Atwood machine

File:Atwoods machine.png

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 Template:Math and Template:Math, connected by an inextensible massless string over an ideal massless pulley.<ref>Template:Cite book Chapter 6, example 6-13</ref>

Both masses experience uniform acceleration. When Template:Math, the machine is in neutral equilibrium regardless of the position of the weights.

Equation for constant acceleration

File:Atwood.svg

The free body diagrams of the two hanging masses of the Atwood machine. Our sign convention, depicted by the acceleration vectors is that Template:Math accelerates downward and that Template:Math accelerates upward, as would be the case if Template:Math

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 (Template:Mvar), and the weight of the two masses (Template:Math and Template:Math). To find an acceleration, consider the forces affecting each individual mass. Using Newton's second law (with a sign convention of Template:Nowrap derive a system of equations for the acceleration (Template:Mvar).

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 m₁: <math display="block"> m_1 g - T = m_1 a</math> Forces affecting m₂: <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>Template:Cite book Section 1-6, example 2</ref>

See also

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Notes

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External links

• 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.
• Professor Greenslade's account on the Atwood Machine
• Atwood's Machine by Enrique Zeleny, The Wolfram Demonstrations Project