Understanding Tensional Forces in Supporting Wires

Tensional forces often act through wires and ropes in mechanical systems. When three forces act at a point and are in equilibrium, Lami’s Theorem helps calculate the unknown forces.

What is Lami’s Theorem?

Lami’s Theorem states that if three coplanar forces keep a body in equilibrium, each force is proportional to the sine of the angle between the other two forces. The formula is:

T₁ / sin(θ₁) = T₂ / sin(θ₂) = T₃ / sin(θ₃)

This relation is essential when solving for tensions in wires holding up a load.

Example Problem: Find Tensions in Wires AC and BC

Let’s calculate the tensile forces in AC and BC wires that support a 430N vertical load.

Step 1: Geometry and Angles

Draw a line QD parallel to support PB through the point C. Given:

• ∠B = 65°, so ∠BCD = 65° (alternate angles rule)
• Line QD is perpendicular to QP, so ∠Q = 90°

Now, in triangle AQC:

• ∠A = 40°
• ∠Q = 90°

So,

∠C = 180° – (90° + 40°) = 50°

Step 2: Free-Body Diagram Angles

The vertical force acts downward, so:

• ∠QCE = ∠DCE = 90°

Calculate:

∠ACE = ∠ACQ + ∠QCE = 50° + 90° = 140°

∠BCE = ∠BCD + ∠DCE = 65° + 90° = 155°

Now, use the angle sum around point C:

∠ACB = 360° – (∠ACE + ∠BCE)
∠ACB = 360° – (140° + 155°) = 65°

Step 3: Applying Lami’s Theorem

Now apply Lami’s Theorem to solve for T₁ (AC) and T₂ (BC):

T₁ / sin(155°) = T₂ / sin(140°) = 430N / sin(65°)

Step 4: Calculate Tension in Wire AC (T₁)

T₁ = [430N ÷ sin(65°)] × sin(155°)
T₁ = [430 ÷ 0.906] × 0.422
T₁ = 474.61 × 0.422 = 200.29N

Step 5: Calculate Tension in Wire BC (T₂)

T₂ = [430N ÷ sin(65°)] × sin(140°)
T₂ = [430 ÷ 0.906] × 0.643
T₂ = 474.61 × 0.643 = 305.08N

Final Result: Tension Forces

• Tension in wire AC = 200.29 N
• Tension in wire BC = 305.08 N

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