The Atwood machine (or Atwood's machine) was invented in 1784 by Rev. George Atwood as a laboratory experiment to verify the mechanical laws of uniformly accelerated motion. Atwood's machine is a common classroom demonstration used to illustrate principles of physics, specifically mechanics. The ideal Atwood Machine consists of two objects of mass m1 and m2, connected by an inelastic massless string over an ideal massless pulley. [1] When <math>m_1 = m_2</math>, the machine is in neutral equilibrium regardless of the position of the weights. When <math>m_2 \; > \ m_1</math> both masses experience uniform acceleration.
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Equation for uniform acceleration
We are able to derive an equation for the acceleration by using force analysis. If we consider a massless, inelastic string and an ideal massless pulley the only forces we have to consider are: tension force (T), and the weight of the two masses (mg). To find <math>\sum F</math> we need to consider the forces affecting each individual mass. forces affecting m1 : <math>T-m_1g</math> forces affecting m2 : <math>m_2g-T</math> <math>\sum F=(m_2g-T)+(T-m_1g)=g(m_2-m_1)</math> Using Newton's second law we can derive an equation for the system's acceleration. <math>\sum F=ma</math> <math>a={\sum F \over m}</math> <math>\sum F=g(m_2-m_1)</math> <math>m=(m_1+m_2)</math> <math>a = g{m_2-m_1 \over m_1+m_2}</math> 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>. The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion. [2]
Equation for tension
It can be useful to know an equation for the tension in the string. To evaluate tension we substitute the equation for acceleration in either of the 2 force equations. <math>a = g{m_2-m_1 \over m_1+m_2}</math> For example substituting into <math>m_1a = T-m_1g</math>, we get <math>T=g{2m_1m_2\over m_1+m_2}</math> The tension can be found in a similar manner from <math>m_2a = m_2g-T</math>
Equations for a non-ideal pulley
For very small mass differences between m1 and m2, the moment of inertia I of the pulley of radius r cannot be neglected. The angular acceleration of the pulley is given by: <math> \alpha = {a\over r}</math> In that case, the total torque for the system becomes: <math>\tau_{Total}=\left(T_2 - T_1 \right)r = I \alpha + \tau_{friction}</math>
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 car — 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.
See also
Notes
- ^ Tipler, Paul A. (1991). Physics For Scientists and Engineers, Third Edition, Extended Version. New York: Worth Publishers. ISBN 0-87901-432-6. Chapter 6, example 6-13, page 160.
- ^ Goldstein, Herbert (1980). Classical Mechanics, second Edition. New Delhi: Addison-Wesley/Narosa Indian Student Edition. ISBN 81-85015-53-8. Section 1-6, example 2, pages 26-27.
