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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."} {}
Got questions? Get instant answers now!

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 does mean ohms law imply
Victoria Reply
what is matter
folajin Reply
Anything that occupies space
Kevin
Any thing that has weight and occupies space
Victoria
Anything which we can feel by any of our 5 sense organs
Suraj
Right
Roben
thanks
Suraj
what is a sulphate
Alo
any answers
Alo
the time rate of increase in velocity is called
Blessing Reply
acceleration
Emma
What is uniform velocity
Victoria
Greetings,users of that wonderful app.
Frank Reply
how to solve pressure?
Cruz Reply
how do we calculate weight and eara eg an elefant that weight 2000kg has four fits or legs search of surface eara is 0.1m2(1metre square) incontact with the ground=10m2(g =10m2)
Cruz
P=F/A
Mira
can someone derive the formula a little bit deeper?
Bern
what is coplanar force?
OLADITI Reply
what is accuracy and precision
Peace Reply
How does a current follow?
Vineeta Reply
follow?
akif
which one dc or ac current.
akif
how does a current following?
Vineeta
?
akif
AC current
Vineeta
AC current follows due to changing electric field and magnetic field.
akif
you guys are just saying follow is flow not follow please
Abubakar
ok bro thanks
akif
flows
Abubakar
but i wanted to understand him/her in his own language
akif
but I think the statement is written in English not any other language
Abubakar
my mean that in which form he/she written this,will understand better in this form, i write.
akif
ok
Abubakar
ok thanks bro. my mistake
Vineeta
u are welcome
Abubakar
what is a semiconductor
Vineeta Reply
substances having lower forbidden gap between valence band and conduction band
akif
what is a conductor?
Vineeta
replace lower by higher only
akif
convert 56°c to kelvin
Abubakar
How does a current follow?
Vineeta
A semiconductor is any material whose conduction lies between that of a conductor and an insulator.
AKOWUAH
what is Atom? what is molecules? what is ions?
Abubakar Reply
What is a molecule
Samuel Reply
Is a unit of a compound that has two or more atoms either of the same or different atoms
Justice
A molecule is the smallest indivisible unit of a compound, Just like the atom is the smallest indivisible unit of an element.
Rachel
what is a molecule?
Vineeta
what is a vector
smith Reply
A quantity that has both a magnitude AND a direction. E.g velocity, acceleration, force are all vector quantities. Hope this helps :)
deage
what is the difference between velocity and relative velocity?
Mackson
Velocity is the rate of change of displacement with time. Relative velocity on the other hand is the velocity observed by an observer with respect to a reference point.
Chuks
what do u understand by Ultraviolet catastrophe?
Rufai
A certain freely falling object, released from rest, requires 1.5seconds to travel the last 30metres before it hits the ground. (a) Find the velocity of the object when it is 30metres above the ground.
Mackson
A vector is a quantity that has both magnitude and direction
Rufus
the velocity Is 20m/s-2
Rufus
derivation of electric potential
Rugunda Reply
V = Er = (kq/r^2)×r V = kq/r Where V: electric potential.
Chuks
what is the difference between simple motion and simple harmonic motion ?
syed
hi
Peace
hi
Rufus
hi
Chip
simple harmonic motion is a motion of tro and fro of simple pendulum and the likes while simple motion is a linear motion on a straight line.
Muinat
a body acceleration uniform from rest a 6m/s -2 for 8sec and decelerate uniformly to rest in the next 5sec,the magnitude of the deceleration is ?
Patricia Reply
The wording not very clear kindly
Moses
6
Leo
9.6m/s2
Jolly
the magnitude of deceleration =-9.8ms-2. first find the final velocity using the known acceleration and time. next use the calculated velocity to find the size of deceleration.
Mackson
wrong
Peace
-3.4m/s-2
Justice
Hi
Abj
Firstly, calculate final velocity of the body and then the deceleration. The final ans is,-9.6ms-2
Muinat
8x6= 48m/-2 use v=u + at 48÷5=9.6
Lawrence
can i define motion like this motion can be define as the continuous change of an object or position
Shuaib Reply
Any object in motion will come to rest after a time duration. Different objects may cover equal distance in different time duration. Therefore, motion is defined as a change in position depending on time.
Chuks
Practice Key Terms 6

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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