Constraint equations need signed directions
The mistake
Using equal acceleration magnitudes where the constraint requires opposite signed accelerations.
Why it feels right
Connected objects often share a single acceleration magnitude, so signs feel like decoration.
How to catch it
Your string equation says a₁ = a₂ even when the objects move in opposite coordinate directions.
The first principle
Pick coordinates first, then write the length constraint. Different coordinate choices can produce a₁ = a₂, a₁ = −a₂, or 2a₁ = a₂.
Two masses hang over a pulley, with upward positive for both masses. If the left mass rises, what relation holds?
Common wrong move
a_left = a_right.
Correct
a_left = -a_right because the right mass accelerates downward in its upward-positive axis.
Make a fresh prediction before referring to the worked example.