Learning Outcomes

Upon completing this lesson, you will be able to:

• Define a force as a vector and identify its key characteristics.
• Calculate the resultant force of a system of concurrent forces by adding vectors.
• Define the condition for the equilibrium of a particle.
• Draw a Free-Body Diagram (FBD) to account for all forces acting on a particle.
• Apply the equations of equilibrium to solve for unknown forces in a 2D system.

Topics Covered

• Force as a Vector
• Resultant Force
• Equilibrium of a Particle
• The Free-Body Diagram
• Common Forces in Statics (Springs and Cables)

I. Force as a Vector

In mechanics, a force is an action that tends to change the state of motion of a body. It is a push or a pull. Because a force has both a magnitude (how strong the push or pull is) and a direction, it is a vector quantity.

For many problems in statics, we can model a physical object as a particle. A particle is an object that has mass but negligible size and shape. This is a valid simplification when all the forces acting on the body are concurrent, meaning they all act through the same point.

II. Resultant Force

When multiple forces act on a single particle, their combined effect can be represented by a single force called the resultant force, $\vec{F}_R$. The resultant force is simply the vector sum of all the individual forces acting on the particle.

Given several forces $\vec{F}_1, \vec{F}_2, \vec{F}_3, \dots$, the resultant force is:

$$ \vec{F}_R = \Sigma \vec{F} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \dots $$

To find the resultant force analytically, we resolve each force into its $x$ and $y$ components and then sum the respective components:

$$ F_{Rx} = \Sigma F_x = F_{1x} + F_{2x} + \dots $$

$$ F_{Ry} = \Sigma F_y = F_{1y} + F_{2y} + \dots $$

The final resultant vector is

$$ \vec{F}R = F{Rx} \mathbf{i} + F_{Ry} \mathbf{j} $$

III. Equilibrium of a Particle

A particle is said to be in equilibrium if it is at rest or moving with a constant velocity. According to Newton's First Law, this condition occurs when the resultant force acting on the particle is zero.

This is the fundamental equation of statics:

$$ \Sigma \vec{F} = 0 $$

For this vector equation to be true, the sum of the components in each direction must also be zero. For a 2D system, this gives us two scalar equations of equilibrium:

$$ \Sigma F_x = 0 $$

$$ \Sigma F_y = 0 $$

These two equations allow us to solve for, at most, two unknown forces (e.g., two magnitudes, or a magnitude and a direction) in a statics problem.

IV. The Free-Body Diagram (FBD)

The most important step in solving any statics problem is to draw a Free-Body Diagram (FBD). An FBD is a simplified, idealized sketch of the particle or body, isolated from its surroundings, that shows all the external forces acting on it.

Steps to Draw a Free-Body Diagram:

1. Isolate the Particle: Draw a simple dot or shape to represent the particle, completely separated from all its supports.
2. Identify All Forces: Identify every force that acts on the particle. These typically include Weight, Tension, Normal Force, Spring Force, and any other Applied Forces.
3. Draw and Label Forces: Draw each force as a vector arrow originating from the particle. Each arrow should point in the correct direction, and each should be clearly labeled.

V. Common Forces in Statics