Statics is the study of objects that are not accelerating, which means the forces and moments on them balance. This cheat sheet helps students organize forces, draw clear free-body diagrams, and solve basic engineering equilibrium problems. It is useful for analyzing structures such as signs, beams, bridges, frames, and simple machines.
A careful setup is often the difference between a correct solution and a confusing one.
The core ideas are force, moment, support reaction, and equilibrium. A force can be split into x and y components using trigonometry, and a moment equals force times perpendicular distance. For a rigid body in 2D equilibrium, the main equations are sum Fx = 0, sum Fy = 0, and sum M = 0.
Beams and trusses use these same rules to find support reactions and internal member forces.
Key Facts
- A body in static equilibrium has no acceleration, so sum F = 0 and sum M = 0.
- For 2D problems, the equilibrium equations are sum Fx = 0, sum Fy = 0, and sum M = 0.
- Force components are Fx = F cos(theta) and Fy = F sin(theta) when theta is measured from the positive x-axis.
- The magnitude of a force from components is F = sqrt(Fx^2 + Fy^2).
- A moment is calculated by M = Fd, where d is the perpendicular distance from the pivot to the force line of action.
- A pin support can provide two reaction components, usually Ax and Ay, while a roller support usually provides one reaction force perpendicular to the surface.
- For a simply supported beam with a center point load P, the support reactions are usually RA = P/2 and RB = P/2 when the load is centered.
- In a truss joint, use sum Fx = 0 and sum Fy = 0 to solve for unknown member forces at that joint.
Vocabulary
- Statics
- Statics is the branch of engineering mechanics that studies objects with balanced forces and no acceleration.
- Free-Body Diagram
- A free-body diagram is a simplified sketch showing one object isolated with all external forces and moments acting on it.
- Equilibrium
- Equilibrium means the net force and net moment on an object are zero.
- Moment
- A moment is the turning effect of a force about a point, calculated as force times perpendicular distance.
- Reaction Force
- A reaction force is a force supplied by a support to keep a structure or object in equilibrium.
- Truss
- A truss is a structure made of connected straight members that are usually assumed to carry only tension or compression.
Common Mistakes to Avoid
- Leaving forces off the free-body diagram makes the equations incomplete, so every external load, weight, reaction, and applied moment must be shown.
- Drawing internal forces on the free-body diagram of the whole object is wrong because internal forces cancel within the isolated body.
- Using the wrong angle for components gives incorrect signs or magnitudes, so check whether theta is measured from the x-axis or y-axis before using sine and cosine.
- Using distance instead of perpendicular distance in M = Fd is wrong because only the shortest distance to the force line of action creates the moment arm.
- Forgetting sign conventions causes equations to conflict, so choose positive directions for forces and moments before solving and keep them consistent.
Practice Questions
- 1 A 50 N force acts 3 m from a pivot at a right angle. What moment does it create about the pivot?
- 2 A cable pulls with a force of 100 N at 30 degrees above the horizontal. Find Fx and Fy.
- 3 A simply supported beam has a 200 N point load at its center. What are the two support reactions?
- 4 Why should a free-body diagram include support reactions but not internal forces when analyzing the entire object?
Understanding Statics, Forces & Free-Body Diagrams
A free body diagram is a model, not a sketch of every physical detail. First choose one object to isolate. Then imagine cutting it away from everything touching it.
Replace each contact with the force that the contact applies. A floor pushes upward on a box. A rope pulls along its own length.
A smooth surface pushes perpendicular to its surface. Friction acts along a surface and opposes the possible relative motion. Weight acts straight downward through the object’s center of mass.
This last point matters because weight is not usually applied at a corner or support. Draw known dimensions and angles clearly. Choose positive directions before doing calculations, then keep that choice throughout the work.
Support symbols describe allowed motion, so they tell you which unknown reactions belong on the diagram. A pin prevents horizontal and vertical movement but permits rotation. A roller prevents motion in one direction while allowing movement along the supporting surface.
A fixed support prevents translation and rotation, so it can exert horizontal force, vertical force, and a turning effect. Students often add forces that a support cannot provide. They may draw a horizontal reaction at a roller on a level surface, for example.
Another common mistake is to draw action and reaction forces on the same diagram. If a beam pushes down on a support, the support pushes up on the beam. Only the force acting on the isolated beam belongs on the beam’s diagram.
Moments explain why the location of a force changes its effect. Pushing a door near its hinges produces little turning, even with a strong push. Pushing near the handle produces much more turning because the perpendicular distance is larger.
Only the part of a force that acts across the lever arm creates the turning effect. A force whose line of action passes directly through the chosen pivot has no moment about that pivot. Choosing a smart pivot can remove unknown reactions from a moment calculation.
This is often the fastest way to solve a beam problem. Distributed loads need extra care.
A uniform load spread across a beam can be replaced by one equivalent force equal to load per length times the loaded length. That equivalent force acts at the center of the loaded region.
Trusses are built from straight members joined at points called joints. In an ideal truss, each member carries either tension or compression, not bending. A member in tension is pulled outward at both ends.
A member in compression is pushed inward. Begin at a joint with no more than two unknown member forces after any known loads are included. Assume unknown member forces pull away from the joint.
A negative result means the real force pushes toward the joint, so that member is in compression. This method appears in roof frames, cranes, bicycle frames, bridge supports, and shelving brackets. Check every answer for physical sense.
Reaction forces should balance the total load. Units must match. A very large answer may signal a missed force, a wrong distance, or an angle measured from the wrong direction.