Rotation About a Fixed Axis

Rotation about a fixed axis describes rigid body motion where every point moves in a circular path about a stationary axis.

This concept is commonly applied in the design and analysis of rotating machinery, such as turbines, gears, shafts, motors, etc.

We can work with angular parameters using both vector and scalar kinematics depending on whether we treat motion of the object as two or three-dimensional.

---

Vector Kinematics

Angular parameters ($\theta ,\omega ,\alpha$) are pseudo-vectors (also known as axial vectors) directed along the axis of rotation. We use unit vector $\hat{k}$ to show that they rotate about the $z$-axis.

• Angular Velocity Vector: $\vec{\omega }=\dot{\theta }\hat{k}$
• Angular Acceleration Vector: $\vec{\alpha }=\dot{\omega }=\ddot{\theta }\hat{k}$

For a position vector $r$ measured from any point on the rotation axis, we can obtain the velocity and acceleration vector using cross-product.

• Velocity Vector:
  $\vec{v}=\vec{\omega }\times \vec{r}$
• Acceleration Vector:
  $\vec{a}={\vec{a}}_t+{\vec{a}}_n=(\vec{\alpha }\times \vec{r})+\vec{\omega }\times (\vec{\omega }\times \vec{r})$

For ${\vec{a}}_n$, the position vector $\vec{r}$ must point toward the axis of rotation.

---

Scalar Kinematics

All points on a rotating body share the same angular parameters:
$\omega =\dot{\theta }\alpha =\dot{\omega }=\ddot{\theta }\omega d\omega =\alpha d\theta$

However, each point’s linear parameters ($s,v,a$) scale linearly based on the distance $(r)$ from the axis of rotation.

Quantity	Angular Domain	Linear Domain	Relationship
Position/Displacement	$\theta$	$s$	$s=r\theta$
Velocity	$\omega =\frac{d\theta }{dt}$	$v=\frac{ds}{dt}$	$v=r\omega$
Tangential Acceleration	$\alpha =\frac{d\omega }{dt}$	$a_t$	$a_t=r\alpha$
Normal / Centripetal Acceleration	N/A	$a_n$	$a_n=r{\omega }^2=\frac{v^2}{r}$

Total Acceleration Magnitude:
$a=\sqrt{a_t^2+a_n^2}=r\sqrt{{\alpha }^2+{\omega }^4}$

---

Constant Angular Acceleration Equations

When angular acceleration $(\alpha )$ is constant, we get a set of simplified equations analogous to 1D linear motion.

• $\omega ={\omega }_0+\alpha t$
• $\theta ={\theta }_0+{\omega }_{0}t+\frac{1}{2}\alpha t^2$
• ${\omega }^2={\omega }_0^2+2\alpha (\theta -{\theta }_0$)

---

const current = dv.current();
 
if (current && current.file.tags.some(t => t.toLowerCase() === "#topic")) {
    const inlinks = current.file.inlinks;
 
    // Split backlinks: general notes vs. lecture notes
    const lectureLinks = inlinks.where(l => l.path.includes("/Lecture Notes/"));
    const otherLinks = inlinks.where(l => !l.path.includes("/Lecture Notes/"));
 
    if (otherLinks.length > 0) {
        dv.header(4, "Links To This Page");
        dv.list(otherLinks);
    }
 
    if (lectureLinks.length > 0) {
        dv.header(4, "Referenced In Lecture Notes");
        dv.list(lectureLinks);
    }
}