Question Video: Using a Triangle of Forces to Solve Equilibrium Problems Mathematics

Video Transcript

Three coplanar forces 𝐅 sub one, 𝐅 sub two, and 𝐅 sub three are acting on a body in equilibrium. The triangle of forces forms a right triangle as shown. Given that the magnitude of 𝐅 sub one equals five newtons and the magnitude of 𝐅 sub two equals 13 newtons, find the magnitude of 𝐅 sub three.

Since the body is in equilibrium, we know that the vector sum of the three forces must be equal to zero. We also know that this means we can represent the system as a triangle, where the lengths of the triangle are proportional to their magnitudes. This means we can treat the two force magnitudes as if they were lengths and use the Pythagorean theorem to find the missing magnitude. That is, five squared plus the magnitude of 𝐅 sub three squared equals 13 squared. Subtracting five squared from both sides and we see that the magnitude of 𝐅 sub three squared is equal to 144.

Finally, we take the positive square root. Remember, we don’t need a negative root since a magnitude, by definition, is positive. The magnitude of this force is 12 newtons. So, we’ve found the magnitude of 𝐅 sub three.