Rotational Equilibrium

Second condition of equilibrium: vanishing net external torque about any pivot point.

General idea of the topic

Rotational equilibrium requires the sum of the moments of the forces about an axis to be zero. For complete static equilibrium, in addition, the sum of forces must be zero. The moment of a force measures its capacity to make a body rotate: it depends on the modulus of the force, on the distance from the axis to its line of action and on the angle at which it is applied.

The topic defines the moment, states the condition for rotational equilibrium and applies it to levers, where a small force with a long arm balances a large force with a short arm. It is the principle of scales, scissors, wheelbarrows and hand tools.

What the student is expected to understand

By the end, the student should be able to calculate the moment of a force about an axis, its sign according to the sense of rotation, and to set up the sum of moments to obtain unknown forces.

The student should also be able to apply the moment condition to levers and to calculate the force that balances a load.

Recommended learning route

1. Moment of a Force: moment as force times arm, with its sign and direction, and how it is calculated about an axis.
2. Net Torque Balance Condition: the condition of a zero sum of moments about an axis, how to choose it and its application to levers.

Essential physical criterion

The essential criterion is to measure the arm as the perpendicular distance from the axis to the line of action of the force, not to the point of application, and to keep a single sign convention. Choosing the axis where the unknown forces act eliminates unknowns.

Connection with real problems

• levers, scales, wheelbarrows and scissors.
• spanners and tools for tightening screws.
• doors, cranes and articulated arms.
• tipping stability of vehicles and structures.

Final goal of the topic

The final goal is to set up and solve the rotational equilibrium of any rigid body, to calculate moments with the correct arm and to analyse the equilibrium of a lever.