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

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

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

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

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

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

The given figure shows the front wheel of a car. The radius of the car wheel, r, is shown as an arrow and the linear velocity, v, is shown with a green horizontal arrow pointing rightward. The angular velocity, omega, is shown with a clockwise-curved arrow over the wheel.
A car moving at a velocity v size 12{v} {} to the right has a tire rotating with an angular velocity ω size 12{ω} {} .The speed of the tread of the tire relative to the axle is v size 12{v} {} , the same as if the car were jacked up. Thus the car moves forward at linear velocity v = size 12{v=rω} {} , where r size 12{r} {} is the tire radius. A larger angular velocity for the tire means a greater velocity for the car.

How fast does a car tire spin?

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


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


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

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

Substituting the knowns,

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


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 . 0 m/s ) / ( 1 . 20 m ) = 12 . 5 rad/s. size 12{ω= \( "15" "." 0`"m/s" \) / \( 1 "." "20"`m \) ="12" "." 5`"rad/s."} {}
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Both ω size 12{ω} {} and v 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 [link] .

Take-home experiment

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.

Questions & Answers

Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
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Practice Key Terms 6

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