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where θ is the angle of rotation ( [link] ). The units of angular velocity are radians per second (rad/s). Angular velocity can also be referred to as the rotation rate in radians per second. In many situations, we are given the rotation rate in revolutions/s or cycles/s. To find the angular velocity, we must multiply revolutions/s by 2 π , since there are 2 π radians in one complete revolution. Since the direction of a positive angle in a circle is counterclockwise, we take counterclockwise rotations as being positive and clockwise rotations as negative.

We can see how angular velocity is related to the tangential speed of the particle by differentiating [link] with respect to time. We rewrite [link] as

s = r θ .

Taking the derivative with respect to time and noting that the radius r is a constant, we have

d s d t = d d t ( r θ ) = θ d r d t + r d θ d t = r d θ d t

where θ d r d t = 0 . Here d s d t is just the tangential speed v t of the particle in [link] . Thus, by using [link] , we arrive at

v t = r ω .

That is, the tangential speed of the particle is its angular velocity times the radius of the circle. From [link] , we see that the tangential speed of the particle increases with its distance from the axis of rotation for a constant angular velocity. This effect is shown in [link] . Two particles are placed at different radii on a rotating disk with a constant angular velocity. As the disk rotates, the tangential speed increases linearly with the radius from the axis of rotation. In [link] , we see that v 1 = r 1 ω 1 and v 2 = r 2 ω 2 . But the disk has a constant angular velocity, so ω 1 = ω 2 . This means v 1 r 1 = v 2 r 2 or v 2 = ( r 2 r 1 ) v 1 . Thus, since r 2 > r 1 , v 2 > v 1 .

Figure shows two particles on a rotating disk. Particle 1 is at the distance r1 from the axis of rotation and moved with the speed v1. Particle 2 is at the distance r2 from the axis of roation and moves with the speed v2.
Two particles on a rotating disk have different tangential speeds, depending on their distance to the axis of rotation.

Up until now, we have discussed the magnitude of the angular velocity ω = d θ / d t , which is a scalar quantity—the change in angular position with respect to time. The vector ω is the vector associated with the angular velocity and points along the axis of rotation. This is useful because when a rigid body is rotating, we want to know both the axis of rotation and the direction that the body is rotating about the axis, clockwise or counterclockwise. The angular velocity ω gives us this information. The angular velocity ω has a direction determined by what is called the right-hand rule. The right-hand rule is such that if the fingers of your right hand wrap counterclockwise from the x -axis (the direction in which θ increases) toward the y- axis, your thumb points in the direction of the positive z -axis ( [link] ). An angular velocity ω that points along the positive z -axis therefore corresponds to a counterclockwise rotation, whereas an angular velocity ω that points along the negative z -axis corresponds to a clockwise rotation.

Figure is a graph that shows the XYZ coordinate system with the counterclockwise rotation in the XY plane. The angular velocity points in the positive Z-direction.
For counterclockwise rotation in the coordinate system shown, the angular velocity points in the positive z- direction by the right-hand-rule.

We can verify the right-hand-rule using the vector expression for the arc length s = θ × r , [link] . If we differentiate this equation with respect to time, we find

d s d t = d d t ( θ × r ) = ( d θ d t × r ) + ( θ × d r d t ) = d θ d t × r .

Questions & Answers

who discover periodic table?
lovet Reply
it wasn't discovered , it was made.and the person who made it was dmitri mendeleev. dobreinier and newland gave their laws before dmitri related periodic table but wasn't successful in their work
Nope, numerous number of scientist had actually contributed in the making of periodic table. Dmitri Mendeleev succeeded making all the elements into the right order in accordance to their atomic number.
what is the Greek name for calcium
Oniyide Reply
different types of wave
Yog Reply
longitudinal and transverse waves
a gun is kept in the state that it cannot move anywhere and the bullet is fired. Then what is the effect on the velocity of bullet and KE of gun ?
Gobinda Reply
want is meant by the term solar system
jafar Reply
it refers to the sun and all heavenly bodies revolving around it.
excatly...for sure
in addition to Danie's, a solar system is a collection of planets and their moons, asteroids, and other objects bound together by the Star's gravitational force directly or indirectly.
what is meant by total internal reflection
Akshay Reply
what iw meant by total internal reflection
Lorentz force?
study fibre optics. .you will get total internal reflection
what is Lorentz force?
a ray of light traveling at an angle of incidence greater than critical angle from denser to rarer medium is totally reflected back into the denser medium is called total internal reflaction
motion in strat line where is this chapter
Vijaybhai Reply
motion in straight line is kinematic's part
this defination isn't correct
motion in one dimension
what is Lorentz force?
jyotirmayee Reply
what is maxwell electromagnetic law?
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
Akshay Reply
what answer fir this
what's the working difference between a dynamo and a pump?
Piyali Reply
a dynamo is basically a dc generator while pump is usually equipped with a motor
a dynamo converts mechanical energy to electrical while a pump is opposite to that
why sea water looks bluish?
Piyali Reply
cuz the sky is blue...
you see the reflexion of the "blue" sky in the water
somewhere sea water turns green why?
never seen bro... are u sure ?
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
waaaww... you re awesome
because of the reflection of the sky
rays coming from the sun consist of all 7 colours ie.VIBGYOR. when the ray strikes surface of water,all colors gets absorbed by it except blue which gets reflected by it.so we find the sea water appearing bluish
how can someone identify sea water from rain water
i think it's not possible as because rainwater consists of water from all kind of water bodies ie.lakes,seas etc but u can predict if u have a sea nearby ur home or city
i need solutions of unuversity pbysucs volume 1
Vimla Reply
me too
Help us if anyone knows
bring questions
actually if u wanted whole book solution then u should buy the solution book
from where
where do u live ,,,,,if u live in Delhi then at bellsarayeiii or from stationary store ,,,,chatarpurrr ,,,,,there is a popular books store,,,,,u have to buy from there
Kota Rajasthan
so strange,,,,,r u preparing for pmt,,,?
for IIT
then concern near book store
in Rajasthan
or ask questions here
thanks for your help
its fine,,most welcome
what is three dimensional coordinate system?
Sachindra Reply
considering the change of vectors in all three dimensions of space
direction co-sign of vector questions
Vipin Reply
opposite mean
what is the different between action and reaction?
Asali Reply
action is external force. Reaction exists because of the external action. Reaction is mostly an internal force
The difference is said in the word itself. There is no existance of reaction without an application of action. So action occur first then reaction. Reaction may produce in body itself or to other body i.e., it may be internal or external.
or we can say , reaction is the result of action
Practice Key Terms 5

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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