Connected Systems, Tension and Constraint Equations

Two bodies, two equations, one constraint

An ideal string is inextensible and massless; an ideal pulley is massless and frictionless. Under those assumptions the tension has the same magnitude everywhere along one continuous string, and a pulley only changes the direction of that tension. (Once the pulley has rotational inertia — Unit 5 — the tensions on the two sides differ, and that is precisely what makes the pulley rotate.)

The method that never fails: write a separate $\sum F=ma$ for each body, taking the positive direction along that body's own motion, then close the system with a constraint equation obtained from the geometry of the string.

For a simple Atwood machine the constraint is $a_1=a_2$ in magnitude. For a movable pulley the string length $\ell$ is constant, so differentiating $\ell=\text{const}$ twice with respect to time gives the acceleration relation directly — typically $a_{\text{free end}}=2a_{\text{block}}$.

Tension is never "equal to the weight" as a rule. $T=mg$ only in the special case of zero acceleration; when the system accelerates, $T$ lies strictly between the two weights of an Atwood machine, and solving for it is a separate scoring point on the rubric.