Forces and Vectors
Forces and Vectors
Engineering is the discipline of making systems behave exactly as intended. Before you can do that, you need a language for describing how things push and pull on each other. That language is vectors.
What is a Force?
A force is an interaction that changes — or tends to change — the motion of an object. Forces have two essential qualities:
- Magnitude — how large the push or pull is, measured in Newtons (N)
- Direction — which way the force points
Because forces have both magnitude and direction, they are vectors.
Vector Notation
We write vectors in bold or with an arrow overhead:
$$\vec{F} = F_x \hat{i} + F_y \hat{j} + F_z \hat{k}$$
Where $\hat{i}$, $\hat{j}$, $\hat{k}$ are unit vectors along the x, y, and z axes respectively.
Resolving Components
Any force in 2D can be decomposed into perpendicular components:
Fx = F · cos(θ)
Fy = F · sin(θ)
Where θ is the angle measured from the positive x-axis.
Example: A 100 N force at 30° from horizontal has:
- Fx = 100 · cos(30°) = 86.6 N
- Fy = 100 · sin(30°) = 50.0 N
The Parallelogram Law
When two forces act at a point, their combined effect (the resultant) is found by the parallelogram law — place the vectors tail-to-tail and draw the diagonal.
For vectors added algebraically:
R = √(Rx² + Ry²)
θ = arctan(Ry / Rx)
Why This Matters for Robotics
Every link in a robot arm experiences forces. Understanding how those forces decompose tells you:
- What loads the actuator must overcome
- Where structural stress concentrates
- Whether a joint will slip, yield, or hold
This is the foundation of everything that follows.
Summary
- Forces are vectors with magnitude and direction
- Resolve forces into orthogonal components using trigonometry
- Add vector components algebraically to find resultants
- This skill directly feeds into free-body diagram analysis