When objects rotate about some axis—for example, when the CD (compact disc) in Figure 1 rotates about its center—each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle ΔθΔθ size 12{Δθ} {} to be the ratio of the arc length to the radius of curvature:

Figure 1: All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle ΔθΔθ size 12{Δθ} {} in a time ΔtΔt size 12{Δt} {}.

Figure 2: The radius of a circle is rotated through an angle ΔθΔθ size 12{Δθ} {}. The arc length ΔsΔs size 12{Δs} {} is described on the circumference.

The arc lengthΔsΔs size 12{Δs} {} is the distance traveled along a circular path as shown in Figure 2 Note that rr size 12{r} {} is the radius of curvature of the circular path.

We know that for one complete revolution, the arc length is the circumference of a circle of radius rr size 12{r} {}. The circumference of a circle is 2πr2πr size 12{2πr} {}. Thus for one complete revolution the rotation angle is

This result is the basis for defining the units used to measure rotation angles, ΔθΔθ size 12{Δθ} {} to be radians (rad), defined so that

A comparison of some useful angles expressed in both degrees and radians is shown in Table 1.

Figure 3: Points 1 and 2 rotate through the same angle (ΔθΔθ size 12{Δθ} {}), but point 2 moves through a greater arc length ΔsΔs size 12{ left (Δs right )} {} because it is at a greater distance from the center of rotation (r)(r) size 12{ \( r \) } {}.

If Δθ=2πΔθ=2π size 12{Δθ=2π} {} rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are 360º360º size 12{"360"°} {} in a circle or one revolution, the relationship between radians and degrees is thus

so that

How fast is an object rotating? We define angular velocity ωω size 12{ω} {} as the rate of change of an angle. In symbols, this is

where an angular rotation ΔθΔθ size 12{Δθ} {} takes place in a time ΔtΔt size 12{Δt} {}. The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity ωω size 12{ω} {} is analogous to linear velocity vv size 12{v} {}. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length ΔsΔs size 12{Δs} {} in a time ΔtΔt size 12{Δt} {}, and so it has a linear velocity

From Δθ=ΔsrΔθ=Δsr size 12{Δθ= { {Δs} over {r} } } {} we see that Δs= r Δ θ Δs= r Δ θ size 12{Δs=rΔθ} {}. Substituting this into the expression for vv size 12{v} {} gives

We write this relationship in two different ways and gain two different insights:

The first relationship in v=rω or ω=vrv=rω or ω=vr size 12{v=rω``"or "ω= { {v} over {r} } } {} states that the linear velocity vv size 12{v} {} is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest rr size 12{r} {}), as you might expect. We can also call this linear speed vv size 12{v} {} of a point on the rim the tangential speed. The second relationship in v=rω or ω=vrv=rω or ω=vr size 12{v=rω``"or "ω= { {v} over {r} } } {} can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed vv size 12{v} {} of the car. See Figure 4. So the faster the car moves, the faster the tire spins—large vv size 12{v} {} means a large ωω size 12{ω} {}, because v=rωv=rω size 12{v=rω} {}. Similarly, a larger-radius tire rotating at the same angular velocity (ωω size 12{ω} {}) will produce a greater linear speed (vv size 12{v} {}) for the car.

Figure 4: A car moving at a velocity vv size 12{v} {} to the right has a tire rotating with an angular velocity ωω size 12{ω} {}.The speed of the tread of the tire relative to the axle is vv size 12{v} {}, the same as if the car were jacked up. Thus the car moves forward at linear velocity v=rωv=rω size 12{v=rω} {}, where rr size 12{r} {} is the tire radius. A larger angular velocity for the tire means a greater velocity for the car.

Both ωω size 12{ω} {} and vv size 12{v} {} have directions (hence they are angular and linear velocities, respectively). Angular velocity has only two directions with respect to the axis of rotation—it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in Figure 5.

Figure 5: As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case.