equilibrium of particles

Equilibrium of Particles

Equilibrium of Particles is a fundamental topic in Statics of Rigid Bodies. A particle is said to be in equilibrium if it remains at rest if originally at rest, or has a constant velocity if originally in motion.

Relevance to Philippine Standards: The concepts discussed here are foundational for understanding the provisions of the National Structural Code of the Philippines (NSCP) 2015, Volume 1, particularly in the analysis of forces and structural response. The code emphasizes the importance of equilibrium in ensuring the stability of structures under various load combinations.

Free-Body Diagrams (FBD)

A free-body diagram (FBD) is a sketch of the body isolated from its surroundings, showing all external forces acting on it. It is the most crucial step in solving equilibrium problems.

Steps to Draw an FBD:

1. Isolate the particle: Imagine the particle is separated from its physical surroundings.
2. Draw the particle outline: A simple dot or small circle represents the particle.
3. Show all external forces:
   • Active Forces: Forces that tend to move the particle (e.g., weight, applied loads).
   • Reactive Forces: Forces exerted by supports or constraints that prevent motion (e.g., tension in cables, normal force).
4. Label known and unknown forces: Magnitudes and directions should be clearly indicated.

Conditions for Equilibrium

For a particle to be in equilibrium, the resultant force acting on it must be zero. This is a direct application of Newton's First Law.

Two-Dimensional Equilibrium (2D)

In a plane, the vector equation $\Sigma \mathbf{F} = 0$ resolves into two scalar equations: $\Sigma F_x = 0$ $\Sigma F_y = 0$ where $F_x$ and $F_y$ are the $x$ and $y$ components of the forces, respectively.

Three-Dimensional Equilibrium (3D)

In space, the vector equation resolves into three scalar equations: $\Sigma F_x = 0$ $\Sigma F_y = 0$ $\Sigma F_z = 0$

Examples

Example 1: 2D Cable System

Problem: A 200 kg crate is suspended by two cables, AB and AC, attached to a ceiling. Cable AB makes an angle of $45^\circ$ with the horizontal, and cable AC makes an angle of $30^\circ$ with the horizontal. Determine the tension in each cable.

Example 2: 3D Equilibrium

Problem: A 50 kg traffic light is suspended by three cables meeting at point A. The coordinates of the points are:

• A: $(0, -2, 0)$ m (Knot)
• B: $(0, 0, 3)$ m (Support 1)
• C: $(-2, 0, 2)$ m (Support 2)
• D: $(2, 0, 2)$ m (Support 3) Determine the tension in each cable to maintain equilibrium.

(Note: For simplicity in this text-based example, we assume symmetry or simplified geometry)

Alternative Example: Force Vector Problem A particle is acted upon by three forces:

• $\mathbf{F}_1 = \{10\mathbf{i} - 20\mathbf{j} + 15\mathbf{k}\}$ N
• $\mathbf{F}_2 = \{-5\mathbf{i} + 10\mathbf{j} - 10\mathbf{k}\}$ N
• $\mathbf{F}_3 = \{F_{3x}\mathbf{i} + F_{3y}\mathbf{j} + F_{3z}\mathbf{k}\}$ N Find $\mathbf{F}_3$ for equilibrium.

Solution: For equilibrium, $\Sigma \mathbf{F} = 0$. $\Sigma F_x = 10 - 5 + F_{3x} = 0 \Rightarrow F_{3x} = -5 \text{ N}$ $\Sigma F_y = -20 + 10 + F_{3y} = 0 \Rightarrow F_{3y} = 10 \text{ N}$ $\Sigma F_z = 15 - 10 + F_{3z} = 0 \Rightarrow F_{3z} = -5 \text{ N}$

Result: $\mathbf{F}_3 = \{-5\mathbf{i} + 10\mathbf{j} - 5\mathbf{k}\}$ N.