Rotational Kinematics - Revision history

Shivali17: Preserved the original page content while improving grammar, clarity, formatting, and physics notation. Added real world examples, expanded applications of rotational motion, improved worked examples, strengthened the history and connectedness sections, and updated references for a more polished final submission.

2026-04-28T01:38:53Z

Preserved the original page content while improving grammar, clarity, formatting, and physics notation. Added real world examples, expanded applications of rotational motion, improved worked examples, strengthened the history and connectedness sections, and updated references for a more polished final submission.

Show changes

Spencer B.: /* A Mathematical Model */

2023-11-29T04:51:33Z

A Mathematical Model

← Older revision		Revision as of 00:51, 29 November 2023
Line 19:		Line 19:
	:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>		:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</math>

	:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\~~omega~~}}{{dt}^{2}}</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 can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:		Here <math>\boldsymbol{\theta}</math> represents angular position, <math>\boldsymbol{\omega}</math> represents angular velocity, and <math>\boldsymbol{\alpha}</math> represents angular acceleration. One can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:

Aayala35: /* The Main Idea */

2019-11-25T03:02:27Z

The Main Idea

← Older revision		Revision as of 23:02, 24 November 2019
Line 4:		Line 4:

	Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular orientation of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Rotational quantities (also called "angular quantities") describe the angular components of an object's motion.		Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular orientation of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object 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 of rotation for which motion is analyzed. Examples of this include rotating about an object's center of mass (as in a wheel spinning on an axle) and a planet about a star.

			[[File:3200 Phaethon orbit dec 2017.png\|thumb\|Rotational Motion outside an object's center of mass]]

Aayala35: /* History */

2019-11-25T02:49:28Z

History

← Older revision		Revision as of 22:49, 24 November 2019
Line 170:		Line 170:

	== History ==		== History ==
	The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot. Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:		The general concept of rotation was well known to those as far back as ancient Egypt. For example, the Egyptians were aware of the fact that when a force was applied to a round object, such as a log, the object would rotate, or roll across the ground. Archimedes also appeared to be aware of a consequence of torque, which he described as "the principle of the lever". He noted in his work that ''"Magnitudes are in equilibrium at distances reciprocally proportional to their weights"'' when discussing objects balancing about a pivot.

			[[File:Archimedes lever.png\|400px\|thumb\|"Give me a lever long enough, and I shall move the world" - Archimedes]]

			Beyond this point, the first time angular velocity and rotational inertia were discussed by name and given precise definitions were by Thomas Bradwardine and Jean Buridan, respectively. Buridan was notable in that he was also the first to roughly define the notion of angular momentum, as he noted that the celestial bodies maintained rotational motion around the sky, not due to an external force, but because their own inertia (or impedus as he called it) carried them forward, encountering no external resistance. Or, in his own words:

	''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''		''"it could be said that God, in creating the world, set each celestial orb in motion… and, in setting them in motion, he gave them an impetus capable of keeping them in motion without there being any need of his moving them any more."'' - '''''Jean Buridan'''''

Aayala35: /* Similarities to linear motion */

2019-11-25T02:44:38Z

Similarities to linear motion

← Older revision		Revision as of 22:44, 24 November 2019
Line 40:		Line 40:

	It should be fairly apparent why each of these variables corresponds with their respective counterparts. For <math> \boldsymbol{\theta} </math>, the angle that defines an object's orientation is analogous to an object's position, <math> \boldsymbol{x} </math>. Once this relation is recognized, the connection between the other kinematic variables should be obvious.		It should be fairly apparent why each of these variables corresponds with their respective counterparts. For <math> \boldsymbol{\theta} </math>, the angle that defines an object's orientation is analogous to an object's position, <math> \boldsymbol{x} </math>. Once this relation is recognized, the connection between the other kinematic variables should be obvious.


	After seeing the connection between linear and rotational kinematic variables, it can be observed their kinematic equations are virtually identical:		After seeing the connection between linear and rotational kinematic variables, it can be observed their kinematic equations are virtually identical:
Line 56:		Line 55:

	:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math> <math> \Longleftrightarrow \boldsymbol{{x}}^{2} = {{x}_{0}}^{2} + 2\boldsymbol{a}(\boldsymbol{x} - {x}_{0})</math>		:<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math> <math> \Longleftrightarrow \boldsymbol{{x}}^{2} = {{x}_{0}}^{2} + 2\boldsymbol{a}(\boldsymbol{x} - {x}_{0})</math>




	==Examples==		==Examples==

Aayala35 at 02:44, 25 November 2019

2019-11-25T02:44:15Z

← Older revision		Revision as of 22:44, 24 November 2019
Line 180:		Line 180:
	== See also ==		== See also ==

	To learn more about Rotation in a more complete context, please refer to Torque or Rigid-Body Objects or Angular Momentum.		Although the equations in rotational kinematics are useful tools, they become much more powerful when used alongside the angular momentum (as one could guess this topic is analogous linear momentum). Thus, interested students could expand upon their growing knowledge by going through the following readings:

			'''Within this student wiki:'''

			To learn more about Rotation in a more complete context, please refer to [[http://www.physicsbook.gatech.edu/Torque Torque]] or Rigid-Body Objects or [[http://www.physicsbook.gatech.edu/Rotational_Angular_Momentum Angular Momentum]].

	===Further reading===		===Further reading===

	Books, Articles or other print media on this topic		Books, Articles or other print media on this topic

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

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

	===External links===		===External links===

Aayala35 at 02:27, 25 November 2019

2019-11-25T02:27:52Z

← Older revision		Revision as of 22:27, 24 November 2019
Line 147:		Line 147:
	:<math>K{\int}{e}^{-{\beta}t}dt = \boldsymbol{\omega}</math>		:<math>K{\int}{e}^{-{\beta}t}dt = \boldsymbol{\omega}</math>

	This integral can then be solved via u substitution (for more information on solving Integrals, ~~please see "Insert Help page here"~~). However, here we will simply give the solution to the integral:		This integral can then be solved via [https://www.mathsisfun.com/calculus/integration-by-substitution.html u substitution] (for more information on solving Integrals, you can click on the following link: [https://www.mathsisfun.com/calculus/integration-by-substitution.html U sub]). However, here we will simply give the solution to the integral:

	:<math>\omega = \frac{-K}{\beta}{e}^{-{\beta}t}</math>		:<math>\omega = \frac{-K}{\beta}{e}^{-{\beta}t}</math>
Line 171:		Line 171:

	==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~~.		Rotation is an extremely important aspect of dynamics (the study of moving objects). Every object can experience rotational motion, thus it is crucial that we understand the tools to analyze this type of motion. Such an example can be found in biomechanics, where 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.

	== History ==		== History ==

Aayala35 at 02:13, 25 November 2019

2019-11-25T02:13:36Z

← Older revision		Revision as of 22:13, 24 November 2019
Line 65:		Line 65:

	A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?		A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?
	[[File:~~0ee~~.png]]		[[File:AllenEarth.png\|thumb\|Calculate the angular velocity]]


	Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:		Given that we know the amount of time it takes for the Earth to complete one revolution, and one revolution is simply <math>2\pi</math> radians, we can state that:

Line 95:		Line 93:
	A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?		A torque is exerted on a disk initially at rest, causing the disk to undergo constant angular acceleration for 20 s. During this time, the disk completes 20 full rotations. What was the angular acceleration of the disk during this time?

			[[File:AllenDrawing.png\|thumb\|The value of the torque is unknown]]
	To begin, this problem, we can start by recognizing that we are given:		To begin, this problem, we can start by recognizing that we are given:

Aayala35 at 01:52, 25 November 2019

2019-11-25T01:52:10Z

@@ Line 3: / Line 3: @@
 ==The Main Idea==
-Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.
+Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular orientation of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Rotational quantities (also called "angular quantities") describe the angular components of an object's motion.
@@ Line 13: / Line 11: @@
 of differential equations which relate the rate of change of a given quantity to some other aspect of rotational motion. These equations are listed below:
 :<math>\boldsymbol{\omega} = \frac{d\boldsymbol{\theta}}{dt}</math>
-:<math>\boldsymbol{{\alpha}} = \frac{{d}^{2}\boldsymbol{\omega}}{{dt}^{2}}</math>
+:<math>\boldsymbol{{\alpha}} = \frac{d\boldsymbol{\omega}}{dt}</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 can also approximate the rotational motion of an object by using the discrete time equations of motion, as listed below:
@@ Line 33: / Line 30: @@
 :<math>\boldsymbol{{\omega}}^{2} = {{\omega}_{0}}^{2} + 2\boldsymbol{\alpha}(\boldsymbol{\theta} - {\theta}_{0})</math>
 ==Examples==
@@ Line 39: / Line 65: @@
 A simple example and application of the concept of rotation is the earth's rotation on it's axis. Given that the Earth completes one rotation every 24 hours, what is it's angular velocity in radians per second?

Aayala35 at 20:21, 24 November 2019

2019-11-24T20:21:16Z

← Older revision		Revision as of 16:21, 24 November 2019
Line 3:		Line 3:
	==The Main Idea==		==The Main Idea==

	Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.		Rotational Motion (also known as curvilinear motion), in contrast to linear ( also known as rectilinear) motion, involves motion of objects where the angular position of an object changes over time (For this reason, it is common practice to use polar coordinates when analyzing any object undergoing rotational motion). Because of this, rotational quantities are often called angular quantities, as they describe the angular component of an object's motion.

			Since we have already kinematics dealing with linear motion, it is easy to identify the rotational counterpart of previously studied linear variables in the rotational equations that we will observe.