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Assume that the arc length is equal to one-half the circumference of the circle. This arc represents a subtended angle of 180 degrees. Then,

s = circumference/2 = pi * radius

angle = (pi * radius)/radius = pi

From this we can see that pi radians is equal to 180 degrees.

Earlier we saw that 2*pi radians equal 360 degrees.

Facts worth remembering

  • One radian is equal to approximately 57.3 degrees
  • Pi radians is equal to 180 degrees
  • 2*Pi radians are equal to 360 degrees

Tangential displacement versus angular displacement

Consider the case of a 1.5-radian angular displacement of a wheel in a given time interval. What is the corresponding displacement of apoint on the circumference of the wheel? Assume that the radius of the wheel is 0.5 meters.

angle = s/r, or

s = angle * r, or

s = 1.5 * 0.5m = 0.75m

A simple solution

Thus, we see that the tangential displacement of a point on the circumference of a wheel due to a given angular displacement of the wheel in radians is simplythe product of the displacement and the radius of the wheel.

Solving for the same result using angular displacement in degrees would be somewhat morecomplicated.

Tangential speed versus angular velocity

A similar simplification occurs when dealing with the angular velocity of a wheel and the tangential speed of a point on the circumference of the wheel.

As in linear measurements, the average angular velocity of a wheel is equal to the angular displacement of the wheel divided by the time interval duringwhich the displacement takes place.

Measurement of angular velocity in radians

w = dA/dT

By substitution,

w = (s/r)/dT = s/(r*dT)


  • s is the length of an arc along the circumference of a circle
  • dA is an angular displacement in a given time interval
  • dT is the time interval
  • r is the radius of the circle
  • w is the angular velocity

Another example

Consider the case of a 1.5-radian/second angular velocity of a wheel. What is the corresponding tangential speed of a point on the circumference ofthe wheel? Assume that the radius of the wheel is 0.5 meters.

The tangential speed is equal to the tangential displacement, s, divided by the time interval over which the displacement occurs. Given the above information,we can write:

w = s/(r*dT)

Given that

v = s/dT

Substitution yields

v; = w*r, or

v = (1.5/s)*(0.5m) = 0.75 m/s

Another simple relationship

Once again, if you keep your units straight, the tangential speed of a point on the circumference of the wheel is simply equal to the angular velocity inradians per second multiplied by the radius of the wheel.

Facts worth remembering

tangential displacement = dA * r

tangential speed = w * r


  • tangential displacement is the distance that a given point travels around the circumference of a circle as a function of anangular displacement in radians.
  • tangential speed is the speed at which a point travels around the circumference of a circle as a function of an angularvelocity in radians.
  • w is the angular velocity in radians/second
  • dA represents angular displacement
  • r represents radius

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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