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Since r is constant, the term θ × d r d t = 0 . Since v = d s d t is the tangential velocity and ω = d θ d t is the angular velocity, we have

v = ω × r .

That is, the tangential velocity is the cross product of the angular velocity and the position vector, as shown in [link] . From part (a) of this figure, we see that with the angular velocity in the positive z -direction, the rotation in the xy -plane is counterclockwise. In part (b), the angular velocity is in the negative z- direction, giving a clockwise rotation in the xy- plane.

Figure A is an XYZ coordinate system that shows three vectors. Vector Omega points in the positive Z direction. Vector v is in the XY plane. Vector r is directed from the origin of the coordinate system to the beginning of the vector v. Figure B is an XYZ coordinate system that shows three vectors. Vector Omega points in the negative Z direction. Vector v is in the XY plane. Vector r is directed from the origin of the coordinate system to the beginning of the vector v.
The vectors shown are the angular velocity, position, and tangential velocity. (a) The angular velocity points in the positive z- direction, giving a counterclockwise rotation in the xy- plane. (b) The angular velocity points in the negative z -direction, giving a clockwise rotation.

Rotation of a flywheel

A flywheel rotates such that it sweeps out an angle at the rate of θ = ω t = ( 45.0 rad / s) t radians. The wheel rotates counterclockwise when viewed in the plane of the page. (a) What is the angular velocity of the flywheel? (b) What direction is the angular velocity? (c) How many radians does the flywheel rotate through in 30 s? (d) What is the tangential speed of a point on the flywheel 10 cm from the axis of rotation?


The functional form of the angular position of the flywheel is given in the problem as θ ( t ) = ω t , so by taking the derivative with respect to time, we can find the angular velocity. We use the right-hand rule to find the angular velocity. To find the angular displacement of the flywheel during 30 s, we seek the angular displacement Δ θ , where the change in angular position is between 0 and 30 s. To find the tangential speed of a point at a distance from the axis of rotation, we multiply its distance times the angular velocity of the flywheel.


  1. ω = d θ d t = 45 rad / s . We see that the angular velocity is a constant.
  2. By the right-hand rule, we curl the fingers in the direction of rotation, which is counterclockwise in the plane of the page, and the thumb points in the direction of the angular velocity, which is out of the page.
  3. Δ θ = θ ( 30 s ) θ ( 0 s ) = 45.0 ( 30.0 s ) 45.0 ( 0 s ) = 1350.0 rad .
  4. v t = r ω = ( 0.1 m ) ( 45.0 rad / s ) = 4.5 m/s .


In 30 s, the flywheel has rotated through quite a number of revolutions, about 215 if we divide the angular displacement by 2 π . A massive flywheel can be used to store energy in this way, if the losses due to friction are minimal. Recent research has considered superconducting bearings on which the flywheel rests, with zero energy loss due to friction.

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Angular acceleration

We have just discussed angular velocity for uniform circular motion, but not all motion is uniform. Envision an ice skater spinning with his arms outstretched—when he pulls his arms inward, his angular velocity increases. Or think about a computer’s hard disk slowing to a halt as the angular velocity decreases. We will explore these situations later, but we can already see a need to define an angular acceleration    for describing situations where ω changes. The faster the change in ω , the greater the angular acceleration. We define the instantaneous angular acceleration     α as the derivative of angular velocity with respect to time:

Questions & Answers

A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
is the eye the same like the camera
I can't understand
same here please
I think the question is that ,,, the working principal of eye and camera same or not?
yes i think is same as the camera
what are the dimensions of surface tension
why is the "_" sign used for a wave to the right instead of to the left?
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
what is the formula for pressure?
Goodness Reply
force is newtom
and area is meter squared
so in SI units pressure is N/m^2
In customary United States units pressure is lb/in^2. pound per square inch
who is Newton?
John Reply
a scientist
that discovered law of motion
but who is Isaac newton?
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
and the color spectrum
what is a scalar quantity
Peter Reply
scalar: are quantity have numerical value
is that a better way in defining scalar quantity
quantity that has magnitude but no direction
upward force and downward force lift
adegboye Reply
upward force and downward force on lift
Yes what about it?
what's the answer? I can't get it
Rachel Reply
what is the question again?
What's this conversation?
what is catenation? and give examples
How many kilometres in 1 mile
1.609km in 1mile
what's the si unit of impulse
Iguh Reply
The Newton second (N•s)
what is the s. I unit of current
Roland Reply
thanks man
u r welcome
the velocity of a boat related to water is 3i+4j and that of water related to earth is i-3j. what is the velocity of the boat relative to earth.If unit vector i and j represent 1km/hour east and north respectively
Pallavi Reply
Practice Key Terms 5

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