Angular velocity is also known as rotational velocity. It is a vector quantity, which means it has both magnitude and direction.
Types of Angular Velocity
Formula
Average and Instantaneous Velocity
Relation Between Linear and Angular Velocity
Angular Velocity of the Earth
Example Problems
Orbital angular velocity refers to how fast an object revolves around a fixed point, typically taken as the origin. Therefore, it is the time rate of change of its angular displacement relative to the origin. For example, the Earth revolves around the Sun. It has an orbital angular velocity. Spin angular velocity refers to how fast a rigid body rotates about an axis, irrespective of the choice of origin. For example, a bicycle wheel rotates about its axis. It has a spin angular velocity. Units and Dimensions The SI unit of angular velocity is radians per second or radˑs-1. Another unit is revolutions per minute or RPM. The conversion between degrees, radians, and revolutions is quite simple. 1 revolution = 3600 = 2π radians and, 1 RPM = 0.10472 rad/s The dimensional formula is [M0 L0 T-1]. Direction The right-hand rule conventionally specifies the direction of angular velocity. According to this rule, if the fingers curl in the direction of rotation, the thumb points toward the angular velocity. For a revolving object, the angular velocity direction is always perpendicular to the plane of revolution. Where θ1 and θ2 are the object’s initial and final angular positions, respectively, and t1 and t2 are the times the object was at these positions. We can also define instantaneous velocity, which is the instantaneous rate of change of angular displacement. It is obtained by taking the limit of the average angular velocity as the time interval (Δt) approaches zero. Mathematically, it is the time derivative of the angular displacement. A disc of radius R rotates counterclockwise about an origin O, as shown below. Consider an arc AB = Δs on its rim such that, in the triangle OAB, The linear velocity v of the disc is always tangential to the points A and B and is given by Therefore, linear velocity is the product of the angular velocity and the radial distance or radius. The angular velocity does not change with radius, but linear velocity does. When an object rotates about an axis, every point on the object has the same angular velocity. However, points farther from the axis move at a greater linear velocity than points closer to it. 1 rotation = 2π radians and 1 hour = 3600 s Therefore, Solution Given R = 40 cm = 0.4 m and v = 40 km/h = 40 x 103 m/(1 x 3600 s) = 11.11 m/s Therefore, the angular velocity is ω = v/R = 11.11 mˑs-1/0.4 m = 27.8 radˑs-1 Problem 2. How many revolutions does a turntable make if it moves at 36 rad/s and for 6 minutes? Solution Given ω = 36 rad/s and Δt = 6 min = 360 s The turntable has been rotated by Δθ = ωΔt = 36 rad/s x 360 s = 12960 rad We know that 2π rad = 1 revolution Therefore, 12960 rad = 1 revolution/2π rad x 12960 rad = 2062 revolutions
