# 6.1 Rotation angle and angular velocity  (Page 2/9)

 Page 2 / 9
$v=\frac{\text{Δ}s}{\text{Δ}t}\text{.}$

From $\text{Δ}\theta =\frac{\text{Δ}s}{r}$ we see that $\text{Δ}s=r\text{Δ}\theta$ . Substituting this into the expression for $v$ gives

$v=\frac{r\text{Δ}\theta }{\text{Δ}t}=\mathrm{r\omega }\text{.}$

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

The first relationship in states that the linear velocity $v$ is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest $r$ ), as you might expect. We can also call this linear speed $v$ of a point on the rim the tangential speed . The second relationship in 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 $v$ of the car. See [link] . So the faster the car moves, the faster the tire spins—large $v$ means a large $\omega$ , because $v=\mathrm{r\omega }$ . Similarly, a larger-radius tire rotating at the same angular velocity ( $\omega$ ) will produce a greater linear speed ( $v$ ) 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 $\text{15}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{m/s}$ (about $\text{54}\phantom{\rule{0.25em}{0ex}}\text{km/h}$ ). See [link] .

Strategy

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

Solution

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

$\omega =\frac{v}{r}\text{.}$

Substituting the knowns,

$\omega =\frac{\text{15}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{m/s}}{0\text{.}\text{300}\phantom{\rule{0.25em}{0ex}}\text{m}}=\text{50}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{rad/s.}$

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

$\omega =\left(\text{15}\text{.}0\phantom{\rule{0.25em}{0ex}}\text{m/s}\right)/\left(1\text{.}\text{20}\phantom{\rule{0.25em}{0ex}}\text{m}\right)=\text{12}\text{.}5\phantom{\rule{0.25em}{0ex}}\text{rad/s.}$

Both $\omega$ and $v$ 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.

If an object has no acceleration in an inertial reference frame, can you conclude that no forces are acting on it?
What are the two major system units in the world
What is physical quantities
Jizel
they are physical properties that can be measured or calculated
ZIFAC
Do you mean the SI system and Imperial system? SI is used by scientists the world over (with notable exception being the USA)
Who is the father of physics
Newton.Geliliyo and Einstein is called father of physics
Neha
Newtin
Jizel
Einstein
Jizel
Ancient - Archimedes Classical Physics - Newton Modern Physics - Einstein But each of these has built on the work of predecessors. No single personality defines physics.
Thanks
Hussain
Galileo Galilie must be. He was the creator of the scientific method
Vincent
Galileo was no doubt inspirational, but the scientific method has existed for millennia. What Galileo did was prove the superiority of experimentation to just thinking (Aristotle's way). Again all noteworthy scientists
Newton galileo and Einstein
Gabriel
what is wave
a wave is a distirbance that transmits energy from one place ro another within or without a medium
Vincent
wave is the transfer of energy from one medium to another without the transfer of particles
ZIFAC
wave is a disturbance which transfer energy from one medium to another without causing any permanent displacement by itself
Joyfulsounds
wave is a disturbance or oscillation that travel through space and matter,accompanied by a transfer of energy
Ridwan
A wave is any disturbances in an elastic medium which carries energy from one point to another through a medium
abdul
what is harmonic motion
Nozyani
is a restoring force
Joyfulsounds
a wave is a disturbance and there are trasfer energy from one medium to another and travel through any space without the trasfer of particles.
Neha
a wave is a disturbance that travels or carries energy from one point to another through a medium
Wisdom
A moving disturbance in the level of a body water, undulation
Jizel
what is thermodynamics
what is thermodynamics
Charity
thermodynamics is a heat and energy significant physics
Neha
Relating to the conversation of heat into other forms of energy
Jizel
It's the science of conversion between heat and other forms of energy
ZIFAC
Are the antimatters of Hadrons also Hadrons?!Does the same rule apply to Leptons?
yes. Hadrons are the elementary particles that take part in stong, electromagnetic and weak interactions. Infact only Hadrons are involved in Strong interactions and when an anti-particle of any hadron is produced, it would be a hadron-conservations laws. Leptons are involved in weak int and follow
Lalita
what is physics
physic is a pure science that deal with behavior of matter,energy & how it related to other physical properties
Ridwan
Owk. But am are Art student.
Hussaini
Thanks a lot,Lalita
Daniel
what are the differences between reflection and refraction ?
Matins
What happens when an aeroplanes window is opened at cruise altitude?
what is the minimum speed for any object to travel in time?
as per theory of relativity, minimum speed will be the speed of light
Mr.
what is physics
it is just a branch of science which deals with the reasons behind the daily activities taking place everyday in our lives. it clearly states the reason in the form of laws.
sandhya
?
lkpostpost2000@yahoo
like Newton's laws , Kepler's laws etc....
sandhya
physics is the study of motion or moving things. Usually the moving things are normal items like vars or planets but sometimes it's electricity or heat that moves.
Jake
physics is one of the most significant diciplines of natural science which describe the nature and its matter
Neha
I would describe it as the science that is interested in the fundamental laws of nature. For example, what is light, what is sound, what is electricity/magentism, what forces are at work on a specific body. The knowledge of the world around us makes it possible to fly, have cell phones, GPS, etc.
Robyn
what happens when an aeroplane takes off?
it flies
Mr.
the lift generated by the wing overcome the weight of the plane(in Newton)and a net force of upward is created
Phebilia
it is a direct application of Magnus effect (which helps in throwing curve balls) the wings of plane are made in such a way that the net flow of air is more below them rather than on their upper side. So when the plane accelerates, the flaps produce the upward lift when enough velocity is obtained
Mr.
then due to lower pressure on upper part of wings helps producing an additional lift because air flows from areaof lower to the area of higher pressure
Mr.
The engines located under the wings generate thrust .. in relation thrust is a force ... which ovwrcomes or becomes greater than the weight of the plane.. remember weight is a force Weight = m x g-2 So therefore F(thrust) becomes greater than F(weight) Even if by 1Newton the plane starts lifting o
Theophilus
what happens when a ship moves
Williams
What is the sign of an acceleration that reduces the magnitude of a negative velocity? Of a positive velocity?
If it reduces the magnitude of the velocity, the acceleration sign is the opposite compared to the velocity.
Nicolas
yes
Williams
what is accerelation
an objects tendency to speed up over time
RayRay
acceleration is the change in velocity over the change in time it would be written delta-v over delta-t.
Shii
the change in velocity V over a period of time T.
Matthew
Delta means "change in"...not period of
Shii
just kidding. it all works mathematically
Shii
except doesn't time really only change if the instantaneous speeds vary...?
Shii
and I assume we are all talking average acceleration
Shii
Hey shiii 😀
the rate of change of velocity is callaed acceleration
Amna
a=delta v/delta t
Amna
the rate of change in velocity with respect to time is acceleration
Nana
nana you r right
Indrajit
good
oguji
what is meant by lost volt
Lost volt. Lol. It is the electrical energy lost due to the nature or the envirommental conditions (temperature and pressure) that affect the cable across which the potential difference is measured.
Theophilus
What is physics?
physics is brance science concerned with nature and properties of matter and energy
George
sure
Okpara
yah....
kashif
physics is study of the natural phenomenon on the basis of certain laws and principles. it's like watching a game of chess and trying to understand its rules how it's played.
Ajit
awesome
Okpara
physics is study of nature and it's law
AMRITA
physics is a branch of science that deals with the study of matter ,properties of matter and energy
Lote
Branch of science (study) of matter, motion and energy
Theophilus
what is a double-slit experiment?Explain.
when you pass a wave of any kind ie sound water light ect you get an interface pattern forming on a screen behind it, where the peaks and troughs add and cancel out due to the diffraction caused by a wave traveling through the slits
Luke
double slit experiment was done by YOUNG. And it's to give out monochromatic coherent, if an incoherent wave is passing through it. And then the waves form interference fringes. The screen placed in front of the double slit is preferably a film and then in the middle where "p=0" a brighter color
navid
is formed and then the constructive interferences occur at 0 (which is the brightest band)... then a sequence of bright band (constructive interference) and dark band (destructive interference) happens and the further from the central band the lower the intensity of bright band(constructive interfe
navid