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
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 (T), 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 {\displaystyle m_{1}>m_{2}} ) derive a system of equations for the acceleration (a). As a sign convention, 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 m1: m 1 g − T = m 1 a {\displaystyle m_{1}g-T=m_{1}a}
Forces affecting m2: T − m 2 g = m 2 a {\displaystyle 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 , {\displaystyle m_{1}g-m_{2}g=m_{1}a+m_{2}a,}
and the concluding formula for acceleration a = g m 1 − m 2 m 1 + m 2 {\displaystyle 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 Frictionless plane Kater's pendulum – Reversible free swinging pendulum Spherical cow – Humorous concept in scientific models Swinging Atwood's machine – Variation of Atwood's machine incorporating a pendulum
Notes
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