Consider a rigid body in pure rotational motion about a fixed axis. Each particle of the rotating rigid body executes uniform circular motion. The centre of the circular path is on the axis of rotation and the plane of the circle is perpendicular to the axis of rotation.
When a rigid body is in pure rotational motion, the angular displacement of all the particles of a rigid body is the same and is equal to the angle through which the rigid body has rotated about the axis of rotation. In the SI System of Units, angular displacement is measured in radian. If the angular displacement of a rigid body is θ during a small time interval of Δt, then the average angular velocity of the particle is defined Δθ/Δt. Angular velocity ω is defined as the rate of change of angular displacement and is measured in radians per second. The angular velocity is a vector quantity.
If the right hand's fingers are curled in the direction of the rotation of the body the direction of the stretched thumb is the direction of the angular velocity, ω. When a rigid body is in pure rotational motion the angular velocity of each particle is the same and is equal to the angular velocity of the whole body. If the angular velocity of a rigid body changes with time, it is said to be moving with angular acceleration. The rate of change of angular velocity is defined as angular acceleration, α. In the SI System, it is measured in radians per second squared. This is also a vector quantity and its direction is along the vector dω which represents the change in angular velocity. Here, the direction of angular acceleration is opposite to that of the direction of angular velocity.
Let v be the linear speed of a particle P that is at a perpendicular distance r from the axis of rotation. Then we have v
. This is a useful relationship between linear and angular speeds of a body.