Rotational Kinematics

Claimed by Shivali Singh (Spring 2025)

The Main Idea

Rotational Motion (also known as curvilinear motion), in contrast to linear motion (also known as rectilinear motion), describes the motion of objects whose angular orientation changes over time. For this reason, it is common practice to use polar coordinates when analyzing systems undergoing rotational motion. Rotational quantities (also called angular quantities) describe the angular components of an object's motion.

When working in the context of rotational kinematics, there is usually a defined point or axis of rotation about which motion is analyzed. Examples include a wheel spinning about its axle, a ceiling fan rotating about a fixed shaft, or a planet orbiting a star.

Rotational motion about a point external to the object's center of mass

Rotational motion is extremely common in both nature and engineering. Examples include gears, turbines, satellites, amusement park rides, and biological joints such as the shoulder and knee.

A Mathematical Model

Similar to linear motion, angular quantities can be described by differential equations which relate the rate of change of one quantity to another aspect of rotational motion.

[math]\displaystyle{ \boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt} }[/math]

[math]\displaystyle{ \boldsymbol{\alpha} = \frac{d\boldsymbol{\omega}}{dt} }[/math]

[math]\displaystyle{ \boldsymbol{\alpha} = \frac{d^{2}\boldsymbol{\theta}}{dt^{2}} }[/math]

Here, [math]\displaystyle{ \boldsymbol{\theta} }[/math] represents angular position, [math]\displaystyle{ \boldsymbol{\omega} }[/math] represents angular velocity, and [math]\displaystyle{ \boldsymbol{\alpha} }[/math] represents angular acceleration.

One may also approximate rotational motion using discrete-time equations:

[math]\displaystyle{ \boldsymbol{\omega} = \frac{\Delta \boldsymbol{\theta}}{\Delta t} }[/math]

[math]\displaystyle{ \boldsymbol{\alpha} = \frac{\Delta \boldsymbol{\omega}}{\Delta t} }[/math]

Angular velocity is commonly measured in radians per second (rad/s)

Objects undergoing rotational motion, like objects in linear motion, can be analyzed using kinematics. However, depending on the complexity of the motion, solving for all quantities may become difficult. As a result, special cases are often considered, such as rotation about a fixed axis or uniform circular motion.

A particularly important case occurs when an object undergoes constant angular acceleration. In this case, equations analogous to the constant-acceleration equations from linear kinematics can be used:

[math]\displaystyle{ \boldsymbol{\omega} = \omega_0 + \boldsymbol{\alpha}t }[/math]

[math]\displaystyle{ \boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}t^2 + \omega_0 t + \theta_0 }[/math]

[math]\displaystyle{ \boldsymbol{\omega}^{2} = \omega_0^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - \theta_0) }[/math]

These equations are especially useful in introductory physics and engineering mechanics.

Similarities to Linear Motion

For students who have already studied linear kinematics, it is useful to recognize that rotational equations closely parallel linear equations.

Rotational [math]\displaystyle{ \Longleftrightarrow }[/math] Linear

[math]\displaystyle{ \boldsymbol{\theta} \Longleftrightarrow \boldsymbol{x} }[/math]

[math]\displaystyle{ \boldsymbol{\omega} \Longleftrightarrow \boldsymbol{v} }[/math]

[math]\displaystyle{ \boldsymbol{\alpha} \Longleftrightarrow \boldsymbol{a} }[/math]

The angular position [math]\displaystyle{ \theta }[/math] is analogous to linear position [math]\displaystyle{ x }[/math], angular velocity corresponds to velocity, and angular acceleration corresponds to acceleration.

The kinematic equations are therefore nearly identical in form:

[math]\displaystyle{ \boldsymbol{\omega}=\frac{d\boldsymbol{\theta}}{dt} }[/math] [math]\displaystyle{ \Longleftrightarrow \boldsymbol{v}=\frac{d\boldsymbol{x}}{dt} }[/math]

[math]\displaystyle{ \boldsymbol{\alpha}=\frac{d\boldsymbol{\omega}}{dt} }[/math] [math]\displaystyle{ \Longleftrightarrow \boldsymbol{a}=\frac{d\boldsymbol{v}}{dt} }[/math]

[math]\displaystyle{ \boldsymbol{\omega}=\omega_0+\boldsymbol{\alpha}t }[/math] [math]\displaystyle{ \Longleftrightarrow \boldsymbol{v}=v_0+\boldsymbol{a}t }[/math]

[math]\displaystyle{ \boldsymbol{\theta}=\frac{1}{2}\boldsymbol{\alpha}t^2+\omega_0 t+\theta_0 }[/math] [math]\displaystyle{ \Longleftrightarrow \boldsymbol{x}=\frac{1}{2}\boldsymbol{a}t^2+v_0 t+x_0 }[/math]

[math]\displaystyle{ \boldsymbol{\omega}^{2}=\omega_0^{2}+2\boldsymbol{\alpha}(\boldsymbol{\theta}-\theta_0) }[/math] [math]\displaystyle{ \Longleftrightarrow \boldsymbol{v}^{2}=v_0^{2}+2\boldsymbol{a}(\boldsymbol{x}-x_0) }[/math]

Examples

Simple

The Earth completes one full rotation every 24 hours. What is its angular velocity in radians per second?

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Calculating Earth's angular velocity

One full revolution corresponds to:

[math]\displaystyle{ \Delta\theta = 2\pi \text{ rad} }[/math]

[math]\displaystyle{ \Delta t = 24 \text{ hr} }[/math]

Using:

[math]\displaystyle{ \omega = \frac{\Delta\theta}{\Delta t} }[/math]

Substitute:

[math]\displaystyle{ \omega = \frac{2\pi}{24}\frac{\text{rad}}{\text{hr}} }[/math]

Convert hours to seconds:

[math]\displaystyle{ \omega = \frac{2\pi}{24}\cdot\frac{1}{3600}\frac{\text{rad}}{\text{s}} }[/math]

[math]\displaystyle{ \omega = \frac{\pi}{43200}\frac{\text{rad}}{\text{s}} }[/math]

Medium

A torque is exerted on a disk initially at rest, causing constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. Find the angular acceleration.

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The applied torque is unknown

Given:

[math]\displaystyle{ \omega_0 = 0 }[/math]

[math]\displaystyle{ \theta = 20(2\pi)=40\pi \text{ rad} }[/math]

[math]\displaystyle{ t = 20 \text{ s} }[/math]

Use:

[math]\displaystyle{ \theta=\frac{1}{2}\alpha t^2+\omega_0 t+\theta_0 }[/math]

Assume [math]\displaystyle{ \theta_0=0 }[/math]:

[math]\displaystyle{ 40\pi=\frac{1}{2}\alpha(20)^2 }[/math]

[math]\displaystyle{ 40\pi=200\alpha }[/math]

[math]\displaystyle{ \alpha=\frac{\pi}{5}\text{ rad/s}^2 }[/math]

Difficult

Given angular acceleration:

[math]\displaystyle{ \alpha = Ke^{-\beta t} }[/math]

where [math]\displaystyle{ K }[/math] and [math]\displaystyle{ \beta }[/math] are constants, determine angular velocity and angular displacement as functions of time.

Since acceleration is not constant, the constant-acceleration equations cannot be used.

Start with:

[math]\displaystyle{ \alpha=\frac{d\omega}{dt} }[/math]

So:

[math]\displaystyle{ Ke^{-\beta t}=\frac{d\omega}{dt} }[/math]

Integrating:

[math]\displaystyle{ \omega=\int Ke^{-\beta t}dt = -\frac{K}{\beta}e^{-\beta t}+C_1 }[/math]

Now use:

[math]\displaystyle{ \omega=\frac{d\theta}{dt} }[/math]

Then:

[math]\displaystyle{ \theta=\int\left(-\frac{K}{\beta}e^{-\beta t}+C_1\right)dt }[/math]

[math]\displaystyle{ \theta=\frac{K}{\beta^2}e^{-\beta t}+C_1 t + C_2 }[/math]

Constants of integration depend on initial conditions.

Connectedness

Rotation is an extremely important aspect of mechanics. Nearly every machine contains rotating components such as wheels, motors, gears, turbines, and fans. Engineers must understand rotational motion when designing safe and efficient systems.

In biomechanics, rotation is also important. For example, the shoulder rotates about multiple axes, and the knee experiences rotational effects during walking and running. Understanding rotational motion helps physicians, therapists, and engineers develop better treatments and prosthetic devices.

Rotational motion is also fundamental in astronomy, where planets rotate on their axes while simultaneously revolving around stars.

History

The concept of rotation was known as far back as ancient Egypt. The Egyptians recognized that applying force to round objects such as logs allowed them to roll across the ground.

Archimedes later studied rotational equilibrium through the principle of the lever. He famously stated:

"Give me a lever long enough, and I shall move the world."

Archimedes and the principle of the lever

Later scholars such as Thomas Bradwardine and Jean Buridan contributed to early ideas involving angular velocity, inertia, and rotational motion.

Buridan suggested that celestial bodies continued moving due to an internal tendency, an early concept related to inertia and angular momentum.

See also

Within this student wiki:

[Torque]

[Angular Momentum]

Rigid Body Motion

Further Reading

https://brilliant.org/wiki/angular-kinematics-problem-solving/

https://courses.lumenlearning.com/physics/chapter/10-2-kinematics-of-rotational-motion/

External Links

http://www.mathwarehouse.com/transformations/rotations-in-math.php

http://demonstrations.wolfram.com/Understanding3DRotation/

References

[1] Basic Biomechanics sources discussing joint rotation and axes of motion.

[2] Van Nostrand's Scientific Encyclopedia. "Angular Velocity and Angular Acceleration." 2005.

[3] Halliday, Resnick, and Walker. Fundamentals of Physics. Rotational Motion chapters.