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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

what is electromagnetism
David Reply
It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
what is units?
Subhajit Reply
units as in how
What is th formular for force
Joseph Reply
F = m x a
State newton's second law of motion
Seth Reply
can u tell me I cant remember
force is equal to mass times acceleration
The acceleration of a system is directly proportional to the and in the same direction as the external force acting on the system and inversely proportional to its mass that is f=ma
The uniform seesaw shown below is balanced on a fulcrum located 3.0 m from the left end. The smaller boy on the right has a mass of 40 kg and the bigger boy on the left has a mass 80 kg. What is the mass of the board?
Asad Reply
Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?
Sohail Reply
how can one calculate the value of a given quantity
Helen Reply
To determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
briefly discuss rocket in physics
Ibrahim Reply
ok let's discuss
What is physics
Nura Reply
physics is the study of natural phenomena with concern with matter and energy and relationships between them
a potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inductance of the inductor that should be connected to the capacitor for the circuit to oscillate at 1125Hza potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inducta
Royalty Reply
L= 0.002H
how did you get it?
is the magnetic field of earth changing
tibebeab Reply
what is thought to be the energy density of multiverse and is the space between universes really space
can you explain it
Energy can not either created nor destroyed .therefore who created? and how did it come to existence?
Suzana Reply
this greatly depend on the kind of energy. for gravitational energy, it is result of the shattering effect violent collision of two black holes on the space-time which caused space time to be disturbed. this is according to recent study on gravitons and gravitational ripple. and many other studies
and not every thing have to pop into existence. and it could have always been there . and some scientists think that energy might have been the only entity in the euclidean(imaginary time T=it) which is time undergone wick rotation.
What is projectile?
Nana Reply
An object that is launched from a device
2 dimensional motion under constant acceleration due to gravity
Not always 2D Awais
no comments
why not? a bullet is a projectile, so is a rock I throw
bullet travel in x and y comment same as rock which is 2 dimensional
no all pf you are wrong. projectile is any object propelled through space by excretion of a force which cease after launch
for awais, there is no such thing as constant acceleration due to gravity, because gravity change from place to place and from different height
it is the object not the motion or its components
where are body center of mass on present.
Balwant Reply
on the mid point
is the magnetic field of the earth changing?
does shock waves come to effect when in earth's inner atmosphere or can it have an effect on the thermosphere or ionosphere?
and for the question from bal want do you mean human body or just any object in space
A stone is dropped into a well of 19.6m deep and the impact of sound heared after 2.056 second ,find the velocity of sound in air.
Sisco Reply
9.53 m/s ?
In this case, the velocity of sound is 350 m/s.
some calculations is need. then you will get exact result.
i mean how? isn't it just a d over t?
calculate the time it takes the stone to hit the ground then minus the stone's time to the total time... then divide the total distance by the difference of the time
awit lenard. Hahahah ari ga to!
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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