{{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 George Atwood as a laboratory experiment to verify the mechanical laws of motion with constant acceleration. Atwood's machine is a common classroom demonstration used to illustrate principles of 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 massless string over an ideal massless 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 regardless of the position of the weights.
== Equation for constant acceleration == [[Image:Atwood.svg|right|thumb|220px|The free body diagrams of the two hanging masses of the Atwood machine. Our sign convention, depicted by the acceleration 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 (with a sign convention of {{nowrap|<math>m_1 > m_2</math>)}} derive a system of 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 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
Category:Mechanics Category:Physics experiments