5.2 Rotational Equilibrium

Condition for rotational equilibrium

An object is in rotational equilibrium when the net torque acting on it about any axis is zero: Στ = 0. This means the object's angular velocity is constant (including zero, i.e., not rotating). Combined with ΣF = 0 (translational equilibrium), the object is said to be in static equilibrium if it is also at rest.

Choosing a pivot point

Since Στ = 0 must hold about *any* axis when in equilibrium, you can strategically choose a pivot point to eliminate unknown forces from the torque equation. Placing the pivot where an unknown force acts makes its torque zero (since r = 0), simplifying the algebra needed to solve for other unknowns like tension or normal force.

Torque sign convention & setup

Torque is τ = r × F = rF sin θ, where θ is the angle between the position vector and force. Assign counterclockwise torques as positive and clockwise as negative (or vice versa, consistently). Sum all torques, set equal to zero, and solve. Remember weight acts at the center of mass for torque calculations.

Key terms

Rotational equilibrium condition

Στ = 0 about any chosen axis; angular velocity remains constant.

Static equilibrium

Both ΣF = 0 and Στ = 0, and the object is at rest (not translating or rotating).

Strategic pivot choice

Placing the pivot at an unknown force's location sets its torque to zero (r = 0), simplifying the equation to solve for other unknowns.

Torque sign convention

Counterclockwise torques are typically positive, clockwise negative (or vice versa, as long as consistent).

Where does weight act for torque?

At the object's center of mass, since gravity effectively acts there for torque calculations.

Try an AP-style question

A uniform 4 m, 20 kg beam is supported at both ends. What force does each support exert? (g ≈ 10 m/s²)

1. 50 N
2. 200 N
3. 100 N
4. 400 N