Connected body problems require relating the motions of multiple bodies through constraint equations.

String Constraint (Inextensible): Total string length L = constant. Differentiate once for velocity constraint, twice for acceleration constraint. For a simple Atwood machine: $a_1$ = -$a_2$.

Movable Pulley: If a pulley is attached to a block with n string segments supporting it, then: $v_{block}$ = v_{free}_{end} / n and $a_{block}$ = a_{free}_{end} / n. For a single movable pulley with 2 segments: $a_{block}$ = a/2.

Problem-Solving Strategy:

1. Identify all bodies and draw separate FBDs
2. Write F = ma for each body along appropriate axes
3. Write constraint equations relating accelerations
4. Solve simultaneous equations

Shortcut (System Approach): If all bodies have the same acceleration (e.g., blocks on a smooth surface pushed together):

• a = F_{external}_{net} / $M_{total}$
• Then use individual FBDs to find internal forces (tension, contact forces, friction)

This shortcut saves significant time in JEE but only works when all parts move with the same acceleration.

Two-Block Test: To check if blocks move together: assume they do, find required friction, compare with mu*N. If required <= max available, they move together. Otherwise, treat separately with kinetic friction.