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

ω=
Δ
θ
Δ
t
,ω=
Δ
θ
Δ
t
, size 12{ω= { {Δθ} over {Δt} } ","} {}

(6)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

v=
Δ
s
Δ
t
.v=
Δ
s
Δ
t
. size 12{v= { {Δs} over {Δt} } "."} {}

(7)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

v=
r
Δ
θ
Δ
t
=rω.v=
r
Δ
θ
Δ
t
=rω. size 12{v= { {rΔθ} over {Δt} } =rω"."} {}

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

v=rω or ω=vr.v=rω or ω=vr. size 12{v=rω``"or "ω= { {v} over {r} } "."} {}

(9)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.

Calculate the angular velocity of a 0.300 m radius car tire when the car travels at 15.0m/s15.0m/s size 12{"15" "." 0`"m/s"} {} (about 54km/h54km/h size 12{"54"`"km/h"} {}). See Figure 4.

*Strategy*

Because the linear speed of the tire rim is the same as the speed of the car, we have
v
=
15.0 m/s
.
v
=
15.0 m/s
.
size 12{v} {}
The radius of the tire is given to be
r
=
0.300 m
.
r
=
0.300 m
.
size 12{r} {}
Knowing
vv size 12{v} {} and rr size 12{r} {}, we can use the second relationship in v=rω, ω=vrv=rω, ω=vr size 12{v=rω,``ω= { {v} over {r} } } {} to calculate the angular velocity.

*Solution*

To calculate the angular velocity, we will use the following relationship:

ω=vr.ω=vr. size 12{ω= { {v} over {r} } "."} {}

(10)Substituting the knowns,

ω=15.0m/s0.300m=50.0rad/s.ω=15.0m/s0.300m=50.0rad/s. size 12{ω= { {"15" "." 0" m/s"} over {0 "." "300"" m"} } ="50" "." 0" rad/s."} {}

(11)
*Discussion*

When we cancel units in the above calculation, we get 50.0/s. But the angular velocity must have units of rad/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would have an angular velocity

ω=(15.0m/s)/(1.20m)=12.5rad/s.ω=(15.0m/s)/(1.20m)=12.5rad/s. size 12{ω= \( "15" "." 0`"m/s" \) / \( 1 "." "20"`m \) ="12" "." 5`"rad/s."} {}

(12)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.

Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the motion. What is the approximate speed of the object? Identify a point close to your hand and take appropriate measurements to calculate the linear speed at this point. Identify other circular motions and measure their angular velocities.

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