Connected Systems and Atwood Machines

A taut string cannot stretch, so the two ends move together: same speed, same acceleration, always. That one fact is what turns two separate blocks into a single system. You still write F_net = ma for each block on its own, using only the forces touching that block, but you plug in the same a for both. For an Atwood machine, two masses hang over a pulley by a string. Pick a positive direction for each mass consistent with the string (if m1 goes up, m2 goes down), write tension T and weight mg for each, then add the two equations so T cancels: a = (m2 - m1)g / (m1 + m2). The heavier side accelerates downward, the lighter side upward, at the same rate.

Blocks pushed or pulled in a line work the same way but trade tension for a contact force, normal to the touching surfaces. Treat the whole stack as one object to get the shared acceleration from the total force and total mass, then isolate just one block and apply Newton's second law to that block alone to solve for the contact force pushing or pulling on it.

This is the first place rotational dynamics sneaks into a force unit: give the pulley real mass and a radius, and the tensions on the two sides can no longer be equal, because a net torque is needed to spin the pulley up. That difference in tension becomes the torque term in I·alpha = T2·r - T1·r, linking straight-line acceleration a = r·alpha to angular acceleration through the pulley's moment of inertia.