3.2 Free-body diagrams and their importance

Free-body diagrams are essential tools in engineering mechanics. They isolate objects and show all external forces acting on them, making it easier to set up equilibrium equations and solve for unknown forces. These diagrams are crucial for analyzing forces in equilibrium problems.

By replacing supports and connections with reaction forces, free-body diagrams provide a clear visual representation of a system. They help identify applied and reaction forces, allowing engineers to classify and understand the forces acting on particles and rigid bodies in various scenarios.

Free-body diagrams for particles and rigid bodies

Graphical representations of isolated objects

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• Free-body diagrams are graphical representations that isolate an object of interest (a particle or rigid body) and show all external forces acting on it
• They provide a clear, visual representation of all external forces acting on a body, making it easier to set up equilibrium equations
• Free-body diagrams help in identifying the unknown forces or reactions that need to be solved for using equilibrium equations
• They also serve as a visual check to ensure that all forces have been accounted for and that the equilibrium equations are set up correctly

Replacing supports and connections with reaction forces

• Drawing a free-body diagram requires replacing all supports and connections (constraints) with the corresponding reaction forces and/or moments
• Forces in a free-body diagram are typically represented as vectors with arrows indicating their direction and magnitude
• For particles, forces are drawn as vectors originating from the particle itself (a point mass)
• For rigid bodies, forces can act at any point on the body and are drawn as vectors originating from their point of application (a distributed mass)
• Moments acting on rigid bodies are represented by curved arrows indicating their direction (clockwise or counterclockwise) and magnitude

Identifying forces on a body

Classifying forces as applied or reaction forces

• Identifying all forces acting on a body is crucial for accurately representing the system in a free-body diagram
• Forces can be classified as applied forces (gravitational force, normal force, friction force) or reaction forces (forces exerted by supports or constraints)
• Applied forces are external forces that act on the body due to its interaction with the environment or other bodies (weight, wind load, tension in a cable)
• Reaction forces are forces exerted by supports or constraints in response to the applied forces and the body's tendency to move (normal force from a surface, tension in a rope, force from a pin or hinge)

Common types of supports and their reaction forces

• Common types of supports and their corresponding reaction forces include:
  • Pinned support: Provides a force in both the horizontal and vertical directions, but no moment (a hinge on a door)
  • Roller support: Provides a force perpendicular to the surface it rests on, but no force parallel to the surface or moment (a wheel on a ramp)
  • Fixed support: Provides both force and moment reactions, fully constraining the body (a bolted connection)
• Friction forces should be included when the body is in contact with a rough surface and the friction is relevant to the problem (a block sliding down an incline)
• Internal forces within the body or system are not shown in a free-body diagram, as they cancel each other out according to Newton's Third Law (tension in a truss member)

Free-body diagrams for equilibrium problems

Equilibrium equations derived from Newton's First Law

• Free-body diagrams are essential tools for analyzing forces and solving equilibrium problems in engineering mechanics
• Equilibrium equations, derived from Newton's First Law, state that the sum of all forces and moments acting on a body in equilibrium must be zero
• For particles in equilibrium, the sum of forces in each coordinate direction (x, y, and z) must be zero:
  • $\sum F_x = 0$
  • $\sum F_y = 0$
  • $\sum F_z = 0$
• For rigid bodies in equilibrium, the sum of forces in each coordinate direction and the sum of moments about any point must be zero:
  • $\sum F_x = 0$
  • $\sum F_y = 0$
  • $\sum F_z = 0$
  • $\sum M_A = 0$ (moments about any point A)

Solving for unknown forces using equilibrium equations

• Free-body diagrams help in identifying the unknown forces or reactions that need to be solved for using equilibrium equations
• To solve for unknown forces, set up equilibrium equations based on the free-body diagram and solve the resulting system of equations
• The number of equilibrium equations needed depends on the number of unknown forces and the type of body (particle or rigid body)
• For particles, a maximum of three equilibrium equations can be written (one for each coordinate direction)
• For rigid bodies, a maximum of six equilibrium equations can be written (three for forces and three for moments)
• If the number of unknown forces exceeds the number of available equilibrium equations, the problem is statically indeterminate and cannot be solved using statics alone (additional information or relationships are needed)