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 m₁ and m₂, connected by an inextensible massless string over an ideal massless pulley.

Both masses experience uniform acceleration. When m₁ = m₂, 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 m₁ accelerates downward and that m₂ accelerates upward, as would be the case if m₁ > m₂

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 (W₁ and W₂). To find an acceleration, consider the forces affecting each individual mass. Using Newton's second law (with a sign convention of $m_1>m_2$) derive a system of equations for the acceleration (Template:Mvar).

As a sign convention, assume that a is positive when downward for $m_1$ and upward for $m_2$. Weight of $m_1$ and $m_2$ is simply $W_1=m_{1}g$ and $W_2=m_{2}g$ respectively.

Forces affecting m₁: $$m_{1}g-T=m_{1}a$$ Forces affecting m₂: $$T-m_{2}g=m_{2}a$$ and adding the two previous equations yields $$m_{1}g-m_{2}g=m_{1}a+m_{2}a,$$ and the concluding formula for acceleration $$a=g\frac{m_1-m_2}{m_1+m_2}$$

The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion.

See also

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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