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 i^{th} particle at point P at a distance r_{i} from point O and at an angle θ_{i} which OP makes with some reference line fixed in space say OX as shown below in the figure
- If particle moves an small distance ds_{i} along the arc of the circle in small amount of time dt then
ds_{i}=v_{i}dt ----(1)
where v_{i} is the speed of the particle
- dθ is the angle subtended by an arc of length ds_{i} on the circumference of a circle of radius r_{i},so dθ( in radians) would be equal to the length of the arc divided by the radius
i.e.
dθ=ds_{i}/r_{i} =v_{i}dt/r_{i} ----(2)
- distance ds_{i} 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
dθ=2π rad
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
v_{i}=r_{i}(dθ/dt) =r_{i}ω ---(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
2π radians =360^{0}
or π radians/180^{0}=1
And this relation can be used for expressing angular velocity in degree to that of angular velocity in terms of radian
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