Lami's Theorem: Which Angle Goes Under Which Force

Lami's theorem says that if three coplanar, concurrent forces keep a point in equilibrium, each force divided by the sine of the angle between the other two forces is the same number. Write it as F₁/sin α = F₂/sin β = F₃/sin γ. Here α sits between F₂ and F₃, β between F₁ and F₃, and γ between F₁ and F₂.

The three angles between the pairs add up to 360° (Wikipedia: Lami's theorem). That sum is your built-in error check.

Lami's theorem is the sine rule (a/sin A = b/sin B = c/sin C in any triangle) applied to the closed triangle of three balanced forces. Forces that cancel can be drawn tip to tail as a closed triangle. The inside angles of that triangle are 180° minus the angles between the forces, and sin(180° − θ) = sin θ. So the angle opposite a side in the force triangle and the angle between the other two forces at the point have the same sine (Wikipedia: Lami's theorem). Books that say "opposite angle" and books that say "angle between the other two" are saying the same thing.

You can use Lami's theorem only when four tests all pass: exactly three forces, all meeting at one point, all in one plane, and the body in equilibrium. If any test fails, resolve the forces into horizontal and vertical components instead. Draw the free body diagram first (a free body diagram is a sketch of the body showing every force on it as an arrow; see our free body diagram routine), because every test below is run on it.

One more trap: no two forces may lie along the same line. Then the angle between them is 0° or 180°, its sine is zero, and the ratio breaks down.

A wrong pairing is not a small slip. In the problem below, all six wrong tensions from three wrong pairings are off by 22.5% to 41.4%. We worked them out ourselves: a 10 kg lamp (weight 98 N, taking g = 9.8 m/s²) hangs from two strings at 30° and 45° to the vertical.

Correct solution. Call the 30° string T₁ and the 45° string T₂.

• Angle between T₁ and T₂: 30° + 45° = 75°.
• Angle between T₁ and the weight W: 180° − 30° = 150°.
• Angle between T₂ and W: 180° − 45° = 135°.
• These add up to 360°.

Each force goes over the sine of the angle between the other two:

T₁ / sin 135° = T₂ / sin 150° = 98 / sin 75° = 101.46

So T₁ = 101.46 × 0.7071 = 71.74 N and T₂ = 101.46 × 0.5 = 50.73 N.

Back-check by components. Horizontal: T₁ sin 30° = 35.87 N and T₂ sin 45° = 35.87 N, so they cancel. Vertical: T₁ cos 30° + T₂ cos 45° = 62.13 + 35.87 = 98.00 N, which equals W. The string closer to the vertical carries more load, as it should.

The three wrong pairings. The first uses 98 ÷ sin 90° = 98 as its base. The other two use the 101.46 base from the correct solution with a wrong angle.

Errors are measured against the correct 71.74 N and 50.73 N. Using g = 9.81 instead of 9.8 raises both correct tensions by about 0.1%, to 71.8 N and 50.8 N. Both are right, so use the g your question gives.

Write it in five steps, each with a pass or fail you can check on paper. This is the routine to copy into an exam.

Our Engineering Mechanics study plan shows where this routine fits in the wider subject.

Resolve into components when you are unsure, and use Lami's theorem when exactly three forces pass all four tests and the angles are given. Components work on every equilibrium problem, including four-force and rough-surface ones. Lami's theorem works on fewer problems but gives each unknown in one line.

Our recommendation: if the question says "using Lami's theorem", use it and show the angle sum. If it does not, use whichever you can finish without an arithmetic slip, and use the other as your back-check. Components cost two equations, but they have no usage conditions beyond equilibrium.

Try these, then check the answers. All take g = 9.8 m/s².

1. A 100 N weight hangs from two strings making 30° and 60° with the ceiling. Find both tensions.
2. A 6 kg board hangs from two equal strings, each at 45° to the ceiling. Find the tension in each.
3. A 20 kg crate is pulled across a rough floor by two ropes and stays in equilibrium. Can you use Lami's theorem?

Answers.

1. The angle between the strings is 90°, so T = 100 × sin 150° / sin 90° = 50 N for the string at 30° to the ceiling and 100 × sin 120° / sin 90° = 86.6 N for the string at 60° to the ceiling.
2. The strings are 90° apart and each is 135° from the weight, so T = 58.8 × sin 135° / sin 90° = 41.6 N each.
3. No. Five forces act on the crate (weight, normal reaction, friction, two tensions), so fall back on components.

Lami's theorem is one tool in a subject where the free body diagram, component resolution and moments carry most of the marks. A correct diagram makes every tool work, which is why our free body diagram routine is worth reading next.

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