Unit 5 Rotation: Understanding Rotational Kinematics (AP Physics C: Mechanics)

Rotational kinematics

The set of concepts/equations used to describe how an object rotates: angular displacement ($\theta$), angular velocity ($\boldsymbol{\beta}$), and angular acceleration ($\boldsymbol{\beta}$).

Angular position (θ)

An object’s rotational position measured as an angle from a chosen reference line; SI angle unit is the radian (rad).

Radian (rad)

Angle unit defined by $\theta = \frac{s}{r}$ (arc length divided by radius); treated as dimensionless but labeled “rad” to indicate an angle.

Arc length (s)

Distance traveled along a circular path; related to angular displacement by $s = r\theta$ ($\theta$ must be in radians).

Counterclockwise-positive sign convention

Common rotation sign convention in AP Physics: CCW angles/ω/α are positive, CW are negative; must be used consistently.

Angular velocity (ω)

Rate of change of angular position: $\boldsymbol{\beta} = \frac{d\boldsymbol{\theta}}{dt}$ (units rad/s); can be positive or negative depending on direction.

Average angular velocity (ω_avg)

Angular displacement per time interval: $\boldsymbol{\bar{\boldsymbol{\beta}}} = \frac{\triangle \theta}{\triangle t}.$.

Instantaneous angular velocity

Angular velocity at a moment in time: ω = dθ/dt (the slope of a θ vs t graph).

Right-hand rule (for ω direction)

In 3D, curl right-hand fingers with rotation direction; thumb points along the angular velocity vector (axis direction).

Angular acceleration ($\alpha$)

Rate of change of angular velocity: $\boldsymbol{\beta} = \frac{d\boldsymbol{\beta}}{dt} = \frac{d^2\boldsymbol{\theta}}{dt^2}$ (units rad/s²).

Average angular acceleration ($\alpha_{avg}$)

Change in angular velocity per time interval: $\boldsymbol{\bar{\beta}} = \frac{\triangle \boldsymbol{\beta}}{\triangle t}.$

Constant angular acceleration

A situation where α is constant; then ω changes linearly with time and θ changes quadratically with time.

Constant-$\alpha$ kinematics: $\omega = \omega_0 + \alpha t$

Relates angular velocity and time when $\boldsymbol{\beta}$ is constant; $\boldsymbol{\beta_0}$ is angular velocity at $t = 0$: $\boldsymbol{\beta} = \boldsymbol{\beta_0} + \boldsymbol{\beta} t$

Constant-$\alpha$ kinematics: $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$

Angular position as a function of time for constant $\boldsymbol{\beta}$; $\theta_0$ is the initial angular position: $\theta = \theta_0 + \boldsymbol{\beta_0} t + \frac{1}{2}\boldsymbol{\beta}t^2$

Constant-$\alpha$ kinematics: $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Time-eliminated rotational kinematics equation valid only when $\boldsymbol{\beta}$ is constant: $\boldsymbol{\beta}^2 = \boldsymbol{\beta_0}^2 + 2\boldsymbol{\beta}(\theta - \theta_0)$.

Average $\omega$ under constant $\alpha$

If $\alpha$ is constant, $\omega_{avg} = \frac{(\omega_0 + \omega)}{2}$, so $\Delta \theta = \omega_{avg} t = \frac{(\omega_0 + \omega)}{2}t$.

Slope of θ vs t graph

The slope of a θ(t) graph equals angular velocity ω.

Area under ω vs t graph

The area under an ω(t) graph over a time interval equals angular displacement Δθ.

Tangential speed ($v$)

Linear speed of a point at radius $r$ on a rotating object: $v = r\omega$ (units m/s).

Tangential acceleration ($a_t$)

Linear acceleration that changes speed along the tangent: $a_t = r\boldsymbol{\beta}$ (units m/s²).

Centripetal (radial) acceleration ($a_c$)

Inward acceleration that changes direction of velocity in circular motion: $a_c = r\omega^2 = \frac{v^2}{r}$.

Total acceleration in circular motion

Vector sum of perpendicular components: $a = \sqrt{a_t^2 + a_c^2}$.

Rigid body rotation about a fixed axis

Model where all points share the same $\theta, \boldsymbol{\beta},$ and $\boldsymbol{\beta},$ but have different linear speeds/accelerations depending on radius $r$.

Degrees-to-radians conversion factor

To use $s = r\theta$, $v = r\omega$, etc., convert degrees to radians: radians = degrees $\times \left(\frac{\pi}{180}\right)$.

Direction reversal condition

An object reverses rotation when angular velocity changes sign; the turning point occurs when $\boldsymbol{\beta} = 0.$