Atwood machine - wikipedia Atwood machine Illustration of 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 m and m, connected by an inextensible massless string over an ideal massless pulley. When m = m, the machine is in neutral equilibrium regardless of the position of the weights. When m ≠ m both masses experience uniform acceleration. Contents 1 Equation for constant acceleration 2 Equation for tension 3 Equations for a pulley with inertia and friction 3.1 Practical implementations 4 See also 5 Notes 6 External links Equation for constant acceleration The free body diagrams of the two hanging masses of the Atwood machine. Our sign convention, depicted by the acceleration vectors is that m accelerates downward and that m accelerates upward, as would be the case if m > m We are able to derive an equation for the acceleration by using force analysis. 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 and W ). To find an acceleration we need to consider the forces affecting each individual mass. Using Newton 's second law ( with a sign convention of m 1 > m 2 ( displaystyle m_ ( 1 ) > m_ ( 2 ) ) ) we can derive a system of equations for the acceleration ( a ). As a sign convention, we assume that a is positive when downward for m 1 ( displaystyle m_ ( 1 ) ) and upward for m 2 ( displaystyle m_ ( 2 ) ). Weight of m 1 ( displaystyle m_ ( 1 ) ) and m 2 ( displaystyle m_ ( 2 ) ) is simply W 1 = m 1 g ( displaystyle W_ ( 1 ) = m_ ( 1 ) g ) and W 2 = m 2 g ( displaystyle W_ ( 2 ) = m_ ( 2 ) g ) respectively. Forces affecting m : m 1 g − T = m 1 a ( displaystyle ; m_ ( 1 ) g-T = m_ ( 1 ) a ) Forces affecting m : T − m 2 g = m 2 a ( displaystyle ; T - m_ ( 2 ) g = m_ ( 2 ) a ) and adding the two previous equations we obtain m 1 g − m 2 g = m 1 a + m 2 a ( displaystyle ; m_ ( 1 ) g - m_ ( 2 ) g = m_ ( 1 ) a + m_ ( 2 ) a ), and our concluding formula for acceleration a = g m 1 − m 2 m 1 + m 2 ( displaystyle a = g ( m_ ( 1 ) - m_ ( 2 ) over m_ ( 1 ) + m_ ( 2 ) ) ) 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 : a = 2 d t 2 ( displaystyle a = ( \frac ( 2d ) ( t ^ ( 2 ) ) ) ). The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion. Equation for tension It can be useful to know an equation for the tension in the string. To evaluate tension, substitute the equation for acceleration in either of the 2 force equations. a = g m 1 − m 2 m 1 + m 2 ( displaystyle a = g ( m_ ( 1 ) - m_ ( 2 ) over m_ ( 1 ) + m_ ( 2 ) ) ) For example, substituting into m 1 a = m 1 g − T ( displaystyle m_ ( 1 ) a = m_ ( 1 ) g-T ), results in T = 2 g m 1 m 2 m 1 + m 2 = 2 g 1 / m 1 + 1 / m 2 ( displaystyle T = ( 2gm_ ( 1 ) m_ ( 2 ) over m_ ( 1 ) + m_ ( 2 ) ) = ( 2g over 1 / m_ ( 1 ) + 1 / m_ ( 2 ) ) ) Equations for a pulley with inertia and friction For very small mass differences between m and m, the rotational inertia I of the pulley of radius r can not be neglected. The angular acceleration of the pulley is given by the no - slip condition : α = a r, ( displaystyle alpha = ( a over r ), ) where α ( displaystyle alpha ) is the angular acceleration. The net torque is then : τ n e t = ( T 1 − T 2 ) r − τ f r i c t i o n = I α ( displaystyle \tau _ ( mathrm ( net ) ) = left ( T_ ( 1 ) - T_ ( 2 ) \right ) r - \tau _ ( mathrm ( friction ) ) = I alpha ) Combining with Newton 's second law for the hanging masses, and solving for T, T, and a, we get : Acceleration : a = g ( m 1 − m 2 ) − τ f r i c t i o n r m 1 + m 2 + I r 2 ( displaystyle a = ( ( g ( m_ ( 1 ) - m_ ( 2 ) ) - ( \tau _ ( mathrm ( friction ) ) over r ) ) over ( m_ ( 1 ) + m_ ( 2 ) + ( ( I ) over ( r ^ ( 2 ) ) ) ) ) ) Tension in string segment nearest m : T 1 = m 1 g ( 2 m 2 + I r 2 + τ f r i c t i o n r g ) m 1 + m 2 + I r 2 ( displaystyle T_ ( 1 ) = ( ( m_ ( 1 ) g ( 2m_ ( 2 ) + ( ( I ) over ( r ^ ( 2 ) ) ) + ( ( \tau _ ( mathrm ( friction ) ) ) over ( rg ) ) ) ) over ( m_ ( 1 ) + m_ ( 2 ) + ( ( I ) over ( r ^ ( 2 ) ) ) ) ) ) Tension in string segment nearest m : T 2 = m 2 g ( 2 m 1 + I r 2 + τ f r i c t i o n r g ) m 1 + m 2 + I r 2 ( displaystyle T_ ( 2 ) = ( ( m_ ( 2 ) g ( 2m_ ( 1 ) + ( ( I ) over ( r ^ ( 2 ) ) ) + ( ( \tau _ ( mathrm ( friction ) ) ) over ( rg ) ) ) ) over ( m_ ( 1 ) + m_ ( 2 ) + ( ( I ) over ( r ^ ( 2 ) ) ) ) ) ) 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 : Acceleration : a = g ( m 1 − m 2 ) m 1 + m 2 + I r 2 ( displaystyle a = ( ( g ( m_ ( 1 ) - m_ ( 2 ) ) ) over ( m_ ( 1 ) + m_ ( 2 ) + ( ( I ) over ( r ^ ( 2 ) ) ) ) ) ) Tension in string segment nearest m : T 1 = m 1 g ( 2 m 2 + I r 2 ) m 1 + m 2 + I r 2 ( displaystyle T_ ( 1 ) = ( ( m_ ( 1 ) g ( 2m_ ( 2 ) + ( ( I ) over ( r ^ ( 2 ) ) ) ) ) over ( m_ ( 1 ) + m_ ( 2 ) + ( ( I ) over ( r ^ ( 2 ) ) ) ) ) ) Tension in string segment nearest m : T 2 = m 2 g ( 2 m 1 + I r 2 ) m 1 + m 2 + I r 2 ( displaystyle T_ ( 2 ) = ( ( m_ ( 2 ) g ( 2m_ ( 1 ) + ( ( I ) over ( r ^ ( 2 ) ) ) ) ) over ( m_ ( 1 ) + m_ ( 2 ) + ( ( I ) over ( r ^ ( 2 ) ) ) ) ) ) Practical implementations Atwood 's original illustrations show the main pulley 's axle resting on the rims of another four wheels, to minimize friction forces from the bearings. Many historical implementations of the machine follow this design. 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, and for the elevators on the Eiffel Tower which counterbalance each other. Ski lifts are another example, where the gondolas move on a closed ( continuous ) pulley system up and down the mountain. The ski lift is similar to the counter-weighted elevator, but with a constraining force provided by the cable in the vertical dimension thereby achieving work in both the horizontal and vertical dimensions. Boat lifts are another type of counter-weighted elevator system approximating an Atwood machine. See also Wikimedia Commons has media related to Atwood 's machine. Frictionless plane Kater 's pendulum Spherical cow Swinging Atwood 's machine Notes Jump up ^ Tipler, Paul A. ( 1991 ). Physics For Scientists and Engineers ( 3rd, extended ed. ). New York : Worth Publishers. p. 160. ISBN 0 - 87901 - 432 - 6. Chapter 6, example 6 - 13 Jump up ^ Goldstein, Herbert ( 1980 ). Classical Mechanics ( 2nd ed. ). New Delhi : Addison - Wesley / Narosa Indian Student Edition. pp. 26 -- 27. ISBN 81 - 85015 - 53 - 8. Section 1 - 6, example 2 Jump up ^ See Wikipedia article : https://en.wikipedia.org/wiki/Boat_lift External links Professor Greenslade 's account on the Atwood Machine Atwood 's Machine by Enrique Zeleny, The Wolfram Demonstrations Project. Retrieved from `` https://en.wikipedia.org/w/index.php?title=Atwood_machine&oldid=792136529 '' Categories : Mechanics Physics experiments Talk Contents About Wikipedia Català Deutsch Español Français 한국어 Hrvatski Italiano Nederlands 日本 語 Polski Português Русский Srpskohrvatski / српскохрватски Türkçe Українська 中文 Edit links This page was last edited on 24 July 2017, at 17 : 58. Text is available under the Creative Commons Attribution - ShareAlike License ; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. About Wikipedia

Question: what is a real world application of an atwood machine
Answer: An elevator with a counterbalance