Run a string over a pulley and hang a weight from each end. The heavier one goes down and the lighter one comes up. Before writing any equations, we settle how the two are tied to each other.
The string does not stretch. So one side rises by exactly as much as the other side falls. The distances moved are equal, so the speeds are equal and the sizes of the accelerations are equal too. The directions are opposite, but the size is a single value, which we write as .
Take the pulley to be smooth and massless. Then no force is spent on turning it, and the pull of the string, the tension, has the same size everywhere along the string. The force lifting the left weight and the force lifting the right weight are the same .
One acceleration, one tension. These two are the ground on which the equations stand. A problem about two bodies has become a problem in two unknowns, and .
The equation of motion is written for one body at a time. We do not lump the two together into a single equation, because the forces on them are different.
Start with the weight that descends. Two forces act on it, gravity downward and the tension upward. Taking the direction of motion as positive gives .
The rising weight is handled the same way. Gravity acts downward and the tension upward, and this one moves up. Taking upward as positive gives .
Two unknowns, and , and now two equations. That is enough to solve. Two bodies joined by a string always come out in this form.
Add the two equations as they stand, left sides together and right sides together.
The tension is in one and in the other, so it vanishes the moment they are added. What is left is .
It is no accident that disappeared. The tension is the force the two weights exert on each other through the string, and within the system of the two together it cancels internally. Seen from outside, it may as well never have existed.
All that remains is to divide, which gives .
Read the result again. The numerator is the difference of the two weights; the denominator is the sum of the two masses.
What drives the motion is the difference in weight; what is being driven is the two masses together. In other words the equation is nothing but for the pair, taken as a single body.
So if we regard them as one body from the start, a single equation will do. The tension is internal and need not be written, and the only external forces are the two weights. Divide the difference by the sum, and that is all.
The figure varies the weights being hung. When the two are equal the numerator is and nothing moves. However heavy one side is made, the acceleration only approaches and never passes it. The lighter weight is holding back the one that wants to fall, and the equation shows it plainly.