# Angular Velocity Definition

## Angular velocity

• Consider a rigid body of arbitrary shape rotating about a fixed axis through point O and perpendicular to the plane of the paper as shown below in the figure-1

• while the body is rotating each and every point in the body moves in a circle with their center lying on the axis of rotation and every point moves through the same angle during a particular interval of time
• Consider the position of a particle say ith particle at point P at a distance ri from point O and at an angle θi which OP makes with some reference line fixed in space say OX as shown below in the figure

• If particle moves an small distance dsi along the arc of the circle in small amount of time dt then
dsi=vidt ----(1)
where vi is the speed of the particle
• dθ is the angle subtended by an arc of length dsi on the circumference of a circle of radius ri,so dθ( in radians) would be equal to the length of the arc divided by the radius
i.e.
dθ=dsi/ri =vidt/ri ----(2)
• distance dsi would vary from particle to particle but angle dθ swept out in a given time remain same for all the particles i.e. if particle at point P moves through complete circle such that
Then all the other particles of the rigid body moves through the angular displacement dθ=2π

• So rate of change of angle w.r.t time i.e. dθ/dt is same for all particles of the rigid body and dθ/dt is known as angular velocity ω of the rigid body so
ω=dθ/dt ----(3)
• Putting equation (3) in equation (2) we find
vi=ri(dθ/dt) =riω ---(4)
This shows that velocity of ith particle of the rigid body is related to its radius and the angular velocity of the rigid body
• Angular velocity of a rotating rigid body can either be positive or negative. It is positive when the body is rotating in anticlockwise direction and negative when the body id rotating in clockwise direction
• Unit of angular velocity radian per second (rad-s-1) and since radian is dimensionless unit of angle so dimension of angular velocity is [T-1]
• Instead of radians angles are often expressed in degrees. So angular velocity can also be expressed in terms of degree per second and degree per minute
• we know that