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Claimed by Aditya Kuntamukkula 2017
'''Claimed by Shivali Singh (Spring 2025)'''


==The Main Idea==
==The Main Idea==


Rotational motion is defined as an object moving around an axis in contrast to translational motion which involves the object moving in a straight trajectory.
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.
 
[[File:3200 Phaethon orbit dec 2017.png|thumb|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===
===A Mathematical Model===
Rotation can be characterized by its angular velocity and angular acceleration. The equations are listed below.


Angular velocity:
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>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>
 
:<math>\boldsymbol{\alpha} = \frac{d\boldsymbol{\omega}}{dt}</math>
 
:<math>\boldsymbol{\alpha} = \frac{d^{2}\boldsymbol{\theta}}{dt^{2}}</math>
 
Here, <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration.
 
One may also approximate rotational motion using discrete-time equations:
 
:<math>\boldsymbol{\omega} = \frac{\Delta \boldsymbol{\theta}}{\Delta t}</math>
 
:<math>\boldsymbol{\alpha} = \frac{\Delta \boldsymbol{\omega}}{\Delta t}</math>
 
[[File:NEW.gif|thumb|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>\boldsymbol{\omega} = \omega_0 + \boldsymbol{\alpha}t</math>
 
:<math>\boldsymbol{\theta} = \frac{1}{2}\boldsymbol{\alpha}t^2 + \omega_0 t + \theta_0</math>
 
:<math>\boldsymbol{\omega}^{2} = \omega_0^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - \theta_0)</math>


:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math> ,
These equations are especially useful in introductory physics and engineering mechanics.
where <math>{\boldsymbol{v}}</math> is the velocity of the object and <math>{\boldsymbol{r}}</math> is the radius of the circle of motion. It can also be represented as the change in angle over the distance traveled in the formula shown below:


:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math> , where <math>{\boldsymbol{d\theta}}</math> is the change in angle and <math>{\boldsymbol{dt}}</math> is the change in time.
===Similarities to Linear Motion===


                                                                                [[File:NEW.gif|thumb|Angular velocity always has units of radians per time (radians/seconds, radians/minutes, or radians/hour)]]
For students who have already studied linear kinematics, it is useful to recognize that rotational equations closely parallel linear equations.


                                                                 
: '''Rotational''' <math>\Longleftrightarrow</math> '''Linear'''
Angular acceleration is equal to alpha:


:<math>\boldsymbol{{\alpha}} = \frac{\boldsymbol{a_t}}{\boldsymbol{r}}</math> ,
:<math>\boldsymbol{\theta} \Longleftrightarrow \boldsymbol{x}</math>
where <math>{\boldsymbol{a_t}}</math> is the tangential acceleration of the object and <math>{\boldsymbol{r}}</math> is the radius of the circle of motion.


:<math>\boldsymbol{\omega} \Longleftrightarrow \boldsymbol{v}</math>


Rotational Kinetic Energy:
:<math>\boldsymbol{\alpha} \Longleftrightarrow \boldsymbol{a}</math>
An object with a center of mass at rest can still have rotational kinetic energy. For example, if a disk is suspended in the air and spun, it has no translational kinetic energy. The position of the disk does not change. However, since it is spinning (rotating), it still has kinetic energy. To account for this, we can relate angular velocity with the moment of inertia of the object to find a value for the rotational kinetic energy.


Rotational Kinetic Energy:
The angular position <math>\theta</math> is analogous to linear position <math>x</math>, angular velocity corresponds to velocity, and angular acceleration corresponds to acceleration.


::<math>{KE}_{rot} = \frac{{1}}{{2}}{I}_{cm}{&omega;^2}</math>
The kinematic equations are therefore nearly identical in form:


:<math>\boldsymbol{\omega}=\frac{d\boldsymbol{\theta}}{dt}</math> <math>\Longleftrightarrow \boldsymbol{v}=\frac{d\boldsymbol{x}}{dt}</math>


Relation to Work and Energy Principle:
:<math>\boldsymbol{\alpha}=\frac{d\boldsymbol{\omega}}{dt}</math> <math>\Longleftrightarrow \boldsymbol{a}=\frac{d\boldsymbol{v}}{dt}</math>


The energy principle states:
:<math>\boldsymbol{\omega}=\omega_0+\boldsymbol{\alpha}t</math> <math>\Longleftrightarrow \boldsymbol{v}=v_0+\boldsymbol{a}t</math>
::<math>{E}_{f} = {E}_{i} + W </math>


We can apply the energy principle to rotational kinetic energy as well to find changes in kinetic energy and work done on the system.
:<math>\boldsymbol{\theta}=\frac{1}{2}\boldsymbol{\alpha}t^2+\omega_0 t+\theta_0</math> <math>\Longleftrightarrow \boldsymbol{x}=\frac{1}{2}\boldsymbol{a}t^2+v_0 t+x_0</math>


===A Computational Model===
:<math>\boldsymbol{\omega}^{2}=\omega_0^{2}+2\boldsymbol{\alpha}(\boldsymbol{\theta}-\theta_0)</math> <math>\Longleftrightarrow \boldsymbol{v}^{2}=v_0^{2}+2\boldsymbol{a}(\boldsymbol{x}-x_0)</math>


==Examples==
==Examples==
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===Simple===
===Simple===


A simple example and application of the concept of rotation is the earth's rotation on it's axis. It rotates once every 24 hours. What is the angular velocity?
The Earth completes one full rotation every 24 hours. What is its angular velocity in radians per second?
 
[[File:AllenEarth.png|thumb|Calculating Earth's angular velocity]]
 
One full revolution corresponds to:
 
:<math>\Delta\theta = 2\pi \text{ rad}</math>
 
:<math>\Delta t = 24 \text{ hr}</math>
 
Using:
 
:<math>\omega = \frac{\Delta\theta}{\Delta t}</math>


:<math>\boldsymbol{{w}} = \frac{\boldsymbol{d\theta}}{\boldsymbol{dt}}</math>
Substitute:
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\delta\theta}}{\boldsymbol{\delta t}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{2\pi}}{\boldsymbol{24}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{12}}\frac{\boldsymbol{rad}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{3600}}\frac{\boldsymbol{hr}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{\pi}}{\boldsymbol{43200}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>


Angular velocity can also be represented as change in angle (theta) over change in time. In this case, the earth rotates 2pi radians in 24 hours which reduces to pi/12 rad/hr and that is the equivalent of pi/43200 rad/s.
:<math>\omega = \frac{2\pi}{24}\frac{\text{rad}}{\text{hr}}</math>
 
Convert hours to seconds:
 
:<math>\omega = \frac{2\pi}{24}\cdot\frac{1}{3600}\frac{\text{rad}}{\text{s}}</math>
 
:<math>\omega = \frac{\pi}{43200}\frac{\text{rad}}{\text{s}}</math>


===Medium===
===Medium===


A cylinder with a 2.5 ft radius is rotating at 120 rpm. Find the angular velocity in rad/sec. Find the linear velocity of a point on its rim in mph.
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.
 
[[File:AllenDrawing.png|thumb|The applied torque is unknown]]
 
Given:
 
:<math>\omega_0 = 0</math>
 
:<math>\theta = 20(2\pi)=40\pi \text{ rad}</math>
 
:<math>t = 20 \text{ s}</math>
 
Use:
 
:<math>\theta=\frac{1}{2}\alpha t^2+\omega_0 t+\theta_0</math>


:<math>\boldsymbol{{w}} = {\boldsymbol{120}}\frac{\boldsymbol{rev}}{\boldsymbol{min}}*\frac{\boldsymbol{2\pi}}{\boldsymbol{1}}\frac{\boldsymbol{rad}}{\boldsymbol{rev}}*\frac{\boldsymbol{1}}{\boldsymbol{60}}\frac{\boldsymbol{min}}{\boldsymbol{s}} =
Assume <math>\theta_0=0</math>:
{\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}</math>


:<math>\boldsymbol{{v}} = {\boldsymbol{w}}{\boldsymbol{r}}</math>
:<math>40\pi=\frac{1}{2}\alpha(20)^2</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{4\pi}}\frac{\boldsymbol{rad}}{\boldsymbol{s}}*{\boldsymbol{2.5}}\frac{\boldsymbol{ft}}{\boldsymbol{rad}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}</math>
:<math>\boldsymbol{{v}} = {\boldsymbol{10\pi}}\frac{\boldsymbol{ft}}{\boldsymbol{s}}*\frac{\boldsymbol{3600}}{\boldsymbol{1}}\frac{\boldsymbol{s}}{\boldsymbol{hr}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} =\frac{\boldsymbol{75\pi}}{\boldsymbol{11}}\frac{\boldsymbol{miles}}{\boldsymbol{hr}}</math>


:<math>40\pi=200\alpha</math>


To find the solution of this problem, rpm (revolutions per minute) should be converted to radians/second. Following this, the linear velocity can be calculated by using the v=wr formula shown above. The angular velocity is 4pi radians per second and the linear velocity is 75pi/11 or 21.42 mph.
:<math>\alpha=\frac{\pi}{5}\text{ rad/s}^2</math>


===Difficult===
===Difficult===


A tire with a 9 inch radius is rotating at 30 mph. Find the angular velocity at a point on its rim. Also express the result in revolutions per minute.
Given angular acceleration:
 
:<math>\alpha = Ke^{-\beta t}</math>
 
where <math>K</math> and <math>\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>\alpha=\frac{d\omega}{dt}</math>
 
So:
 
:<math>Ke^{-\beta t}=\frac{d\omega}{dt}</math>
 
Integrating:
 
:<math>\omega=\int Ke^{-\beta t}dt = -\frac{K}{\beta}e^{-\beta t}+C_1</math>
 
Now use:
 
:<math>\omega=\frac{d\theta}{dt}</math>
 
Then:
 
:<math>\theta=\int\left(-\frac{K}{\beta}e^{-\beta t}+C_1\right)dt</math>
 
:<math>\theta=\frac{K}{\beta^2}e^{-\beta t}+C_1 t + C_2</math>


:<math>\boldsymbol{{w}} = \frac{\boldsymbol{v}}{\boldsymbol{r}}</math>
Constants of integration depend on initial conditions.
:<math>\boldsymbol{{r}} = {\boldsymbol{9}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{12}}\frac{\boldsymbol{ft}}{\boldsymbol{in}}*\frac{\boldsymbol{1}}{\boldsymbol{5280}}\frac{\boldsymbol{miles}}{\boldsymbol{ft}} = \frac{\boldsymbol{1}}{\boldsymbol{7040}}{\boldsymbol{miles}}</math>
:<math>\boldsymbol{{w}} = \frac{\boldsymbol{30}}{\frac{\boldsymbol{1}}{\boldsymbol{7040}}}</math>


==Connectedness==
==Connectedness==
Rotation is an extremely important aspect of dynamics (the study of moving objects) which plays a big role in biomechanics. Rotation relates to several important body parts such as the shoulder where there are two axis of rotation, the medial-lateral axis and the anterior-posterior axis. A study of the movement of the shoulder helps to treat medical conditions that may affect this area. Dynamics is also very important in many other disciples, including mechanical engineering and aerospace engineering.


== See also ==
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.<ref>[1]</ref>
 
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."''


To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.
[[File:Archimedes lever.png|400px|thumb|Archimedes and the principle of the lever]]


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


Books, Articles or other print media on this topic
Buridan suggested that celestial bodies continued moving due to an internal tendency, an early concept related to inertia and angular momentum.


===External links===
==See also==


Some other resources to further understand rotation are the following:
Within this student wiki:
 
[[http://www.physicsbook.gatech.edu/Torque Torque]]
 
[[http://www.physicsbook.gatech.edu/Rotational_Angular_Momentum 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://www.mathwarehouse.com/transformations/rotations-in-math.php
Line 99: Line 203:
==References==
==References==


[1] Biomechanics, Basic. <i>“It Is Important When Learning about</i> (n.d.): n. pag. Web.
[1] Basic Biomechanics sources discussing joint rotation and axes of motion.
 
[2] Van Nostrand's Scientific Encyclopedia. "Angular Velocity and Angular Acceleration." 2005.


[2] "Angular Velocity and Angular Acceleration." Van Nostrand's Scientific Encyclopedia (2005): n. pag. Web
[3] Halliday, Resnick, and Walker. ''Fundamentals of Physics.'' Rotational Motion chapters.

Latest revision as of 21:38, 27 April 2026

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?

Error creating thumbnail: sh: /usr/bin/convert: No such file or directory Error code: 127
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.

Error creating thumbnail: sh: /usr/bin/convert: No such file or directory Error code: 127
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.[1]

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.

  1. ↑ [1]
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