Rotational Equilibrium
Two equilibrium conditions, not one. $\sum F = 0$ keeps the center of mass from accelerating. $\sum \tau = 0$ keeps the body from spinning up. They're independent: a body can satisfy one without the other. A couple, two equal-and-opposite forces at offset points, is the cleanest case. Net force cancels, but each force twists the body the same rotational way, so the center holds still and the body spins.
Two traps. First: reading $\sum F = 0$ as if it implied $\sum \tau = 0$. It doesn't. The two equilibria are independent and one can hold while the other fails. Second: letting force magnitude stand in for torque. A small force far from the pivot can outproduce a much larger force close in, because torque is force times lever arm.
The work
3 ways in · any order
Lesson
Rotational Equilibrium
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Build $\sum \tau = 0$ as a separate condition from $\sum F = 0$. Worked examples on balanced beams, plank reactions, and a couple. Closes with a ten-scenario applet targeting the rotational-vs-translational trap and the force-vs-torque trap.
Diagnostic
10-item topic check
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Ten multiple-choice scenarios: independence of the two equilibria, signed torques, lever-arm geometry, and the special case of a couple. Each wrong answer maps to a named misconception. The report at the end tells you which one tripped you up.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the misconception and you move on.