone discussion of this next chapter, after I have introduced angular momentum.)
When ω changes with time, we can introduce an angular acceleration α, defined, again, in the
obvious way:
Δω
α = lim
Δt→0 Δt = dω
(8.33)
dt
Then for motion with constant angular acceleration we have the formulas
ω(t) =ω i + α(t − t i ) or Δω = αΔt (constant α)
θ(t) =θ i + ω i (t − t i )+ 1 2 α(t − t i) 2 or Δθ = ω i Δt + 1 2 α(Δt)2 (constant α) (8.34)
Equations (8.32) and(8.34) completely parallel the corresponding equations for motion in one
dimension that we saw in Chapter 1. In fact, of course, a circle is just a line that has been bent
in a uniform way, so the distance traveled along the circle itself is simply proportional to the angle
swept by the position vector ⃗r. As already pointed out in connection with Fig. 8.5, if we expressed
θ in radians then the length of the arc corresponding to an angular displacement Δθ would be
s = R|Δθ| (8.35)
so multiplying Eqs. (8.32) or(8.34) byR directly gives the distance traveled along the circle in
each case.
Δθ
r(t+Δt)
r(t)
Δr
s
Figure 8.7: A small angular displacement. The distance traveled along the circle, s = RΔθ, is almost
identical to the straight-line distance |Δ⃗r| between the initial and final positions; the two quantities become
the same in the limit Δt → 0.
Figure 8.7 shows that, for very small angular displacements, it does not matter whether the distance
traveled is measured along the circle itself or on a straight line; that is, s ≃|Δ⃗r|. Dividing by Δt,
using Eq. (8.35) and taking the Δt → 0 limit we get the following useful relationship between the
angular velocity and the instantaneous speed v (defined in the ordinary way as the distance traveled
per unit time, or the magnitude of the velocity vector):
|⃗v| = R|ω| (8.36)