Atwood machine explained

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 and, connected by an inextensible massless string over an ideal massless pulley.^[1]

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

Equation for constant acceleration

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

As a sign convention, assume that a is positive when downward for

and upward for . Weight of and is simply and 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 $$a\; =\; g\; \backslash frac$$

The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion.^[2]

Notes

1. Book: Tipler, Paul A. . 1991 . Physics For Scientists and Engineers . registration . 3rd, extended . Worth Publishers . New York . 0-87901-432-6 . 160. Chapter 6, example 6-13
2. Book: Goldstein, Herbert . Herbert Goldstein

   . Herbert Goldstein . 1980 . Classical Mechanics . 2nd . Addison-Wesley/Narosa Indian Student Edition . New Delhi . 81-85015-53-8 . 26–27. Section 1-6, example 2

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