Rotational Kinematics

Overview

Rotational kinematics describes the motion of rotating objects using angular quantities. The concepts parallel those of linear kinematics.

Angular Quantities

Linear	Angular	Relationship
Position ($x$)	Angle ($\theta$)	$\theta = s/r$
Velocity ($v$)	Angular velocity ($\omega$)	$\omega = v/r$
Acceleration ($a$)	Angular acceleration ($\alpha$)	$\alpha = a_t/r$

Angular Position

Measured in radians:

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

Where $s$ is arc length and $r$ is radius

Conversions

$1 \text{ revolution} = 2\pi \text{ radians} = 360°$

$1 \text{ radian} = 57.3°$

Angular Velocity

Average Angular Velocity

$\omega_{\text{avg}} = \frac{\Delta\theta}{\Delta t}$

Instantaneous Angular Velocity

$\omega = \frac{d\theta}{dt}$

Units

• rad/s (SI)
• rpm (revolutions per minute)
• Conversion: $\omega \text{ (rad/s)} = \text{rpm} \times \frac{2\pi}{60}$

Direction

• Use right-hand rule
• Curl fingers in direction of rotation
• Thumb points in direction of $\vec{\omega}$

Angular Acceleration

Average Angular Acceleration

$\alpha_{\text{avg}} = \frac{\Delta\omega}{\Delta t}$

Instantaneous Angular Acceleration

$\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$

Units

• rad/s²

Rotational Kinematic Equations

For constant angular acceleration:

$\omega = \omega_0 + \alpha t$

$\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$

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

$\theta = \theta_0 + \frac{1}{2}(\omega_0 + \omega)t$

Compare to linear equations:

Linear	Angular
$v = v_0 + at$	$\omega = \omega_0 + \alpha t$
$x = x_0 + v_0 t + \frac{1}{2}at^2$	$\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$
$v^2 = v_0^2 + 2a(x-x_0)$	$\omega^2 = \omega_0^2 + 2\alpha(\theta-\theta_0)$

Relating Linear and Angular Motion

For a point at distance $r$ from the rotation axis:

Tangential Velocity

$v = r\omega$

Tangential Acceleration

$a_t = r\alpha$

(Due to changing speed)

Centripetal Acceleration

$a_c = \frac{v^2}{r} = r\omega^2$

(Due to changing direction, points toward center)

Total Linear Acceleration

$a = \sqrt{a_t^2 + a_c^2}$

Period and Frequency

Period ($T$)

Time for one complete revolution:

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

Frequency ($f$)

Revolutions per second:

$f = \frac{1}{T} = \frac{\omega}{2\pi}$

Rolling Motion

For an object rolling without slipping:

Velocity of Center

$v_{cm} = r\omega$

Condition for Rolling Without Slipping

$v_{cm} = r\omega \quad \text{(no relative motion at contact point)}$

Distance Traveled

$d = r\theta$

Examples

Example 1: Spinning Wheel

A wheel accelerates from rest at 2 rad/s² for 5 seconds. Find final angular velocity and angle turned.

$\omega = \omega_0 + \alpha t = 0 + 2(5) = 10 \text{ rad/s}$

$\theta = \frac{1}{2}\alpha t^2 = \frac{1}{2}(2)(25) = 25 \text{ rad} = 3.98 \text{ revolutions}$

Example 2: CD Player

A CD spins at 500 rpm. Find angular velocity and period.

$\omega = 500 \times \frac{2\pi}{60} = 52.4 \text{ rad/s}$

$T = \frac{2\pi}{\omega} = 0.12 \text{ s}$

Example 3: Tangential Speed

A point on the edge of a 15 cm radius disk rotating at 10 rad/s. Find tangential speed.

$v = r\omega = 0.15 \times 10 = 1.5 \text{ m/s}$

Example 4: Stopping Time

A flywheel rotating at 300 rpm slows uniformly to rest in 20 s. Find angular acceleration.

$\omega_0 = 300 \times \frac{2\pi}{60} = 31.4 \text{ rad/s}$

$\alpha = \frac{\omega - \omega_0}{t} = \frac{0 - 31.4}{20} = -1.57 \text{ rad/s}^2$

Example 5: Rolling Wheel

A wheel of radius 0.5 m rolls along the ground. If the center moves at 3 m/s, find angular velocity.

$\omega = \frac{v}{r} = \frac{3}{0.5} = 6 \text{ rad/s}$

Vector Nature of Angular Quantities

• Angular velocity $\vec{\omega}$ and angular acceleration $\vec{\alpha}$ are vectors
• Direction given by right-hand rule
• For 2D problems, often treated as scalars (positive = counterclockwise)