Angular Motion Formula Sheet

Note: many of these formulas— specifically projectile motion ones— can also be found in Physics 1 Review, but in slightly different forms as those are not specifically put into angular terms.

Linear vs Angular Equations §

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

Frequency §

• $f$, frequency, in Hz
• $T$, time, in seconds

$$f=\frac{1}{T}$$

Angular Kinematics §

(Really cringey) application-based explanations at Rotational Kinematics Basics

Rotations §

One rotation is equal to $2\pi$ radians.

Angular Speed §

• $\omega$, angular speed, in radians per second (OR rotations per minute)
• $\Delta \theta$, change in angle, in radians
• $\Delta t$, change in time, in seconds
• $f$, frequency, in hertz
• $T$, time period, in seconds

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

$$\begin{array}{rl}{\omega }_f& =w_i+\alpha \Delta t\\ {w_f}^2& ={w_i}^2+2\alpha ({\theta }_f-{\theta }_i)\end{array}$$

Angular Acceleration §

• $\alpha$, angular acceleration, in radians per second squared
• ${\omega }_f$, final angular speed, in radians per second
• ${\omega }_i$, initial angular speed, in radians per second
• $\Delta t$, change in time, in seconds
• $a_t$, tangential acceleration

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

Angular Motion §

• ${\omega }_f$, final angular speed, in radians per second
• ${\omega }_i$, initial angular speed, in radians per second
• $\Delta t$, change in time, in seconds
• $\alpha$, angular acceleration, in radians per second squared
• ${\theta }_f$, final angle, in radians
• ${\theta }_i$, initial angle, in radians

$$\begin{array}{rl}{\omega }_f& =w_i+\alpha \Delta t\\ & \\ \text{OR}& \\ & \\ {w_f}^2& ={w_i}^2+2\alpha ({\theta }_f-{\theta }_i)\end{array}$$

Angular Distance §

• $\theta$, angular distance, in radians
• $\omega$, angular speed, in radians per second
• $\Delta t$, change in time, in seconds
• $\alpha$, angular acceleration, in radians per second squared

$$\theta =\omega \Delta t+\frac{\alpha (\Delta t{)}^2}{2}$$

Torque §

Torque is equal to the distance $r$ of a force from a pivot point/fulcrum times the magnitude of the that force $F$.

• $\tau$, torque, in Newton meters (Nm)
• $r$, distance, in meters
• $F$, force, in Newtons

$$\tau =\vec{r}\times \vec{F}$$