Rigid body equilibrium describes objects that do not translate or rotate because all external forces and torques balance. It matters whenever engineers design beams, bridges, shelves, ladders, cranes, and supports that must hold steady under load. Unlike a particle, a rigid body can rotate, so both force balance and torque balance are required.
The main goal is to turn a physical drawing into equations that predict unknown support forces, weights, or friction forces.
Key Facts
- Static equilibrium requires ΣFx = 0, ΣFy = 0, and Στ = 0.
- Torque magnitude is τ = rF sin θ, where r is distance from the pivot to the force application point.
- For a force perpendicular to a beam, torque is τ = Fd, where d is the lever arm.
- A force acting through the chosen pivot produces zero torque about that pivot.
- Clockwise and counterclockwise torques must be assigned opposite signs consistently.
- For a uniform beam, its weight acts at its center of mass, usually the geometric center.
Vocabulary
- Rigid body
- A rigid body is an object whose shape and size are assumed not to change when forces act on it.
- Static equilibrium
- Static equilibrium is the condition in which an object remains at rest because its net force and net torque are both zero.
- Torque
- Torque is the rotational effect of a force about a pivot or axis.
- Lever arm
- The lever arm is the perpendicular distance from the pivot to the line of action of a force.
- Center of mass
- The center of mass is the point where an object's weight can be treated as acting for equilibrium calculations.
Common Mistakes to Avoid
- Using only ΣF = 0 and ignoring Στ = 0 is wrong because a rigid body can have balanced forces but still rotate.
- Choosing a pivot at a random point without purpose makes the algebra harder because forces with unknown values may create unnecessary torques.
- Using the full distance instead of the perpendicular lever arm is wrong because torque depends on the shortest distance from the pivot to the force's line of action.
- Forgetting the beam's own weight gives an incomplete force diagram because the weight often creates a major torque at the center of mass.
Practice Questions
- 1 A 6.0 m uniform beam of weight 200 N is supported at both ends. A 300 N crate is placed 2.0 m from the left end. Find the upward support forces at the left and right ends.
- 2 A 4.0 m horizontal sign of weight 120 N is hinged to a wall at one end and supported by a vertical cable at the far end. A 60 N lamp hangs 3.0 m from the hinge. Find the cable tension and the hinge's vertical force.
- 3 A ladder leans against a smooth wall while its base rests on a rough floor. Explain why friction at the floor is necessary for equilibrium and identify which forces create clockwise and counterclockwise torques about the base.