# Atwood machine

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Atwood_machine
> Markdown URL: https://mediated.wiki/source/Atwood_machine.md
> Source: https://en.wikipedia.org/wiki/Atwood_machine
> Source revision: 1303262400
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{short description|Classroom demonstration used to illustrate principles of classical mechanics}}
thumb|153px|right|Illustration of the Atwood machine, 1905.

The '''Atwood machine''' (or '''Atwood's machine''') was invented in 1784 by the English [mathematician](/source/mathematician) [George Atwood](/source/George_Atwood) as a laboratory experiment to verify the [mechanical laws of motion](/source/Newton's_laws_of_motion) with constant [acceleration](/source/acceleration). Atwood's machine is a common classroom demonstration used to illustrate principles of [classical mechanics](/source/classical_mechanics).

The ideal Atwood machine consists of two objects of mass {{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}}, connected by an [inextensible](/source/Kinematics) massless string over an ideal massless [pulley](/source/pulley).<ref><!-- This is a fairly old edition, but it is the one I have. A cite to a newer edition would be better-->{{cite book |last=Tipler |first=Paul A. |year=1991 |title=Physics For Scientists and Engineers |url=https://archive.org/details/physicsforscient00tipl |url-access=registration |edition=3rd, extended |publisher=Worth Publishers |location=New York |isbn=0-87901-432-6 |page=[https://archive.org/details/physicsforscient00tipl/page/160 161]}} Chapter 6, example 6-13</ref>

Both masses experience uniform acceleration. When {{math|1=''m''<sub>1</sub> = ''m''<sub>2</sub>}}, the machine is in [neutral equilibrium](/source/static_equilibrium) regardless of the position of the weights.

== Equation for constant acceleration ==
[[Image:Atwood.svg|right|thumb|220px|The [free body diagram](/source/free_body_diagram)s of the two hanging masses of the Atwood machine. Our [sign convention](/source/sign_convention), depicted by the [acceleration](/source/acceleration) [vectors](/source/Euclidean_vectors) is that {{math|''m''<sub>1</sub>}} accelerates downward and that {{math|''m''<sub>2</sub>}} accelerates upward, as would be the case if {{math|''m''<sub>1</sub> > ''m''<sub>2</sub>}}]]

An equation for the acceleration can be derived by analyzing forces.
Assuming a massless, inextensible string and an ideal massless pulley, the only forces to consider are: tension force ({{mvar|T}}), and the weight of the two masses ({{math|''W''<sub>1</sub>}} and {{math|''W''<sub>2</sub>}}). To find an acceleration, consider the forces affecting each individual mass. 
Using [Newton's second law](/source/Newton's_second_law) (with a [sign convention](/source/sign_convention) of {{nowrap|<math>m_1 > m_2</math>)}} derive a [system of equations](/source/Simultaneous_equations) for the acceleration ({{mvar|a}}).

As a sign convention, assume that ''a'' is positive when downward for <math>m_1</math> and upward for <math>m_2</math>. Weight of <math>m_1</math> and <math>m_2</math> is simply <math>W_1 = m_1 g</math> and <math>W_2 = m_2 g</math> respectively.

Forces affecting m<sub>1</sub>: <math display="block"> m_1 g - T = m_1 a</math>
Forces affecting m<sub>2</sub>: <math display="block"> T - m_2 g = m_2 a</math>
and adding the two previous equations yields <math display="block"> m_1 g - m_2 g = m_1 a + m_2 a,</math>
and the concluding formula for acceleration <math display="block">a = g \frac{m_1 - m_2}{m_1 + m_2}</math>

The Atwood machine is sometimes used to illustrate the [Lagrangian method](/source/Lagrangian_mechanics) of deriving equations of motion.<ref><!-- Again a cite to the most recent edition would be preferable -->{{cite book |last=Goldstein |first=Herbert |authorlink=Herbert Goldstein |year=1980 |title=Classical Mechanics |edition=2nd |publisher=Addison-Wesley/Narosa Indian Student Edition |location=New Delhi |isbn=81-85015-53-8 |pages=26–27}} Section 1-6, example 2</ref>

==See also==
*{{annotated link|Frictionless plane}}
*{{annotated link|Kater's pendulum}}
*{{annotated link|Spherical cow}}
*{{annotated link|Swinging Atwood's machine}}

==Notes==
<references/>

==External links==
{{commons category|Atwood's machine}}
*[https://archive.org/details/b28764821 A treatise on the rectilinear motion and rotation of bodies; with a description of original experiments relative to the subject] by George Atwood, 1764.  Drawings appear on page 450.
*[http://physics.kenyon.edu/EarlyApparatus/Mechanics/Atwoods_Machine/Atwoods_Machine.html Professor Greenslade's account on the Atwood Machine]
*[http://demonstrations.wolfram.com/AtwoodsMachine/ Atwood's Machine] by Enrique Zeleny, [The Wolfram Demonstrations Project](/source/The_Wolfram_Demonstrations_Project)

Category:Mechanics
Category:Physics experiments

---
Adapted from the Wikipedia article [Atwood machine](https://en.wikipedia.org/wiki/Atwood_machine) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Atwood_machine?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.