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Equilibrium of concurrent forces
Intermediate
Lami's theorem links three forces in equilibrium.
F₁/sinα = F₂/sinβ = F₃/sinγThree forces in equilibrium on a point form a closed triangle, so Lami's theorem applies: F₁/sinα = F₂/sinβ = F₃/sinγ, with each angle opposite its force.
A 100 N weight hanging from two ropes at 45° each: each tension ≈ 70.7 N.
Lami's angles are between the other two forces, opposite the one in the numerator.
In more detail
Concurrent forces all act through the same point. An object is in equilibrium when the vector sum of these forces is zero, which can be checked by resolving forces into components.
Real-world uses
- Rigging and lifting engineers check that all cables balance at a hook.
- Structures such as trusses are analysed using concurrent force equilibrium.
- A sign hung from two cables at different angles has three forces in balance, which can be solved with Lami's theorem.
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Frequently asked questions
How do I check equilibrium with three forces?
Use a closed triangle of forces, or resolve each force into components and check both sums are zero.
Do forces need to be at right angles?
No. Components allow any angles to be handled.
When is Lami's theorem useful?
When three forces are in equilibrium and you know the angles between them, it gives the forces without resolving components.
Related topics in Newton's Laws & Forces
Spring force · Springs in series and parallel · Hooke's law limits · Torque and equilibrium · Center of mass · Motion of center of mass