Rotational Kinematics Basics

List of equations at Angular Formulas.

Measuring Rotational Motion §

Rotational motion is characterized by the rotation of an object about an axis. The measure of this rotation can be in radians or degrees.

1. Rotational measurement is typically measured in radians.
2. Angular speed ($\omega$): the rate of change of angular displacement. It can be in radians/second or rotations/minute (rpm).
   • $1$ rotation = 2$\pi$ radians

Finding $\omega$ §

A wheel complete one full rotation in 4 seconds. What is its angular speed?

$$\omega =\frac{2\pi }{T}=2\pi f=\frac{v}{r}$$

We can find this using $w=\frac{2\pi }{T}$. $T$ is our period of rotation, or $T=4$ seconds.

$$\omega =\frac{2\pi }{4}=1.57\text{ rad/s}$$

Angular Acceleration §

If angular speed is the rate of change of angular displacement, then angular acceleration is the rate of change of the rate of change… but we can just call it the rate that angular speed increases or decreases.

Finding $\alpha$ §

We can bring back the wheel in the previous question. If at first it completes a rotation in 4 seconds ($T_i=4$), but Jason boosts it and five seconds after $\Delta t=5$ that time decreases to 3 seconds ($T_f=3$), what is the angular acceleration?

$$\alpha =\frac{w_f-w_i}{\Delta t}$$

Since we know $T_i$ and $T_f$, we can calculate ${\omega }_f$ and ${\omega }_i$. We already know $w_i$, so let’s focus on $\omega f$.

$${\omega }_f=\frac{2\pi }{3}=2.09\text{ rad/s}$$

Then we can plug in the values:

$$\alpha =\frac{2.09-1.57}{5}=0.104{\text{ rad/s}}^2$$

Rotational Relationships §

Angular Speed §

$$\omega =\frac{2\pi }{T}=2\pi f$$

Angular Acceleration §

$$\alpha =\frac{\Delta \omega }{\Delta t}=\frac{a_T}{r}$$

Tangential Speed §

$$v_t=r\omega$$

Linear vs Angular §

Linear Equation ($a$ = constant)	Angular Equation ($\alpha$ = constant)
$v=v_i+at$	$\omega ={\omega }_i+\alpha t$
$x=x_i+\frac{1}{2}(v_i+v)t$	$\theta ={\theta }_i+\frac{1}{2}({\omega }_i+\omega )t$
$x=x_i+v_{i}t+\frac{1}{2}a{t}^2$	$\theta ={\theta }_i+{\omega }_{i}t+\frac{1}{2}\alpha t^2$
$v^2=v_i^2+2a(x-x_i)$	${\omega }^2=w_i^2+2\alpha (\theta -{\theta }_i)$

Centripetal and Tangential Acceleration §

$$\begin{array}{rlrl}{v}_t& =r\omega & & \\ & & & \\ a_c& =\frac{v^2}{r}& =\frac{(r\omega {)}^2}{r}& \\ & & & \\ \text{ so }{a}_c& =(r\omega {)}^2& & \\ & & & \\ a_t& =\frac{\Delta v_t}{\Delta t}& =\frac{r\Delta \omega }{\Delta t}& =r\alpha \\ & & & \\ a_t& =r\alpha & & \end{array}$$