Messenger_creation_6613CC4D-FA77-440F-A3AD-34AC891AEEA0.jpeg

# Statics ## Chapter 3 Equilibrium ### Equilibrium of a Particle #### Introduction * **Equilibrium of a Particle:** A particle is in equilibrium if the resultant force acting on it is zero. * **Newton's First Law of Motion:** If the resultant force on a particle is zero, the particle will rem...

# Statics ## Chapter 3 Equilibrium ### Equilibrium of a Particle #### Introduction * **Equilibrium of a Particle:** A particle is in equilibrium if the resultant force acting on it is zero. * **Newton's First Law of Motion:** If the resultant force on a particle is zero, the particle will remain at rest or continue to move in a straight line at a constant velocity. * **Statics:** In statics, we deal with particles that are at rest or move with constant velocity (i.e., zero acceleration). #### Free-Body Diagram To solve equilibrium problems, it is essential to consider all the forces acting on the particle. The complete diagram showing the particle and all forces acting on it is called a **free-body diagram (FBD)**. ##### Procedure for Drawing a FBD 1. **Isolate the particle:** Imagine the particle to be isolated or cut free from its surroundings. 2. **Show all forces:** Indicate all the forces acting on the particle. These forces can be active forces or reactive forces * **Active forces:** Tend to set the particle in motion * **Reactive forces:** Result from constraints or supports that tend to prevent motion 3. **Identify each force:** The forces are known or represented by letters. Specify the angles showing the direction of the forces. #### Coplanar Force Systems ##### Cartesian Vector Notation * For equilibrium in 2D, the forces can be resolved into two perpendicular components (i.e., x and y). * Each force can be expressed in Cartesian vector form: $\mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j}$ * Where $F_x = F\cos\theta$ and $F_y = F\sin\theta$ ##### Equilibrium Equations * A particle is in equilibrium if the resultant force acting on it is zero. Therefore, in 2D: $\mathbf{F} = \Sigma \mathbf{F} = 0$ * In Cartesian vector form: $\Sigma F_x \mathbf{i} + \Sigma F_y \mathbf{j} = 0$ * This equation is satisfied only if: $\Sigma F_x = 0$ $\Sigma F_y = 0$ ##### Procedure for Analysis 1. **Free-Body Diagram.** Establish the x, y axes, and draw a FBD of the particle. 2. **Coplanar Force Equilibrium Equations.** Apply the equilibrium equations $\Sigma F_x = 0$ and $\Sigma F_y = 0$. #### Three-Dimensional Force Systems ##### Cartesian Vector Notation * For equilibrium in 3D, the forces can be resolved into three perpendicular components (i.e., x, y, and z). * Each force can be expressed in Cartesian vector form: $\mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} + F_z \mathbf{k}$ * Where $F_x = F\cos\theta_x$, $F_y = F\cos\theta_y$ and $F_z = F\cos\theta_z$ ##### Equilibrium Equations * A particle is in equilibrium if the resultant force acting on it is zero. Therefore, in 3D: $\mathbf{F} = \Sigma \mathbf{F} = 0$ * In Cartesian vector form: $\Sigma F_x \mathbf{i} + \Sigma F_y \mathbf{j} + \Sigma F_z \mathbf{k} = 0$ * This equation is satisfied only if: $\Sigma F_x = 0$ $\Sigma F_y = 0$ $\Sigma F_z = 0$ ##### Procedure for Analysis 1. **Free-Body Diagram.** Establish the x, y, z axes, and draw a FBD of the particle. 2. **Coplanar Force Equilibrium Equations.** Apply the equilibrium equations $\Sigma F_x = 0$, $\Sigma F_y = 0$ and $\Sigma F_z = 0$.