# Rotational Kinematics

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

• To date we have considered the kinematics and dynamics of particles, including translational and circular motion as well as the translational motion of systems of particles (in particular rigid bodies) in terms of the motion of the centre of mass of the system (body).  In the latter case we are able to imagine that all the mass of the object is located at the centre of mass as far as external translational forces are concerned.

• The next step is to consider the rotation of a rigid body about a fixed axis of rotation.  Note that, because we are considering a rigid body, every particle in the body remains fixed relative to the others.  This means that in such a rotational motion every particle moves in a circle whose centre lies on the axis of rotation.  In the diagram at right the object rotates about the z axis; the two sample particles move in circles with radii r1 and r2.  If we can describe the circular motion of a particle, without direct reference to its radius, then all particles in the system will be described by the same set of equations.  Although the radii of the particles are different their angular rotations are identical.  Therefore it is necessary to introduce angular variables.

• Angular Velocity (Speed) and Angular Acceleration

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.

• Rotational Kinematic Equations

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.

• Relation between Angular and Translational Variables

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.

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
email: c.l.davis@louisville.edu