AP® Physics 1: Algebra-Based · Unit 5 · Topic 5.5
Static Equilibrium of a Loaded Beam
A beam resting on two supports is in static equilibrium when the net force and net torque on it are both zero. Picking the right pivot point lets you solve for each support force separately.
A beam sitting still isn't just balanced in force, it's balanced in rotation too. Every weight on the beam, including the beam's own weight acting at its center, creates a torque about any point you choose. For the beam to stay put, those torques must cancel, and the two upward support forces plus gravity must also sum to zero. That's two conditions, one for forces and one for torques, and together they let you solve for two unknown support forces.
The trick is choosing your pivot smartly. Torque depends on force times distance from the pivot, so if you place the pivot right at one support, that support's force has zero lever arm and vanishes from the torque equation. That leaves one equation with one unknown, the other support force, which you solve directly. Then plug that answer into the vertical force equation, F_left + F_right = total weight, to get the second support force. Pivoting at the opposite support for a second calculation is a useful check.
This is Units 1 and 2 catching up to rotation: force balance is the old sum-of-forces-equals-zero idea, and torque balance is its rotational twin, using rotational inertia's cousin concept, the lever arm, instead of mass. As a load slides along the beam, the torque equation shows exactly how the weight shifts from one support to the other, which is why a beam tips the instant a center of mass crosses past the last support: there's no force left on that side to generate a counter-torque.
Key terms
- Static equilibrium
- A state where an object has zero net force and zero net torque, so it neither accelerates nor starts rotating.
- Torque
- A twisting effect of a force, equal to force times the perpendicular distance from the pivot to the force's line of action.
- Lever arm
- The perpendicular distance from the pivot point to the line along which a force acts.
- Pivot point
- The reference point chosen for summing torques; any point works, but a smart choice eliminates an unknown force.
- Normal (support) force
- The upward force a support exerts on the beam, found using the force and torque equations together.
Watch it explained
- How it worksWalk me through choosing a clever pivot point that makes an unknown force vanish from the equation.
- Worked exampleFind the two support forces on a 4 m uniform 30 kg beam with a 50 kg load 1 m from the left end.
- How it worksHow do the force equations and the torque equation get used together on the same beam?
- Going furtherWhat happens to the far support force as the load is walked toward the near end?
Check yourself
End of topic.