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| [[Image:Atwoods machine.png|thumb|150px|right|Illustration of Atwood machine, 1905.]]
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| The '''Atwood machine''' (or '''Atwood's machine''') was invented in 1784 by Rev. [[George Atwood]] as a laboratory experiment to verify the [[Newton's laws of motion|mechanical laws of motion]] with constant [[acceleration]]. Atwood's machine is a common classroom demonstration used to illustrate principles of [[classical mechanics]].
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| The ideal Atwood Machine consists of two objects of mass ''m''<sub>1</sub> and ''m''<sub>2</sub>, connected by an [[Kinematics#Inextensible_cord|inextensible]] massless string over an ideal massless [[pulley]].
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| <ref><!-- This is a fairly old edition, but it is the one I have. A cite to a newer edition would be better-->
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| {{cite book | last = Tipler | first = Paul A.
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| | title = Physics For Scientists and Engineers, Third Edition, Extended Version | publisher = Worth Publishers | year = 1991 | location = New York
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| | isbn = 0-87901-432-6}} Chapter 6, example 6-13, page 160.
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| </ref>
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| When m<sub>1</sub> = m<sub>2</sub>, the machine is in [[static equilibrium|neutral equilibrium]] regardless of the position of the weights.
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| When m<sub>1</sub> ≠ m<sub>2</sub> both masses experience uniform acceleration.
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| == Equation for constant acceleration ==
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| [[Image:Atwood.svg|right|thumb|220px|The [[free body diagram]]s of the two hanging masses of the Atwood machine. Our [[sign convention]], depicted by the [[acceleration]] [[Euclidean vectors|vectors]] is that ''m<sub>1</sub>'' accelerates downward and that ''m<sub>2</sub>'' accelerates upward, as would be the case if ''m<sub>1</sub>'' > ''m<sub>2</sub>'']]
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| We are able to derive an equation for the acceleration by using force analysis.
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| If we consider a massless, inextensible string and an ideal massless pulley, the only forces we have to consider are: tension force (''T''), and the weight of the two masses (''W<sub>1</sub>'' and ''W<sub>2</sub>''). To find an acceleration we need to consider the forces affecting each individual mass.
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| Using [[Newton's second law]] (with a [[sign convention]] of <math>m_1>m_2</math>) we can derive a [[Simultaneous equations|system of equations]] for the acceleration (''a'').
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| As a sign convention, we assume that ''a'' is positive when downward for <math>m_1</math>, and that ''a'' is positive when 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.
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| Forces affecting m<sub>1</sub>:
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| <math>\; m_1g-T=m_1a</math>
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| Forces affecting m<sub>2</sub>:
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| <math>\; T-m_2g=m_2a</math>
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| and adding the two previous equations we obtain
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| <math>\; m_1g-m_2g=m_1a+m_2a</math>,
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| and our concluding formula for acceleration
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| <math>a = g{m_1-m_2 \over m_1+m_2}</math>
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| Conversely, the acceleration due to gravity, ''g'', can be found by timing the movement of the weights, and calculating a value for the uniform acceleration ''a'': <math> d = {1 \over 2} at^2 </math>.
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| The Atwood machine is sometimes used to illustrate the [[Lagrangian mechanics|
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| Lagrangian method]] of deriving equations of motion.
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| <ref><!-- Again a cite to the most recent edition would be preferable -->
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| {{cite book | last = Goldstein | first = Herbert | authorlink = Herbert Goldstein
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| | title = Classical Mechanics, second Edition
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| | publisher = Addison-Wesley/Narosa Indian Student Edition | year = 1980
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| | location = New Delhi | isbn = 81-85015-53-8}}
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| Section 1-6, example 2, pages 26-27.</ref>
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| == Equation for tension ==
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| It can be useful to know an equation for the [[tension (physics)|tension]] in the string. To evaluate tension we substitute the equation for acceleration in either of the 2 force equations.
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| <math>a = g{m_1-m_2 \over m_1 + m_2}</math>
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| For example substituting into <math>m_1 a = m_1 g-T</math>, we get
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| <math>T={2 g m_1 m_2 \over m_1 + m_2}</math>
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| == Equations for a pulley with inertia and friction ==
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| For very small mass differences between ''m''<sub>1</sub> and ''m''<sub>2</sub>, the [[moment of inertia|rotational inertia]] ''I'' of the pulley of radius r cannot be neglected. The angular acceleration of the pulley is given by the no-slip condition:
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| <math> \alpha = {a\over r},</math>
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| where <math> \alpha</math> is the angular acceleration. The net [[torque]] is then:
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| <math>\tau_{net}=\left(T_1 - T_2 \right)r - \tau_{friction} = I \alpha </math>
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| Combining with Newton's second law for the hanging masses, and solving for ''T<sub>1</sub>'', ''T<sub>2</sub>'', and ''a'', we get:
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| Acceleration:
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| :<math> a = {{g (m_1 - m_2) - {\tau_{friction} \over r}} \over {m_1 + m_2 + {{I} \over {r^2}}}}</math>
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| Tension in string segment nearest ''m<sub>1</sub>'':
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| :<math> T_1 = {{m_1 g (2 m_2 + {{I} \over {r^2}} + {{\tau_{friction}} \over {r g}})} \over {m_1 + m_2 + {{I} \over {r^2}}}}</math>
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| Tension in string segment nearest ''m<sub>2</sub>'':
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| :<math> T_2 = {{m_2 g (2 m_1 + {{I} \over {r^2}} + {{\tau_{friction}} \over {r g}})} \over {m_1 + m_2 + {{I} \over {r^2}}}}</math>
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| Should bearing friction be negligible (but not the inertia of the pulley and not the traction of the string on the pulley rim), these equations simplify as the following results:
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| Acceleration:
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| :<math> a = {{g (m_1 - m_2)} \over {m_1 + m_2 + {{I} \over {r^2}}}}</math>
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| Tension in string segment nearest ''m<sub>1</sub>'':
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| :<math> T_1 = {{m_1 g (2 m_2 + {{I} \over {r^2}})} \over {m_1 + m_2 + {{I} \over {r^2}}}}</math>
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| Tension in string segment nearest ''m<sub>2</sub>'':
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| :<math> T_2 = {{m_2 g (2 m_1 + {{I} \over {r^2}})} \over {m_1 + m_2 + {{I} \over {r^2}}}}</math>
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| === Practical implementations ===
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| Atwood's original illustrations show the main pulley's axle resting on the rims of another four wheels, to minimize friction forces from the [[rolling-element bearing|bearings]]. Many historical implementations of the machine follow this design.
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| An elevator with a counterbalance approximates an ideal Atwood machine and thereby relieves the driving motor from the load of holding the elevator cab — it has to overcome only weight difference and inertia of the two masses. The same principle is used for [[funicular]] railways with two connected railway cars on inclined tracks.
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| == See also ==
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| {{commons category|Atwood's machine}}
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| *[[Frictionless plane]]
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| *[[Kater's pendulum]]
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| *[[Spherical cow]]
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| *[[Swinging Atwood's machine]]
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| ==Notes==
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| <references/>
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| ==External links==
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| *[http://physics.kenyon.edu/EarlyApparatus/Mechanics/Atwoods_Machine/Atwoods_Machine.html Professor Greenslade's account on the Atwood Machine]
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| *"[http://demonstrations.wolfram.com/AtwoodsMachine/ Atwood's Machine]" by Enrique Zeleny, [[The Wolfram Demonstrations Project]].
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| [[Category:Mechanics]]
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| [[Category:Physics experiments]]
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