# Angular velocity Definition and 109 Discussions

In physics, angular velocity or rotational velocity (

ω

{\displaystyle {\boldsymbol {\omega }}}
or

Ω

{\displaystyle {\boldsymbol {\Omega }}}
), also known as angular frequency vector, is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time.
There are two types of angular velocity. Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. Spin angular velocity refers to how fast a rigid body rotates with respect to its center of rotation and is independent of the choice of origin, in contrast to orbital angular velocity.
In general, angular velocity has dimension of angle per unit time (angle replacing distance from linear velocity with time in common). The SI unit of angular velocity is radians per second, with the radian being a dimensionless quantity, thus the SI units of angular velocity may be listed as s−1. Angular velocity is usually represented by the symbol omega (ω, sometimes Ω). By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.
For example, a geostationary satellite completes one orbit per day above the equator, or 360 degrees per 24 hours, and has angular velocity ω = (360°)/(24 h) = 15°/h, or (2π rad)/(24 h) ≈ 0.26 rad/h. If angle is measured in radians, the linear velocity is the radius times the angular velocity,

v
=
r
ω

{\displaystyle v=r\omega }
. With orbital radius 42,000 km from the earth's center, the satellite's speed through space is thus v = 42,000 km × 0.26/h ≈ 11,000 km/h. The angular velocity is positive since the satellite travels eastward with the Earth's rotation (counter-clockwise from above the north pole.)
Angular velocity is a pseudovector, with its magnitude measuring the angular speed, the rate at which an object rotates or revolves, and its direction pointing perpendicular to the instantaneous plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the right-hand rule.

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1. ### Angular Velocity in the Rotating systems

Summary: Consider a body which is rotating with constant angular velocity ω about some axis passing through the origin. Assume the origin is fixed, and that we are sitting in a fixed coordinate system ##O_{xyz}## If ##\rho## is a vector of constant magnitude and constant direction in the...
2. ### I Find the final angular velocity of two rotating discs after collision (game coding problem)

Hello to everyone, first of all shame on me I has to ask and can not figure out it by myself... The problem is I am trying to code game where two homogenous discs with same mass and same diameter, no fricition due to gravitational forces, can collide. I can figure out the speed and direction...
3. ### How to determine angular velocity about a certain axis?

If the crawling insect were stationary at a certain instant of time, then it would have the same angular velocity as that of disk, which is w in a clockwise direction. But now it's velocity at any instant is the vector sum of velocity due to rotation and the velocity it crawls at. My attempt is...
4. ### Finding Angular Velocity in Rotational Motion Problems

53 rpm equals 5.55 rad/sec multiply 5.55 by 2pi to get angular velocity of 34.8717 Is the answer 34.8717? What should I have done to more accurately solve the problem with a better understanding? What other steps should I take when solving similar problems? and lastly, Is the mass relevant...
5. ### What is a "Torsional Constant"?

The question was: I will also include the solution: So, what is the justification of the first formula [ω=√(C/I)]? I know how to derive simple harmonic equations, this one as I guess is probably similar? But I cannot connect as to how C is used exactly. And the second formula [ω'=ωβ], I...
6. ### Require help with angular velocity and people flying off the planet!

My solutions (attempts) : a> w=v/r | r=6.35x10^6m | therefore V=7.04x10^-5 m/s b> speed of rotation is faster at the equator than the pole as w=v/r. As w remains constant, as r increases towards the pole V has to decrease. c> F = W - R d> Stuck here. I presume that I have to use the equation...
7. ### Mechanics: Angular Velocity Vector Questions

Answers are the following : (i) v=(2cost)i - (2sint)j -(1/2)k (ii)2.06m/s (iii)2m/s^2 horizontally towards the vertical axis, making an angle of pi/4 with both the I and j axes.
8. ### Vector potential ##\vec A## in terms of magnetic field ##\vec B##

My solution is making an analogy of the ##\text{Relevant equations}## as shown above, starting from the equation ##\vec \omega = \frac{1}{2} \vec \nabla \times \vec v##. We have ##\vec B = \vec \nabla \times \vec A = \frac{1}{2} \vec \nabla \times 2\vec A \Rightarrow 2\vec A = \vec B \times...
9. ### Pulley angular velocity problem

Further given: - every beam is infinite stiff - pulleys are massless - cables don't stretch, no slip, and frictionless. -Every pulley has a diameter D except pulley Q. Pulley Q has diameter 0.5*D So what I don't understand is how to calculate/determine the velocity at R and S. Can someone help...
10. ### Rotation of two cylinders inclined at an angle

A single pair of points will be in contact between P and Q. The frictional force will try to make the velocity of these points equal. Say the final angular velocity of Q is ωq. The velocity of points in contact can never be equal because of difference in directions of ωq and ωp. If I break...
11. ### Angular Velocity of a Car going around a curve

θ=90°= π /2 so the instantaneous angular velocity dθ/dt= lim∆ t -> 0 (θ(t + ∆ t)-θ(t))/(∆ t) When I calculate it out it is π /2 radians per second. Is this correct?
12. ### Angular speed, acceleration, and angle of a Ferris wheel

ω(10)=(1.3)∗(1.0−e^(−10/22) )= 0.475 rad/s 0.475 rad/s=0 +α(10second) α=0.0475 rad/s^2 ∫ω(t)=Θ =1.3t + 28.6e^(-t/22) | (t=10s, t=0) total angle by which the wheel rotates over this period of t=10 seconds = 2.55 rad Θ= 2(pi)(8m)= 1.3t + 28.6e^(-t/22) 0=1.3t + 28.6e^(-t/22) - 2(pi)(8m) t=34...
13. ### Angular acceleration and velocity in a circle

33.33 RPM (1min/60second)(6.28 radian/1 revolution) = 3.45 rad/s linear velocity=3.45 rad/s * 0.1524m=0.526 m/s linear acceleration= (0.526 m/s)^2 /(0.1524m)=1.814 m/s^2 1.814 m/s^2=(0.1524m)(rotational acceleration) rotational acceleration=11.9 rad/s^2 ω1= (3.45 rad/s)+(11.9rad/s^2)(10s)...
14. ### Write an expression w1 of the angle shown in the first picture

A bicycle wheel rolls at a constant speed along a circular path on a horizontal surface. The wheel has a constant angle of inclination to the vertical direction and the distance from its center of mass G to the fixed Z axis is R. Determine the relationship between the angular velocity w1 around...
15. ### Rotating with slipping to rotating without slipping?

It seems to me that this transition implies going from kinetic friction to static friction. The kinetic friction would apply a torque that would slow down the object's angular velocity, but I'm not sure how this connects to the object suddenly transitioning into rotating without slipping.
16. ### A Index juggling by example of angular momentum tensor

Lets consider the angular momentum tensor (here ##m=1##) \begin{equation} L^{ij} = x^iv^j - x^jv^i \end{equation} and rortational velocity of particle (expressed via angular momentum tensor) \begin{equation} v^j = \omega^{jm}x_m. \end{equation} Then \begin{equation} L^{ij} =...
17. ### Are angular & vertical velocity the same if the objects are connected

think of a engine. it has a flywheel and a rod connected to it. a string had been totally wrapped around the rod and a mass is hanged from the very end of the rod. the system is in equilibrium. but as the engine starts to rotate, the rod with rotate as well and cause the hanged object to go...
18. ### Collision with a ball spinning about its vertical axis

Homework Statement A ball A is rotating on a table with an angular velocity ω about its vertical axis. An identical ball B collides with the ball A elastically. After collision the ball A starts sliding over the table. The coefficient of friction is µ. Find: 1) the angle α between the angular...
19. ### Rigid Bodies/ Angular Velocity

Homework Statement Homework Equations ##v=\omega r## The Attempt at a Solution So, using the equation, one can work out the velocity at point ##B##. ##v_B=\omega_{AB} \cdot r_B## ##v_B=6(0.4)=2.4~ ms^{-1}## I then tried working out the angular velocity at point ##C## using the...
20. ### Calculate the relative angular velocity

Homework Statement AB is a rod of length 10 m that is leaning against the wall. Given variables are shown in the diagram. Find angular velocity of A wrt B. https://imgur.com/a/8bEdYhN I have a doubt in one step that I will highlight in "The attempt at a solution" part. Homework Equations...
21. ### Difference between angular velocity and angular frequency

I have seen so many questions and confusion about the difference between angular velocity/speed and angular frequency. Usually, answers were always given in the context of uniform circular motion (angular speed) and simple harmonic oscillation (angular frequency), but this is what causes the...
22. ### Why is the specific angular momentum equal to this?

From a wiki's vis-viva equation page, it is given that the specific angular momentum h is also equal to the following: h = wr^2 = ab * n How can ab * n be derived to be equal to the angular momentum using elliptical orbit energy/momentum/other equations without having to use calculus or...
23. ### Tyres of a car lifting when turning left sharply...

Homework Statement Why do the left wheels of a car rise when it takes a sharp left turn (that is it lurches towards the right)? Homework Equations $$a_c= V^2/R$$ The Attempt at a Solution I started by imagining the car as being a part of a very large ring, dx. Since it's taking a left turn...
24. ### Kinematics chase problem

Homework Statement Starting from the center of a circular path of radius R, a particle P chases another particle Q that is moving with a uniform speed v on the circular path. The chaser P moves with a constant speed u and always remains collinear with the centre and location of the chased Q...
25. ### Motor driven hydraulic cylinder design

I am trying to design a mechanism which has a hydraulic cylinder driven by a step motor. The motor shaft is connected to the hydraulic cylinder piston via two rods of lengths r and L as shown in the below figure. I want the cylinder's piston to be driven at constant velocity v, so I am trying to...
26. ### Two particle system - find their angular velocities

Hi! I want to start solving problems from the text 'Orbital Mechanics for Engineering students' by Curtis 2nd edition. Is this the right place to post? Homework Statement 2.1 Two particles of identical mass m are acted on only by the gravitational force of one upon the other. If the distance d...
27. ### Effects of centripetal force on longer objects

I have a question. What will happen if you have a long object let’s say a person was lassoed by their feet and spun around by a super strong machine or person. Since the persons head of who is being spun is moving at a much faster tangential velocity then let’s say their feet. If the rope is cut...
28. A

### Angular Velocity in Simple Harmonic Motion

I am very confused about angular velocity ω and why its used in simple harmonic motion. ω is described as θ/τ but when it comes to masses on springs, there is no angle - it is zero. Angular velocity comes from circular motion but the motion of SHM is not circular. My confusion is even greater...
29. S

### Radius and centripetal relation

Homework Statement the file given Homework Equations F=mv^2/r The Attempt at a Solution I do not understand why the centripetal force is 2a and not 2/a since the radius of X is twice longer. When I use the equation above, raidius is inversely proportional to the acceleration. Is radius...
30. M

### Angular Velocity & Acceleration for a Series of Connected Objects

PIC: [https://drive.google.com/file/d/0B0NXDy0RMDe7MXhMcjZBdkhoSDg/view?usp=sharing] 1. A rotating disk is connected with two arms AD and DB which are rotating with the rate of 0.2 rad/s^2 and -0.3 rad/s^2 respectively about X or x' axis. Disk itself is rotating about small z axis with the rate...
31. ### I Left translate back to I in SO(3)

Hello I am hoping someone can explain a sentence to me. Unfortunately, I do not even recall where I read it. I wrote it down years ago and long since lost the source. (Now I think some of it is making sense, but I don't remember the source.) Consider R(t) as an orthogonal rotation matrix...
32. A

### Velocity of a piston in a piston-shaft mechanism

Homework Statement In the figure, a piston P is connected to a cylinder. The piston is connected to a rotating wheel with two shafts AB and BC. The shaft AB is connected on the periphery of the wheel. The wheel is rotating with angular speed ω= 100 rad s-1. At the moment A,C and the center of...
33. C

### Controlling the speed of an AC motor?

For my IB Physics Extended Essay, I have just changed my topic from momentum to this and I have really little time of doing it..... (Due in Oct) :cry::cry::cry::cry::cry::cry: My research question would be The effect of angular velocity of circular motion of a moving sound source on the sound...
34. ### Finding angular speed by conservation of angular momentum

Homework Statement Homework Equations Angular momentum = moment of inertia * angular velocity Change in angular momentum = impulsive moment v^2=u^2+2as The Attempt at a Solution First I used v^2=u^2+2as to find the velocity of the particle the moment the string goes taut. I got v=\sqrt{8ag}...
35. J

### Determine the angular acceleration and angular velocity

Homework Statement A thin uniform rod (of mass 10.0 Kg and length of 1.20 m) is attached to a friction-free pivot. Initially, the rod is balanced vertically above the pivot (position A in the figure attached). If the rod falls from rest, calculate a. the angular acceleration at position B...
36. ### I Relativistic angular velocity

Let's assume that a disk is rotating with relativistic speed in a frame. We can find the velocity of a particle using v=rw formula. However, what is the r in this formula? is it the radius of the disk in rest frame or in the lab frame?? And Is the magnitude of velocity same for all points of...
37. ### Angular Velocity of a wheel problem

The question: A wheel was rotating at 12.5 rad/s when a torque was applied for 15.7 s. The angular velocity increased to 54.0 rad/s. What angle did the wheel turn through in that time? My Solution: theta = Wo*t+1/2*a*t^2 = (12.5 rad/s)(15.7)+(1/2)(41.5)(15.7)^2 therefore, theta =...
38. ### Angular veolcity in rotating frame

Hello everyone, I have some conceptual problems understanding the rotating frame transformation. Take the center of the Earth as inertial frame's origin and another point in Hawaii as rotating frame's origin. In many lecture notes from internet, or Marion chapter 10. The vector describing the...
39. ### Acceleration of center of mass

Homework Statement "Represent (linear) acceleration of center of mass of link OA in terms of variables shown." OmegaOA is given to be some constant value, I have assigned it 'omegaOA'. Homework Equations Already derived equations to previous parts of the problem, fairly certain these are...
40. ### How fast is the edge of the yo-yo spinning in mph

Homework Statement Hello! I am trying to solve the exercise, but I don't see my mistakes - I can't come up with a correct answer. The exercise from the begging of trigonometry, so it has to be solved using concepts of a speed and angular velocity. A yo-yo which is 2.25 inches in diameter...
41. ### Center of mass on plank problem

Problem A 60kg woman walking at a speed of 1.0 m/s steps onto a long plank that has a mass of 60kg. Upon stepping on the plank, both she and the plank begin to slide with speed v and spin with angular rate ω on a frictionless surface. 1. How far from the woman is the center of mass of the woman...
42. ### Can I use this solution? Angular motion

Homework Statement Starting from rest, a wheel has constant α = 3.0 rad/s2. During a certain 4.0 s interval, it turns through 120 rad. How much time did it take to reach that 4.0 s interval? ω0 = 0 α = 3.0 rad/s2 θf = 120 rad Homework Equations Δθ = ω0⋅t + ½αt2 The Attempt at a Solution 120...
43. ### Finding angular acceleration from revolutions and velocity

Homework Statement an object starts from rest and has a final angular velocity of 6 rad/s. the object makes 2 complete revolutions. find the object's angular acceleration. Homework Equations wf^2=wi^2+2αd The Attempt at a Solution Not sure what to do with the revolutions, would it take act as...
44. ### Rotational Motion - Understanding the Idea

1- I am trying to understand the rotational motion and In most books theres ##\vec θ## unit vector which its, ##\vec θ=(-sinθ,cosθ)## I can see that If ##\vec r## unit vector is,##\vec r=(cosθ,sinθ)## then , ##\frac {d\vec r} {dt}=\vec θ## Like calculating ##\vec v=\frac {d\vec r}...
45. ### Rod+mass rotating clockwise

Homework Statement The 3.0 kg rod has a length of 80 cm. The 4.0 kg rectangle attached to the rod has negligible dimensions. The entire object rotates counterclockwise about the bottom of the rod. Determine the angular speed of the apparatus at the instant the rod is horizontal...
46. ### Classical Mechanics - Find angular velocity of two rods

Homework Statement Hello! I apologize for my poor setup, first post. I am given a system of two rods, hope you can see my image.http://file:///C:/Users/Mikkel/Downloads/Mek2_exam_Jan2016_final.pdf [Broken] One along the x-axis with mass = 2m and length = 2l Another perpendicular with the other...
47. ### What is the velocity of a falling object with air resistance?

Homework Statement I am trying to develop simulation for a falling object subject to air resistance. Object is similar to Samara seed. object is considered to be under steady vertical descend. know variable : surface area of object (A) weight of object (W) cd: drag coefficient object rotates...
48. ### Acceleration in circular motion

Homework Statement In the case of circular motion,the direction of the velocity is changing in every moment.So,there is a angular accelleration here.What is the direction of this accelleration? Homework Equations The Attempt at a Solution
49. ### Angular Velocity of a gear

Homework Statement Gear 2 in the figure below is driven at a speed of 80 rpm in the CCW direction viewed from the right end. Gear 4 meshes with a fixed ring gear (gear 7) and with gear 5. Find the angular velocity of gear 5. N2= 16 T, N3= 32 T, N4= 30 T, N5= 22 T and N7= 77 T. Homework...
50. ### Angular Velocity Help

Homework Statement A speed skater increases his speed from 10 m/s to 12.5 m/s over a period of 3 seconds while coming out of a curve of 20 m radius. a. What is the magnitude of his angular velocity as he leaves the curve? Be sure to include proper units for your answer. b. What is his average...