The physics of pulley weights can be determined using Newton's Second and Third Laws of motion. Newton's Second Law states that the net force acting an object is equal to the product of its acceleration and mass. Newton's Third Law states that every force has an equally strong counterforce in the opposite direction. You can calculate the physics of a pulley system by determining the forces acting on the weights and their directions.

Draw arrows projecting directly downward (or toward the ground) from each of the pulley weights. This represents the force of gravity, and is equal to the mass of the pulley weight times 9.8 (the gravitational constant, or g). Label the forces "m1g" and "m2g."

Calculate the angle between the force arrow for each weight and the continuation of the pulley line (if the pulley line is vertical, this angle is zero; if it's ten degrees off vertical, the angle is ten degrees). To find the component of the force acting against the pulley, calculate the product of the mass, g, and the cosine of the angle. Do this for both weights, and label the resulting force vectors "f1" and "f2."

Draw arrows projecting directly along the pulley rope from each of the pulley weights. This represents the tension force of the rope pulling on the weights. According to Newton's Third Law, the tension force on the weights is equal and opposite to the force of the weights acting on the pulley, but we can't calculate that yet, so for now label the force "T."

Calculate the net force acting on one of the weights: Net Force = f1  T (you subtract because the forces are in opposite directions). By Newton's Second law, the net force is also equal to the mass of the weight times its acceleration. By substitution: m1a = f1  T (a is the acceleration, m is the mass of the object). So T = m1a + f1.

Use the same process to find the net force acting on the other weight: m2*a = f2  T. The tension is the same on both weights because of Newton's Third Law. Every fiber of the rope pulls with the same force with which it is being pulled.

Substitute the equation for Tension from Step 4 into the equation from Step 5: m2a = f2  m1a + f1.

Solve for "a" to find the acceleration of the system. m1a + m2a = f1 + f2; a(m1 + m2) = f1 + f2; a = (f1 + f2) / (m1 + m2). In other words, the acceleration of the system is equal to the sum of the component of the gravitational forces acting against the pulley divided by the sum of the mass of the weights.

Compare f1 and f2. The greater of the two will tell you the direction the system is accelerating (if f1 is bigger, the pulley is accelerating on the first weight's end).
Tips & Warnings
 The above calculations assume frictionless surfaces and massless pulleys, which are common for basic physics problems. If the problem states that there is friction or that the pulley has a mass, the calculations become much more complex.
References
 Photo Credit Jupiterimages/Photos.com/Getty Images