CELE Engineering Mechanics — Force Systems and ResultantsCheat Sheet
One-page cheat sheet for CELE Engineering Mechanics — Force Systems and Resultants. Every formula, definition, and key fact you need for this chapter, condensed to a single printable page. Designed for the final review session before the CELE 2026.
Exam context
On the CELE 2026, the Engineering Mechanics subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Civil Engineering's pattern. Force Systems and Resultants lands at position 1st out of 8 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Engineering Mechanics on a typical CELE paper.
Force Systems and Resultants - Cheat Sheet
Your 30-minute final review companion covering every formula, definition, and exam pitfall in Force Systems and Resultants. This is a condensed, direct-fire reference — no fluff, all critical content.
Sections
Formulas
Formula
Fx = F cos θ Fy = F sin θ
Meaning
Fx, Fy = rectangular components; F = magnitude; θ = angle from +x axis (counterclockwise positive)
Watch Out
θ must be measured from +x axis, counterclockwise. Watch quadrants — if force is in 3rd quadrant, θ > 180°. Use atan2(Fy, Fx) or check signs carefully.
When To Use
Any time a single force needs to be broken into x and y parts for summation
Formula
F = √(Fx² + Fy²) θ = tan⁻¹(Fy/Fx)
Meaning
Magnitude from components; direction angle from components
Watch Out
tan⁻¹ only returns angles in quadrants I and IV (±90°). If Fx < 0, add 180° to atan result to get true angle.
When To Use
After summing all components, to find the resultant magnitude and direction
Section Title
Force Resolution & Components (2D)
Important Facts
- Always resolve oblique forces into x and y components first — never use law of cosines on non-right triangles involving force directions.
- Counterclockwise is positive; clockwise is negative (standard sign convention in mechanics).
- cos²θ + sin²θ = 1 — use to verify component calculations or solve for unknown angles.
- In 2D, use rectangular (Cartesian) components; in 3D, use direction cosines with the identity cos²θx + cos²θy + cos²θz = 1.
Key Definitions
Term
Force
Example
A 500 N force acting 60° above horizontal
Definition
A vector quantity with magnitude, direction, and line of action; SI unit: Newton (N)
Term
Component
Example
A 100 N force at 30° has Fx = 86.6 N, Fy = 50 N
Definition
The projection of a force onto a coordinate axis (x or y direction)
Term
Resultant
Example
Two perpendicular 100 N forces combine to give 141.4 N resultant at 45°
Definition
Single force equivalent to a system of forces; replaces all forces without changing effect
Diagrams To Know
- Force in 2D Cartesian plane with Fx and Fy components labeled
- Force vector tail at origin, making angle θ with +x axis
- Parallelogram rule for vector addition (two forces forming adjacent sides)
Formulas
Formula
Rx = ΣFx Ry = ΣFy
Meaning
Rx, Ry = sum of all x and y components respectively
Watch Out
Do NOT add magnitudes directly. You must resolve each force into components, then sum. Ignoring this is the #1 student error.
When To Use
For any concurrent force system, sum components first
Formula
R = √(Rx² + Ry²) θR = tan⁻¹(Ry/Rx)
Meaning
R = magnitude of resultant; θR = direction of resultant from +x axis
Watch Out
Check the quadrant of the resultant. If Rx = −50 and Ry = 100, resultant is in quadrant II (θ > 90°).
When To Use
Always, after finding Rx and Ry
Section Title
Resultant of Concurrent Forces (All Pass Through One Point)
Important Facts
- Concurrent systems have NO moment effect about the point of concurrency (moment arm = 0).
- Equilibrium requires R = 0, which means both Rx = 0 AND Ry = 0 (two independent equations).
- Graphically, the polygon of forces closes if and only if the system is in equilibrium.
- For concurrent forces in 2D, maximum of 2 unknown forces can be solved (ΣFx = 0, ΣFy = 0 give two equations).
Key Definitions
Term
Concurrent Forces
Example
Three ropes pulling on a knot from different directions
Definition
All forces act through a common point; resultant passes through that same point
Term
Equilibrium of Concurrent Forces
Example
Three-force equilibrium: R = 0 means ΣFx = 0 and ΣFy = 0 both true
Definition
Resultant R = 0; body remains at rest or moves at constant velocity
Diagrams To Know
- Particle (point) with multiple forces radiating outward from a single point
- Force polygon (head-to-tail vector addition) closing to form resultant
- Free-body diagram (FBD) of concurrent system
Formulas
Formula
Mo = F × d
Meaning
Mo = moment about point O; F = force magnitude; d = perpendicular distance from O to line of action
Watch Out
d MUST be perpendicular distance to the LINE OF ACTION, not to the point of application. This is the single most common error. If you measure along the line of action, d = 0 and Mo = 0.
When To Use
When calculating rotational effect of a force about a specific point
Formula
Mo = xFy − yFx (Varignon form)
Meaning
x, y = coordinates of force application point; Fx, Fy = force components
Watch Out
Sign is critical: use right-hand rule. Counterclockwise (out of page) = positive. If Fx, Fy, x, y have correct signs, this formula handles sign automatically.
When To Use
When you already have components and want to avoid calculating perpendicular distance; ALWAYS use this in exams to avoid geometry errors
Section Title
Moment (Torque) About a Point
Important Facts
- Moment is a vector perpendicular to the plane of the force (in 2D, pointing in or out of page).
- Sign convention: counterclockwise = positive, clockwise = negative (standard in mechanics).
- Moment about a point is independent of where on the line of action the force is applied — only the perpendicular distance matters.
- Varignon's theorem: moment of a force = sum of moments of its components (allows you to replace complex geometry with simple arithmetic).
Key Definitions
Term
Moment (Torque)
Example
A 100 N force perpendicular to a 2 m moment arm creates 200 N·m moment
Definition
Measure of a force's tendency to rotate a body about a point; units: N·m (or kN·m)
Term
Moment Arm
Example
If force passes through the point, moment arm = 0 and moment = 0
Definition
Shortest (perpendicular) distance from a point to the line of action of a force
Term
Line of Action
Example
A force at (2,3) going at 45° has line of action y − 3 = 1·(x − 2)
Definition
Infinite line along which a force acts; defined by the force's direction and point of application
Diagrams To Know
- Force with perpendicular distance d drawn from point O to line of action (not to point of application)
- Right-hand rule diagram showing moment direction (out of page = positive/counterclockwise)
- Moment arm illustration: shortest distance from pivot to force line
Reactions Or Equations
Note
If ΣFx = 0 and ΣFy = 0 but ΣMo ≠ 0, system is a couple (pure rotation, no translation).
Equation
ΣMo = Σ(xFy − yFx)
Conditions
For any system of coplanar forces; sum taken about point O
Formulas
Formula
Mcouple = F × d = constant (independent of reference point O)
Meaning
F = magnitude of each force; d = perpendicular distance between the two parallel forces
Watch Out
A couple's moment is THE SAME about EVERY point in the plane — it's a free vector, not bound to a point. Do NOT use moment = xFy − yFx for couples as a total; that only works for individual forces.
When To Use
When two equal and opposite forces are separated by distance d
Section Title
Couples (Two Equal, Opposite, Parallel Forces)
Important Facts
- Couple has resultant force R = 0 but resultant moment ≠ 0 (pure rotation).
- Couples can be replaced by other couples of equal moment (same magnitude and direction, any d and F such that F×d is constant).
- In 3D, couple moment is parallel to the vector connecting the two forces.
- If ΣF = 0 but ΣM ≠ 0, the system is a couple and cannot be simplified to a single resultant force.
Key Definitions
Term
Couple
Example
Two 50 N forces: one up at x=0, one down at x=2 m, creates couple of 50 × 2 = 100 N·m
Definition
Two equal, opposite, parallel forces separated by perpendicular distance d; produces pure rotation (no translation)
Term
Free Vector
Example
Couple moment is a free vector; add it anywhere in the system without changing rotational effect
Definition
A vector (like couple moment) whose effect is the same regardless of where it is applied
Diagrams To Know
- Two parallel arrows of equal length in opposite directions, separated by distance d
- Curved arrow showing rotational effect (moment) with label indicating magnitude and direction
- Couple replacing a force system: moment vector perpendicular to plane
Formulas
Formula
Rx = ΣFx Ry = ΣFy
Meaning
Sum all force components in x and y directions
Watch Out
Same component rule applies. Resolve each force fully before summing.
When To Use
First step for ANY force system (concurrent or not)
Formula
R = √(Rx² + Ry²) θR = tan⁻¹(Ry/Rx)
Meaning
Resultant magnitude and direction
Watch Out
Check quadrant of resultant.
When To Use
After summing components
Formula
ΣMo = xRy − yRx = Σ(xFy − yFx) OR ΣMo = Σ Mo,individual
Meaning
Sum of all moments about reference point O; must equal moment of resultant R about O
Watch Out
Moment of resultant is calculated as if R acts at its line of action. If R does not pass through O, then R will produce a moment about O equal to ΣMo.
When To Use
To verify consistency or to locate resultant
Formula
d = ΣMo / R (perpendicular distance from O to resultant line of action)
Meaning
Locates the resultant; d is perpendicular distance from O to R's line of action
Watch Out
If R = 0, this formula breaks down. If R = 0 but ΣMo ≠ 0, the system IS a couple, and d is undefined (couple acts everywhere).
When To Use
When you need to know WHERE the resultant acts (its line of action)
Section Title
Resultant of Non-Concurrent (General) Force Systems
Important Facts
- Resultant R must equal ΣFx and ΣFy; its line of action is determined by the moment condition ΣMo,resultant = ΣMo,original.
- If R ≠ 0: system reduces to single resultant force at unique location d = ΣMo / R from reference point.
- If R = 0 and ΣMo = 0: system is in equilibrium; body at rest.
- If R = 0 and ΣMo ≠ 0: system is a couple; pure rotation, no translation.
- The line of action of a resultant is UNIQUE and independent of the choice of reference point (moment always satisfied).
Key Definitions
Term
Non-Concurrent Forces
Example
Distributed load on a beam, or parallel forces on different points
Definition
Forces whose lines of action do NOT all pass through a single point; can produce both translation and rotation
Term
Line of Action
Example
Resultant must be located such that its moment about any point equals the sum of moments of original forces
Definition
The infinite line along which a force acts; concurrent forces' lines all meet; non-concurrent lines may be skew or parallel
Term
Resultant Reduction
Example
A beam with distributed and concentrated loads reduces to one resultant force at a specific location
Definition
Replacing a complex force system with a single resultant force (if R ≠ 0) or a couple (if R = 0 but ΣM ≠ 0)
Diagrams To Know
- Force system with multiple forces at different points, replaced by single resultant at calculated location
- Moment about reference point O shown separately from force resultant
- Line of action of resultant shown perpendicular to moment vector
Reactions Or Equations
Note
Ensures resultant's moment about O equals original system's moment. Line of action of R is found by solving for d.
Equation
R·d = ΣMo (moment condition)
Conditions
For any non-concurrent planar force system; R ≠ 0
Formulas
Formula
R = ΣF (algebraic sum of all parallel forces)
Meaning
For parallel forces, resultant magnitude is simply the algebraic sum (add downward, subtract upward, or vice versa by convention)
Watch Out
Still must assign signs consistently (down = negative, up = positive, or vice versa). R = ΣF is only true because Rx or Ry perpendicular to the parallel direction equals zero.
When To Use
Simplification when all forces are parallel (e.g., vertical loads on a beam)
Formula
x̄ = ΣMo / R (location of resultant from reference point O)
Meaning
x̄ = distance from O to resultant; Σ Mo = sum of moments of all forces about O
Watch Out
Denominator R is the magnitude of resultant. If R = 0 (equilibrium), formula breaks down — this means no single resultant, but either zero forces or a couple.
When To Use
To locate where the single resultant force acts along a line (e.g., distance along a beam)
Formula
ΣMo = Σ(Fi × xi) where forces and distances are measured from reference point O
Meaning
Moment of each force = force × its distance from O (with signs). Sum all moments.
Watch Out
Include the sign of each force and moment consistently. If a force is down and distance is to the right, moment is typically negative (clockwise).
When To Use
Computing total moment before finding resultant location
Section Title
Parallel Force Systems (Special Case of Non-Concurrent)
Important Facts
- For parallel forces, ΣFx or ΣFy perpendicular to the direction is always zero.
- Resultant is parallel to the original forces and located at the 'center of force' (analogous to center of gravity).
- Moment calculation: for vertical forces on a horizontal beam with origin at left end, Mo = Σ(Force × distance from left) at each point.
- If all forces point the same way, resultant is at the geometric weighted-mean location: x̄ = Σ(Fi × xi) / ΣFi.
Key Definitions
Term
Parallel Forces
Example
Vertical loads on a horizontal beam, or all forces at 45° going the same way
Definition
All forces act in the same direction (all up, all down, or all at the same angle); lines of action are parallel
Diagrams To Know
- Horizontal beam with multiple vertical forces (loads) acting downward at different positions
- Single resultant force shown at calculated location on the beam
- Moment diagram: showing clockwise and counterclockwise effects about a reference point
Formulas
Formula
Fx = F cos αx Fy = F cos αy Fz = F cos αz
Meaning
F = magnitude; αx, αy, αz = angles force makes with +x, +y, +z axes respectively; cos αx, cos αy, cos αz are direction cosines
Watch Out
Angles are measured from the POSITIVE axis. Angle ranges 0° to 180°. cos αx is NOT the same as cos θ in 2D.
When To Use
Breaking down a 3D force into rectangular components
Formula
cos²αx + cos²αy + cos²αz = 1 (identity for direction cosines)
Meaning
Direction cosines satisfy this identity; can find missing angle if two are known
Watch Out
This is a CONSTRAINT — you cannot independently specify three direction cosines. If you know two angles, the third is determined (up to sign ambiguity, which direction sense resolves).
When To Use
To verify direction cosines or find unknown angle; also to solve for F if components are known
Formula
F = √(Fx² + Fy² + Fz²)
Meaning
Magnitude of 3D force from components (Pythagorean in 3D)
Watch Out
This is straightforward; common error is forgetting the Fz term.
When To Use
Finding magnitude after resolving a 3D force into three components
Section Title
3D Force Systems & Direction Cosines
Important Facts
- 3D concurrent forces sum the same way as 2D: Rx = ΣFx, Ry = ΣFy, Rz = ΣFz, then R = √(Rx² + Ry² + Rz²).
- Moment in 3D is a vector: Mo = r × F (cross product); magnitude = |r| |F| sin θ, direction by right-hand rule.
- The constraint cos²αx + cos²αy + cos²αz = 1 means direction cosines are NOT independent; only two can be freely chosen.
- If a force is specified as 'from point A to point B', direction cosines are proportional to (Bx − Ax, By − Ay, Bz − Az) / distance AB.
Key Definitions
Term
Direction Cosines
Example
A force at angles 60°, 45°, and 120° from x, y, z axes has direction cosines cos 60° = 0.5, cos 45° = 0.707, cos 120° = −0.5
Definition
The cosines of angles between a force and the three coordinate axes: cos αx, cos αy, cos αz
Term
3D Force Components
Example
A 100 N force with direction cosines (0.6, 0.8, 0) has components (60, 80, 0) N
Definition
Rectangular components of a force in x, y, z directions (analogous to 2D Fx, Fy)
Diagrams To Know
- 3D Cartesian axes with force vector showing angles to each axis
- Force along space diagonal of a box (e.g., corner at origin to point (3,4,5))
- Direction cosines table: cos αx, cos αy, cos αz labeled on 3D force diagram
Formulas
Formula
Mo = Mo,Fx + Mo,Fy (2D form) OR Mo,total = Σ(xFy − yFx) for individual forces
Meaning
Moment of a force F about point O equals moment of its x-component plus moment of its y-component
Watch Out
This is not a shortcut for lazy students — it's a RIGOROUS theorem and the most reliable method for exam problems. Always prefer Mo = xFy − yFx over perpendicular-distance guessing.
When To Use
To avoid calculating perpendicular distance; break force into components and calculate moment of each component about O
Section Title
Varignon's Theorem
Important Facts
- Varignon eliminates the need to find perpendicular distance geometrically — use arithmetic instead.
- Works for any point O and any decomposition of force (Cartesian, or any other basis).
- In 3D, Mo = r × F (cross product) inherently applies Varignon; component moments automatically sum to total.
- Exam advantage: students who use Varignon (xFy − yFx) make fewer errors than those who measure perpendicular distances.
Key Definitions
Term
Varignon's Theorem
Example
A 500 N force at 60° applied at (4,0): Fx = 250 N (horizontal), Fy = 433 N (vertical); Mo = 4(433) − 0(250) = 1732 N·m
Definition
The moment of a force about a point equals the algebraic sum of the moments of its components about that same point
Diagrams To Know
- Force F decomposed into components Fx and Fy; each component's moment arm shown separately
- Moment addition: Mo,total = Mo,Fx + Mo,Fy illustrated with curved arrows
Must Remember
- ALWAYS resolve forces into components (Fx, Fy) BEFORE summing — never add magnitudes directly. ΣFx and ΣFy are the foundation of every calculation.
- Moment = Force × PERPENDICULAR distance (NOT distance along the force line). If perpendicular distance = 0, moment = 0 regardless of how far the point is from the force's line of action.
- Use Varignon form Mo = xFy − yFx to calculate moments in exams. It is more reliable than measuring perpendicular distances, avoids geometry errors, and is standard on the PRC exam.
- For non-concurrent forces, R ≠ 0 alone is insufficient — you MUST locate where R acts using d = ΣMo / R. An unlocated resultant is an incomplete answer.
- A couple has R = 0 but moment ≠ 0 (pure rotation). If ΣF = 0 and ΣM ≠ 0, the system IS a couple; you cannot reduce it to a single resultant force.
- Quadrant errors on tan⁻¹(Ry/Rx) lose easy marks. Always check the signs of Rx and Ry; if both negative, θ is in quadrant III (add 180°).
- For parallel forces (common in beam problems), resultant location is x̄ = Σ(Fi × xi) / ΣFi. Forget to divide by ΣFi and you lose the entire point-location answer.
- Equilibrium in 2D requires THREE independent equations: ΣFx = 0, ΣFy = 0, ΣMo = 0. Missing any one means the system is not in equilibrium (even if two are satisfied).
- In 3D, direction cosines satisfy cos²αx + cos²αy + cos²αz = 1. This is a constraint — you cannot independently specify three direction cosines; if two are known, the third is determined.
- Sign convention matters: adopt one (e.g., counterclockwise positive) and stick to it throughout a problem. Mixing signs or changing convention mid-solution is a common source of exam errors.
Last Minute Tips
- Before touching a problem, DRAW A CLEAR FREE-BODY DIAGRAM (FBD). Every force must be shown with magnitude, angle, and position. FBDs prevent 80% of errors and are expected by the PRC graders.
- If a problem asks for 'resultant' of a non-concurrent system, give THREE pieces of information: (1) magnitude R, (2) direction θR, and (3) location d from reference point. Missing location is an incomplete answer — you will lose marks.
- For parallel force problems (beams, distributed loads), always use ΣMo about a convenient point (e.g., left support) to find resultant location. This is faster and more accurate than perpendicular distances.
- If you see 'equilibrium' or 'at rest' in the problem, write ΣFx = 0, ΣFy = 0, ΣMo = 0 first. THREE equations, THREE unknowns (or choose reference point O to eliminate one unknown moment).
- Exam boards (PRC) test Varignon's theorem indirectly — they ask you to find moment of an angled force. Always decompose into components and use Mo = xFy − yFx. This is the fastest, most reliable method and shows you understand the theorem.
Comparison Tables
Rows
Values
- All pass through one point
- Do NOT all pass through one point
- All parallel (never intersect)
Property
Lines of action
Values
- R = √(ΣFx)² + (ΣFy)²; acts at point of concurrency
- R = √(ΣFx)² + (ΣFy)²; acts at specific location d = ΣMo / R
- R = ΣF (algebraic sum); acts at x̄ = ΣMo / R
Property
Resultant force R
Values
- Always = 0 (moment arm = 0)
- Typically ≠ 0 (must be calculated and used to locate R)
- Depends on location; use to find resultant position
Property
Moment about point of concurrency
Values
- ΣFx = 0, ΣFy = 0 (2 equations)
- ΣFx = 0, ΣFy = 0, ΣMo = 0 (3 equations)
- ΣF = 0 (1 equation), ΣMo = 0 (1 equation) along different axes
Property
Equilibrium equations (2D)
Values
- Three ropes pulling on a ring at the ring's center
- Beam with concentrated load at midspan and reaction at support
- Vertical loads on a horizontal beam; all gravity forces
Property
Example
Columns
- Property
- Concurrent
- Non-Concurrent
- Parallel
Table Title
Concurrent vs. Non-Concurrent vs. Parallel Force Systems
Rows
Values
- Rotational effect of a single force about a point; Mo = F × d
- Rotational effect of two equal, opposite, parallel forces; M = F × d
Property
Definition
Values
- The original force F acts; moment is secondary effect
- Resultant force = 0; ONLY moment exists (pure rotation)
Property
Resultant force
Values
- Moment CHANGES as O moves (depends on perpendicular distance to line of action)
- Moment is INDEPENDENT of O; same value everywhere (free vector)
Property
Dependence on reference point O
Values
- Mo = F × d (perpendicular distance) OR Mo = xFy − yFx (Varignon)
- M = F × d (perpendicular distance between the two forces); or replace with equivalent couple
Property
How to calculate
Values
- Tends to rotate AND translate (unless forces are concurrent)
- Tends ONLY to rotate; no net translation
Property
Effect on body
Columns
- Aspect
- Moment of a Force
- Couple
Table Title
Moment vs. Couple: Key Differences
Rows
Values
- +
- +
- 0° to 90°
- 100 N at 30°: Fx = 86.6 N, Fy = 50 N
Property
Quadrant I (up-right)
Values
- −
- +
- 90° to 180°
- 100 N at 135°: Fx = −70.7 N, Fy = 70.7 N
Property
Quadrant II (up-left)
Values
- −
- −
- 180° to 270°
- 100 N at 225°: Fx = −70.7 N, Fy = −70.7 N
Property
Quadrant III (down-left)
Values
- +
- −
- 270° to 360°
- 100 N at 315°: Fx = 70.7 N, Fy = −70.7 N
Property
Quadrant IV (down-right)
Columns
- Quadrant / Direction
- Fx sign
- Fy sign
- Angle θ from +x axis
- Example: 100 N force
Table Title
Sign Conventions & Quadrant Reference
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