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Rotational Equilibrium

▶︎  Watch it animatedinteractive step-through · ~3 min · optional

A body is in rotational equilibrium when the net torque about the pivot is zero: Στ = 0. This is Newton's first law for spinning, so a balanced body stays still or keeps turning at a steady rate. Each force adds a torque equal to its size times its lever arm, the perpendicular distance from the pivot to the force, with a sign for which way it turns. Full balance needs two separate conditions: ΣF = 0 and Στ = 0, and one does not give you the other.

Balance is torque, not weight. a body holds still when the net torque about any point is zero: Στ = 0 heavy and near balances light and far d₁ d₂ heavy block, short arm light block, long arm w₁d₁ = w₂d₂ the bar does not turn ΣF = 0 forces balance Στ = 0 torques balance τ = F × d force × lever arm any pivot Στ = 0 everywhere Equal weights balance only at equal arms. A force at the pivot adds no torque. Force balance alone does not stop spin.
Rotational equilibrium means the net torque about the pivot is zero: Στ = 0. A heavy weight on a short arm balances a light weight on a long arm, because balance compares torque (force times lever arm), not weight. Force balance and torque balance are separate conditions; when both hold, the torque sum is zero about every point.
Rotational Equilibrium Builder · Open the sandbox →

Three things go wrong in balance problems. Setting up the torque sum badly — dropping a force's torque, flipping a sign so opposing torques add, or using the wrong distance. Mixing up force balance and torque balance, expecting equal weights to balance no matter the arm. And mishandling the pivot, thinking the answer depends on it or measuring an arm from the wrong spot. Keep the sign, compare torques not weights, and measure every arm from the pivot.

The work

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Lesson
Rotational Equilibrium ›

Builds rotational equilibrium as Στ = 0: torque as force times lever arm, the sign rule, and the two separate conditions ΣF = 0 and Στ = 0. Shows why opposing torques subtract, why equal weights balance only at equal arms, and why any pivot works. Ends with a ten-scenario skill check.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items on the three misconceptions for Topic 5.5: setting up the Στ = 0 torque sum wrong with a missing term or a flipped sign, mixing up force balance and torque balance, and mishandling the pivot. Take it cold to find what is still shaky, or after the lesson to check it is not.

Not started · 10 items · ~15 min
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Targeted Practice
Drill a single misconception ›

Pick one misconception you keep missing and drill it on its own. The round adapts: two correct in a row clears it and you move on.

Take the diagnostic to identify your misconceptions
Diagnostic first