Rotation about a fixed axis is a special case of rotational motion. It is very common to analyze problems that involve this type of rotation — for example, a wheel.

The figure below illustrates rotational motion of a rigid body about a fixed axis at point

In the figure, the angle

Every point in the rigid body rotates by the same angle

Given the angular position of the rigid body we can calculate the angular displacement, angular velocity, and angular acceleration. These are important quantities to consider when evaluating the rotational kinematics of a problem.

A common assumption, which applies to numerous problems involving rotation about a fixed axis, is that angular acceleration is constant. With angular acceleration as constant we can derive equations for the angular position, angular displacement, and angular velocity of a rigid body experiencing rotation about a fixed axis.

The easiest way to derive these equations is by using Calculus. The derivations that follow are of the exact same form as the equations derived for rectilinear motion, with constant acceleration.

The angular acceleration is given by

where

Integrate the above equation with respect to time, to obtain angular velocity. This gives us

where

Integrate the above equation with respect to time, to obtain angular position. This gives us

where

The constants

At time

At time

Substituting these two initial conditions into the above two equations we get

Therefore

This gives us

For convenience, set

Angular displacement is defined as Δ

If we wish to find an equation that doesn’t involve time

Equations (1), (2), (3), and (4) fully describe the rotational motion of rigid bodies (or particles) rotating about a fixed axis, where angular acceleration

For the cases where angular acceleration is not constant, new expressions have to be derived for the angular position, angular displacement, and angular velocity. If the angular acceleration is known as a function of time, we can use Calculus to find the angular position, angular displacement, and angular velocity, in the same manner as before.

Alternatively, if we are given the angular position

For example, let's say the angular position

Thus, the angular velocity

The angular acceleration

The figure below shows a particle

Where:

We can use the equations for curvilinear motion to derive expressions for

Note that there is no radial component of velocity pointing towards, or away from the center of the circle. This is because the radius

If you want to test your understanding and solve some problems go to Circular Motion Problems.

Rotational motion in three dimensions is mathematically more complicated than planar rotation about a fixed axis, since the axis of rotation can change direction. This type of rotation applies to bodies experiencing three-dimensional motion. However, it is generally only necessary (and practical) to account for such rotation when determining the velocity and acceleration of a point on a body that is experiencing three-dimensional motion. As a result, three-dimensional rotation does not lend itself to a stand-alone discussion here.

For an explanation of three-dimensional motion see A closer look at velocity and acceleration.

For an example of a solved problem involving three-dimensional rotation see Gyro Top.

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