Rotation of a Rigid Body About a Fixed Axis

Rotation of a rigid body about a fixed axis is motion in which two points of the body, and therefore the line through them, remain fixed. This line is the axis of rotation. Every other point moves on a circle whose center lies on the axis.

Rotation law

Body orientation is specified by the angular position $\varphi=\varphi(t)$. A positive angular direction is selected in advance and determines the signs of angular velocity and angular acceleration.

Angular velocity and angular acceleration

If rotation about a fixed axis is described by the angular position $\varphi=\varphi(t)$, then:

$$\omega=\frac{d\varphi}{dt},\qquad \varepsilon=\frac{d\omega}{dt}=\frac{d^2\varphi}{dt^2}.$$

The signs of $\omega$ and $\varepsilon$ depend on the selected positive angular direction. Equal signs mean the magnitude of angular velocity increases; opposite signs mean it decreases.

In SI, angular position is measured in radians. Angular velocity is commonly expressed in rad/s and angular acceleration in rad/s²; the radian is dimensionless in SI, but retaining its symbol is useful for identifying angular quantities.

Vector description

The vector $\vec\omega$ lies along the rotation axis according to the right-hand rule. For a fixed axis, $\vec\varepsilon=d\vec\omega/dt$ also lies along that axis; its direction relative to $\vec\omega$ indicates whether the angular-speed magnitude is increasing or decreasing.

Uniform rotation

If $\omega=const$, then $\varepsilon=0$ and $\varphi=\varphi_0+\omega t$. One complete revolution corresponds to an angular change of magnitude $2\pi$ rad.

Constant angular acceleration

If $\varepsilon=const$, then $\omega=\omega_0+\varepsilon t$ and $\varphi=\varphi_0+\omega_0t+\varepsilon t^2/2$. These relations are analogous to those for uniformly accelerated rectilinear motion.

Period and frequency

For uniform rotation, the period $T$ is the time for one revolution and the frequency $f=1/T$ is the number of revolutions per unit time. Angular velocity is related by $\omega=2\pi/T=2\pi f$.

Example

A disk rotates uniformly at frequency $f=5$ Hz. Its angular velocity is $\omega=2\pi f=10\pi\approx31.4$ rad/s. In 2 s, the disk completes 10 revolutions.

Common mistakes

• confusing angular velocity with the linear velocity of a point on the body;
• using degrees in formulas intended for radians;
• ignoring the sign of $\omega$ or $\varepsilon$;
• assuming points on the rotation axis have nonzero linear velocity.