1. The problem statement, all variables and given/known data Note that the FAQ is not clear if I am required to use LaTeX for formatting formulas. I have not used LaTeX, but I will retype my question if I am violating the rules. Please let me know :-) Non-extensible cable attached to ceiling. Cable goes down, around a pulley and then back up, then over another pulley and down. At the end of the cable is mass 1. Hanging from the first pulley is mass 2. I am trying to find a formula for the acceleration of M1 as a function of the mass of 1 and 2. 2. Relevant equations I have set g = -9.8m/s^2 (positive acceleration is up, negative is down) I have a free-body diagram for mass 1: Total force on mass 1: f1 = f1(up) + f1(down) f1(up) is the force imparted by the cable pulling up f1(down) = m1 * g And for mass 2: Total force on mass 2: f2 = 2 * f2(up) + f2(down) f2(up) is the force imparted by each of the two cables lifting on the pulley above mass 2. f2(down) = m2 * g I set also f2(up) = f1(up) since the same cable that holds mass 1 also goes to the pulley for mass 2. This is my first question - is this assumption of the forces upward correct? I also say that because of the pulley arrangement the acceleration of mass 1 (a1) = 2 * the acceleration of mass 2 (a2). I say this because the velocity of mass 1 is double the velocity of mass 2 (due to the pulley arrangement). So we have: a1 = 2 * a2 f2(up) = f1(up) a1 = (f1(up) + f1(down)) / m1 a2 = (2 * f2(up) + f2(down)) / m2 Is this correct so far? 3. The attempt at a solution Then I try to solve for a1 in terms of m1 and m2. First, solve for f1(up) in terms of a1: a2 = (2 * f2(up) + f2(down)) / m2 a2 * m2 = 2 * f2(up) + f2(down) a2 * m2 = 2 * f1(up) + m2 * g (a2 * m2) - (m2 * g) = 2 * f1(up) ((a2 * m2) - (m2 * g)) / 2 = f1(up) (((a1 / 2) * m2) - (m2 * g)) / 2 = f1(up) Now plug this in to the original a1 equation: a1 = (f1(up) + f1(down)) / m1 a1 = ((((a1 / 2) * m2) - (m2 * g)) / 2 + f1(down)) / m1 a1 = ((((a1 / 2) * m2) - (m2 * g)) / 2 + m1 * g) / m1 a1 * m1 = (((a1 / 2) * m2) - (m2 * g)) / 2 + m1 * g 2 * a1 * m1 = (a1 / 2) * m2 - (m2 * g) + 2 * m1 * g 2 * a1 * m1 - (a1 / 2) * m2 = 2 * m1 * g - (m2 * g) a1 * (2 * m1 - m2 / 2) = g * (2 * m1 - m2) a1 = (g * (2 * m1 - m2)) / (2 * m1 - m2 / 2) Now, as a sanity check, I try some examples: Example 1: m1 = 1 kg m2 = 2 kg (I would expect this to show no acceleration) a1 = (-9.8 (2 * 1 - 2)) / (2 * 1 - 2 / 2) = 0 good! Example 2: m1 = 1.1 kg m2 = 2 kg (I would expect this to show a slow negative acceleration as m1 falls to the floor) (-9.8 (2 * 1.1 - 2)) / (2 * 1.1 - 2 / 2) = -1.96 / 1.2 = -1.63 m/s^2 good! Example 3: m1 = 0.9 kg m2 = 2 kg (I would expect this to show a slow positive acceleration as m1 rises to the ceiling) (-9.8 (2 * 0.9 - 2)) / (2 * 0.9 - 2 / 2) = 1.96 / 0.8 = 2.45 m/s^2 good! Example 4: m1 = 0.1 kg m2 = 2 kg (I would expect this to show a fast positive acceleration as m1 rises quickly to the ceiling) (-9.8 (2 * 0.1 - 2)) / (2 * 0.1 - 2 / 2) = 17.64 / -0.8 = -22.05 Huh? I have been through this several (many!) times and cannot figure out what I'm doing wrong. Can anyone offer some insight? Additionally, with m1 = 0.5 kg the equation produces zero in the denominator, which is probably not right :-/ Thanks very much in advance!