Introductory Mechanics | Fixed-Axis Rotation

Angular Position

The angle that an object has rotated about a specific axis, usually measured in radians. It describes the orientation of a rotating object in a rotational reference frame.

Angle related to Radius and Arc Length

$$\theta=\frac{s}{r}$$

Instantaneous Angular Velocity

The rate of change of angular position with respect to time, typically measured in radians per second. It indicates how quickly an object is rotating about a fixed axis at a specific instant.

It is defined mathematically as $\frac{d\theta}{dt}$ , where $\theta$ is the angular position.

$$\omega=\lim_{\Delta t \to 0}\frac{\Delta \theta}{\Delta t}=\frac{d \theta}{dt}$$

Arc Length of the Circle

Equation:$s=r\theta$

Tangential Speed

Relationship | Conditions: Constant Angular Velocity $\left(w=constant\right)$

• Tangential Speed is proportional to the radius

  • Increased Radius: More Speed

  • Decreased Radius: Less Speed

Equation: $v=r\omega$

Vector Angular Velocity $\overrightarrow{w}$

Unit: $\frac{rad}{s}$

The angular velocity $\overrightarrow{w}$ has a direction determined by what is called the right-hand rule.

Counter-Clockwise (CCW): Positive $\left( +\omega \right)$

Clockwise (CW): Negative$\left(-\omega\right)$

Angular Acceleration

Unit: $\frac{rad}{s^2}$

Angular Acceleration is the rate at which an object’s angular velocity changes over time.

Equation: $\alpha=\frac{\Delta w}{\Delta t}=\frac{dw}{dt}=\frac{d^2\theta}{dt^2}$ , $a_{t}=r\alpha$

Average Angular Velocity

Condition: If the system is rotating under constant acceleration, the angular velocity changes linearly over time

Equation:$\overline{w}=\frac{\omega_{o}+\omega_{f}}{2}$

This will also let you calculate the total angular displacement $\left( \Delta\theta \right)$

$$\Delta\theta=\overline{w}\Delta t=\left(\frac{\omega_{o}+\omega_{f}}{2}\right)t$$

Angular Acceleration Vector $\left( \overrightarrow{\alpha} \right)$

$z-$ axis:

• $+z-$ axis: positive angular velocity

• $-z-$ axis: negative angular velocity

Speeding: If $\pm\omega$ and $\pm\alpha$ have the same sign

Slowing: If $\pm\omega$ and $\mp\alpha$ have opposite signs

Rotational Kinematics

Angular displacement from average angular velocity	$$\theta_f=\theta_o+\overline{\omega}t$$
Angular velocity from angular acceleration	$$\omega_f=\omega_o+\alpha t$$
Angular displacement from angular velocity and angular acceleration	$$\theta_f=\theta_o+\omega_ot+\frac{1}{2}\alpha t^2$$
Angular velocity from angular displacement and angular acceleration	$$\omega^{2}_f=\omega^{2}_o+2\alpha \left(\Delta\theta \right)$$