How do you find the resultant force of a force system

Daily Dose 008 | STATICS

How do you find the resultant force of a set of forces acting on a hook? [PART 1]

How do you find the resultant force?

This will most definitely be a statement you read, in some form, on the day of your FE Exam.

Without a doubt, aside from the pure Mathematics, the most broadly applied subject you will encounter in the practice of engineering is Statics.

The theories found within this subject are foundational to much of the material you will encounter later. Dynamics, Mechanics of Materials, and Fluid Mechanics all build on it — so let’s get to work.

In this video, we dive in to a FE Exam Practice Problem in the subject of STATICS, specifically revolving around determining the RESULTANT OF A FORCE SYSTEM acting on a hook…this is PART 1 of 2 for this particular problem.

Key Definition

What is a FORCE SYSTEM?

A FORCE is most commonly defined as “the cause of change in the state of motion of a particle or body”.

Formulaically, it can be defined using:

• F = ma

A FORCE is a vector quantity, having a magnitude, direction, point of application and line of action.

When MULTIPLE FORCES are acting on an object, the collection of FORCES can be referred to as a SYSTEM OF FORCES.

In this case, this FORCE SYSTEM will have a SINGLE RESULTANT FORCE manifesting itself as an action on the object.

The Process

How can you find the resultant force of a force system?

Finding the resultant force in a force system can certainly present itself as difficult, but in practice, it involves some simple, straightforward steps that can be applied across the most complicated scenarios.

Let’s run through a simplified step by step process to getting these problems done quickly and efficiently on the FE Exam.

We will assume we are dealing with forces acting in a two-dimensional plane, but the principles extend to three dimensions just as well.

• Step 1: Understand the ProblemRead the problem statement carefully and identify the forces involved. Sketch a diagram labeling the forces and the angles they make with a reference axis (often the x-axis).
• Step 2: Break Forces into ComponentsFor each force, break it down into its x and y components.
  • F_x = F cos(θ)
  • F_y = F sin(θ)
• Step 3: Sum up the ComponentsAdd up all the x-components to find the total x-component (Σ F_x), and all the y-components to find the total y-component (Σ F_y).
  • Σ F_x = F_1x + F_2x + …
  • Σ F_y = F_1y + F_2y + …
• Step 4: Calculate the Resultant ForceThe magnitude of the resultant force comes from the Pythagorean theorem:
  • R = √((Σ F_x)² + (Σ F_y)²)
• Step 5: Find the Direction of the Resultant ForceCalculate the angle (φ) the resultant makes with the reference axis using the tangent function:
  • tan(φ) = Σ F_y / Σ F_x
  • φ = arctan(Σ F_y / Σ F_x)
• Step 6: Interpret the ResultsUnderstand what the magnitude and direction imply in the context of the problem. Make sure the units are consistent and the answer makes sense.

A few things to keep in mind when finding the resultant force of a force system:

• Confirm the coordinate system and reference line for your angles — horizontal vs. vertical changes your components.
• Make sure all units are consistent before calculating. If not, convert to a common system first.
• Pay close attention to the direction each force acts — this is what determines the sign of each component.
• Watch for forces that aren’t stated outright, like friction or tension.
• Answer exactly what’s asked — sometimes only the magnitude, sometimes magnitude and direction.
• Use a quick sanity check: does the resultant point where the forces would physically push the object?

By following these steps, and keeping these in mind, you will quickly and efficiently solve these problems while avoiding the common pitfalls that tend to pop up.

Worked Example

Let’s find a resultant force

Say two forces act at a point: 300 N along the positive x-axis, and 400 N straight up along the positive y-axis. Find the resultant.

• Components: already split for us — Σ F_x = 300 N, Σ F_y = 400 N
• Magnitude: R = √(300² + 400²) = √250,000 = 500 N
• Direction: φ = arctan(400 / 300) = 53.1° above the x-axis

So the resultant force is 500 N at 53.1°. That 3-4-5 relationship is worth recognizing — it shows up constantly, and spotting it saves you the calculator.

Now check out the video and see how to find the resultant force of a force system in the most efficient manner come your FE Exam day.

Video Review

[VIDEO] How to find the resultant force of a system of forces

Next Step

What’s next?

This was Part 1 — the setup and the method. Part 2 finishes the exact problem from the video, start to answer.

Statics is the foundation everything else on the FE Exam leans on, so getting fast and confident here pays off across the whole test — or let us build the plan around you.

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