Atwood machine problems involve two weights connected by a string hung on opposite sides of a pulley. For the sake of simplicity, the string and pulley are assumed to be massless and frictionless, therefore reducing the problem to an exercise in Newton's laws of physics. Solving the Atwood machine problem requires that you calculate the acceleration of the system of weights. This is achieved using Newton's 2nd law: Force equals mass times acceleration. The difficulty of Atwood machine problems lies in determining the tension force on the string.
Label the lighter of the two weights "1" and the heavier "2."
Draw arrows emanating from the weights representing the forces acting on them. Both weights have a tension force "T" pulling up, as well as the gravitational force pulling down. The force of gravity is equal to the mass (labeled "m1" for weight 1 and "m2" for weight 2) of the weight times "g" (equal to 9.8). Therefore, the gravitational force on the lighter weight is m1g, and the force on the heavier weight is m2g.
Calculate the net force acting on the lighter weight. The net force is equal to the tension force minus the gravitational force, since they pull in opposite directions. In other words, Net force = Tension force - m1*g.
Calculate the net force acting on the heavier weight. The net force is equal to the gravitational force minus the tension force, so Net force = m2*g - Tension force. On this side, Tension is subtracted from mass times gravity rather than the other way around because the direction of tension is opposite on opposite sides of the pulley. This makes sense if you consider the weights and string laid out horizontally -- the tension pulls in opposite directions.
Substitute (tension force - m1g) in for the net force in the equation net force = m1acceleration (Newton's 2nd law states that Force = mass * acceleration; acceleration will be labeled "a" from here on). Tension force - m1g = m1a, or Tension = m1g + m1a.
Substitute the equation for tension from Step 5 into the equation from Step 4. Net force = m2g - (m1g + m1a). By Newton's 2nd law, Net Force = m2a. By substitution, m2a = m2g - (m1g + m1a).
Find the acceleration of the system by solving for a: a(m1 + m2) = (m2 - m1)g, so a = ((m2 - m1)*g) / (m1 + m2). In other words, the acceleration is equal to 9.8 times the difference of the two masses, divided by the sum of the two masses.