Rotational Kinematics

"Men talk of killing time, while time quietly kills them”
Dion BoucicaultLondon Assurance (1841)




Angular velocity and angular acceleration are defined in a similar way to velocity and acceleration.  There are average and instantaneous values of each.

 Angular acceleration is not the same as centripetal acceleration.  Centripetal acceleration is due to a change in the direction of the velocity, angular acceleration is due to a change in the magnitude of the velocity (through the angle of rotation).


 Exactly as in the translational case, the difference between angular speed and angular velocity is direction.  Angular velocity must include a direction of rotation about the axis in question.  For example, 10 rad/s clockwise about the x axis is an angular velocity, 10 rad/s about the x axis is an angular speed.



By direct analogy with the translational kinematic equations, circular motion about a single axis under constant angular acceleration may be described by the following four equations,


where we have made the substitutions,

Note that just as +x is defined arbitrarily to the right, the positive value of theta can be defined as clockwise or anticlockwise.



Starting from the definition of radian measure, by differentiating with respect to time, we can show that,


where v is the tangential velocity and a is the tangential acceleration.


  A particle executing circular motion, with a varying angular velocity (non uniform circular motion), will experience two components of acceleration, a tangential component due to the changing magnitude of its velocity and a radial (centripetal) component due to the changing direction of its velocity

The net acceleration of the particle is the vector sum of these two components as indicated below.






      Simultaneous rotation about more than one axis can be considered in a similar manner to projectile motion, where we extended our 1D translational discussion to 2D motion.  In aeronautic   applications rotations about the three axes are described as Roll, Pitch and Yaw.




wobbly divider

Example Problem

“I don’t want to achieve immortality through my work…I want to achieve it through not dying

Woody Allen – Woody Allen and his Comedy (1975)


Dr. C. L. Davis
Physics Department
University of Louisville