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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] .

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 . 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.

Solution

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."} {}

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

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

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
Berger describes sociologists as concerned with
Mueller Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, College physics arranged for cpslo phys141. OpenStax CNX. Dec 23, 2014 Download for free at http://legacy.cnx.org/content/col11718/1.4
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