What Does It Mean for a Body to Be in Rotational Equilibrium?

A body is in rotational equilibrium when the resultant torque about any axis is zero. Therefore, its angular acceleration is zero, although it may be stationary or rotating with constant angular velocity.

This is an HL-only concept in IB Physics topic A.4, Rigid body mechanics.

The Mechanism

A torque is the turning effect of a force about an axis. Its magnitude is

$$\tau = Fr\sin\theta$$

where $F$ is the force, $r$ is the distance from the axis to the point where the force acts, and $\theta$ is the angle between the force and the position vector. Equivalently, torque equals force multiplied by the perpendicular distance from the axis to the force's line of action.

For rotational equilibrium,

$$\sum \tau = 0.$$

Using anticlockwise torque as positive, this means

$$\sum \tau_{\text{anticlockwise}}=\sum \tau_{\text{clockwise}}.$$

Torque must be calculated about the same chosen axis for every force. A force whose line of action passes through that axis has zero lever arm and therefore contributes no torque, even if the force is large. Choosing the pivot strategically can eliminate unknown reaction forces from the calculation.

For example, a perpendicular $30\,\text{N}$ force acting $0.40\,\text{m}$ from a pivot produces a torque of

$$\tau=30\times0.40=12\,\text{N m}.$$

To balance it, a perpendicular force acting $0.60\,\text{m}$ from the pivot must satisfy $F(0.60)=12$, giving $F=20\,\text{N}$ in the opposite rotational direction.

A common misconception is that rotational equilibrium means the body must be stationary. This is incorrect: zero resultant torque means constant angular velocity, which may be zero or non-zero. Also, rotational equilibrium alone does not require zero resultant force, so a body could have no angular acceleration while still accelerating linearly.

Exam Technique

For an IB Explain question, state $\sum\tau=0$, identify the axis, and show that clockwise and anticlockwise torques balance. Include perpendicular distances and a consistent sign convention; do not simply state that the forces are equal.