Orbiting planets and pendulum illustrating classical mechanics principles

Rotational kinematics

PHYS 201 · Rotational Mechanics

Rotational kinematics describes angular position, angular velocity, and angular acceleration. This lesson develops the rotational analogs of linear motion equations.

Key equations

s=r\theta\omega=\frac{d\theta}{dt}\alpha=\frac{d\omega}{dt}=\frac{d^2\theta}{dt^2}\omega=\omega_0+\alpha t\theta=\theta_0+\omega_0t+\frac{1}{2}\alpha t^2\omega^2=\omega_0^2+2\alpha\Delta\thetav=r\omegaa_t=r\alphaa_c=r\omega^2=\frac{v^2}{r}\omega=2\pi f=\frac{2\pi}{T}v_{cm}=R\omegaa_{cm}=R\alpha

Learning objectives

Define angular position, angular velocity, and angular acceleration.
Use constant-angular-acceleration equations.
Relate angular quantities to linear quantities using radius.
Apply rolling-without-slipping constraints.

Angular position

Rotational motion is described using angular variables. Angular position, usually written $heta$ , tells how far an object has rotated from a reference direction. In physics, angles are usually measured in radians.

Radians connect angle to arc length:

$s=r heta$

where $s$ is arc length and $r$ is radius. This formula works only when $heta$ is measured in radians.

Angular velocity

Angular velocity is the rate of change of angular position:

$omega= rac{d heta}{dt}$

Its SI unit is radians per second. Since radians are dimensionless ratios, the unit is often written as $s^{-1}$ , but $rad/s$ helps preserve meaning.

If an object rotates counterclockwise, angular velocity is often taken as positive. Clockwise may be negative, depending on convention.

Angular acceleration

Angular acceleration is the rate of change of angular velocity:

$alpha= rac{domega}{dt}= rac{d^2 heta}{dt^2}$

If angular speed increases, angular acceleration has the same sign as angular velocity. If angular speed decreases, angular acceleration has the opposite sign.

Constant angular acceleration

If angular acceleration is constant, the rotational kinematics equations mirror the linear constant-acceleration equations:

$omega=omega_0+alpha t$

$heta= heta_0+omega_0t+ rac{1}{2}alpha t^2$

$omega^2=omega_0^2+2alphaDelta heta$

These equations apply only when $alpha$ is constant.

Linear and angular quantities

For a point on a rotating object at radius $r$ , linear speed is related to angular speed by

$v=romega$

Tangential acceleration is

$a_t=ralpha$

Centripetal acceleration is

$a_c=romega^2= rac{v^2}{r}$

Tangential acceleration changes speed around the circle. Centripetal acceleration changes direction of velocity.

Period and frequency

For uniform rotation, the period $T$ is the time for one revolution, and frequency $f$ is the number of revolutions per second:

$f= rac{1}{T}$

Angular speed is

$omega=2pi f= rac{2pi}{T}$

This is essential for wheels, gears, motors, rotating planets, and oscillations.

Rigid body rotation

A rigid body is an ideal object whose shape does not change. In pure rotation about a fixed axis, every point of the body has the same angular velocity and angular acceleration, but points farther from the axis have larger linear speed because $v=romega$ .

This explains why the outer edge of a spinning disk moves faster than a point near the center.

Rolling without slipping

For a wheel rolling without slipping, the center-of-mass speed is related to angular speed by

$v_{cm}=Romega$

where $R$ is the wheel radius. This constraint connects translational and rotational motion.

If the wheel accelerates without slipping,

$a_{cm}=Ralpha$

The big idea

Rotational kinematics mirrors linear kinematics, with angular position, angular velocity, and angular acceleration replacing position, velocity, and acceleration. The bridge between angular and linear quantities depends on radius. These tools prepare us to study torque, rotational energy, and angular momentum.

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