Equilibrium of Particles

Learning Objectives

  • Construct a concurrent-force free-body diagram.
  • Solve two-dimensional and three-dimensional particle equilibrium equations.
  • Determine unknown cable tensions from explicit unit direction vectors.
  • Relate pulley supporting-segment count and aggregate efficiency to cable pull.
  • Detect singular geometry, negative cable tension, and impossible configurations.
  • Verify every solution using equilibrium and independent analytical checks.
Concurrent-force particle free-body diagramCable tensions act away from the isolated particle, while applied weight acts downward. Feasible equilibrium requires force balance and nonnegative cable tension.T₁T₂WΣFₓ = 0 · ΣFᵧ = 0 · T ≥ 0
Concurrent-force particle free-body diagram
Cable tensions act away from the isolated particle, while applied weight acts downward. Feasible equilibrium requires force balance and nonnegative cable tension.

Governing principle and sign convention

A particle has no size for moment analysis, so equilibrium requires only ∑F=0\sum\mathbf F=\mathbf 0. Use +x+x to the right, +y+y upward, and +z+z according to the right-handed coordinate system. A cable force is directed away from the particle and must satisfy T≥0T\ge0 because a cable cannot carry compression.

For the two-cable ring and traffic-signal simulations, the left cable angle α\alpha is measured above the leftward horizontal and the right cable angle β\beta is measured above the rightward horizontal. Both are restricted to acute values from 1∘1^\circ to 89∘89^\circ, preserving the stated left and right anchor positions.

Particle equilibrium

Independent scalar equations in two and three dimensions.

∑F=0;{∑Fx=0∑Fy=02D;{∑Fx=0∑Fy=0∑Fz=03D\sum\mathbf F=\mathbf0; \qquad \begin{cases}\sum F_x=0\\\sum F_y=0\end{cases}_{2D}; \qquad \begin{cases}\sum F_x=0\\\sum F_y=0\\\sum F_z=0\end{cases}_{3D}

Variables

SymbolDescriptionUnit
TiT_iCable tension acting along cable ikN
ui\mathbf u_iCable unit direction vector-
R\mathbf RForce residual after substitutionkN

Two-cable closed-form check

Independent analytical oracle for a downward load W and acute cable angles.

−T1cos⁡α+T2cos⁡β=0,T1sin⁡α+T2sin⁡β=W-T_1\cos\alpha+T_2\cos\beta=0, \qquad T_1\sin\alpha+T_2\sin\beta=WT1=Wcos⁡βsin⁡(α+β),T2=Wcos⁡αsin⁡(α+β)T_1=\frac{W\cos\beta}{\sin(\alpha+\beta)}, \qquad T_2=\frac{W\cos\alpha}{\sin(\alpha+\beta)}

Guided free-body-diagram method

  1. Isolate the ring, joint, signal, pulley block, or connection as a particle.
  2. Draw every applied load and every cable tension away from the particle.
  3. Define coordinate axes and convert cable geometry to unit vectors.
  4. Assemble one independent scalar equation per coordinate direction.
  5. Solve the linear system and inspect its rank.
  6. Reject negative cable tensions.
  7. Substitute the solution and report ∥∑F∥\lVert\sum\mathbf F\rVert.
  8. Where available, compare with an independent closed-form solution.

Guided example: symmetric ring

A 20 kN20\text{ kN} load is supported by two cables each inclined 45∘45^\circ above its corresponding horizontal direction. Horizontal components cancel and 2Tsin⁡45∘=202T\sin45^\circ=20, giving T=14.14 kNT=14.14\text{ kN}. Both tensions are positive, the matrix and closed-form solutions agree, and the force residual is zero.

Common misconceptions

Two-cable ring

Cable-Supported Ring

Concept and model scope

Solve two cable tensions using acute angles measured above the leftward and rightward horizontal axes.

Simulation purpose: Exact cable directions, tension-only admissibility, and independent closed-form verification.

Model scope: Two-dimensional concurrent-force equilibrium. The left angle is measured above the leftward horizontal and the right angle above the rightward horizontal. Both angles are restricted to 1°–89° so the named anchors remain on their intended sides. Cables are tension-only.

Verification: The SVG unit vectors, linear-system coefficients, closed-form solution, displayed tensions, physical verdict, and assessment answer are cross-checked from the same inputs. Near-vertical labels are positioned independently to remain readable.

Load

Load

Load is an adjustable model parameter. Move the slider to change it from 1 to 200 kN; the simulation should update its geometry or engineering response directly from this value.

60 kN
Left cable angle

Left cable angle

Left cable angle is an adjustable model parameter. Move the slider to change it from 1 to 89 °; the simulation should update its geometry or engineering response directly from this value.

45 °
Right cable angle

Right cable angle

Right cable angle is an adjustable model parameter. Move the slider to change it from 1 to 89 °; the simulation should update its geometry or engineering response directly from this value.

60 °
Tension-only cable ringThe rendered unit vectors are exactly the coefficient vectors used in equilibrium.ABT₁ 31.06 kNT₂ 43.92 kNα = 45°β = 60°W 60.0 kN
Left tension
31.0583 kN
Right tension
43.9230 kN
Force residual
0.0000000000
Closed-form difference
0.0000000000

Closed-form difference

Maximum difference between matrix and analytical solutions.

equilibrium independently verified
Engineering model scope

Category

Particle equilibrium

Idealization

Concurrent forces act at an idealized particle; cables carry tension only.

Acceptance check

Require ΣF = 0 and reject negative cable tension or singular geometry.

Interpretation question

Why does one cable tension grow rapidly when that cable approaches the horizontal?

Suspended traffic signal

Suspended Traffic Signal

Concept and model scope

Resolve unequal supporting cable forces with the same angle convention used by the diagram and solver.

Simulation purpose: Exact cable directions, tension-only admissibility, and independent closed-form verification.

Model scope: Two-dimensional concurrent-force equilibrium. The left angle is measured above the leftward horizontal and the right angle above the rightward horizontal. Both angles are restricted to 1°–89° so the named anchors remain on their intended sides. Cables are tension-only.

Verification: The SVG unit vectors, linear-system coefficients, closed-form solution, displayed tensions, physical verdict, and assessment answer are cross-checked from the same inputs. Near-vertical labels are positioned independently to remain readable.

Signal weight

Signal weight

Signal weight is an adjustable model parameter. Move the slider to change it from 1.0 to 50.0 kN; the simulation should update its geometry or engineering response directly from this value.

8.0 kN
Left cable angle

Left cable angle

Left cable angle is an adjustable model parameter. Move the slider to change it from 1 to 89 °; the simulation should update its geometry or engineering response directly from this value.

25 °
Right cable angle

Right cable angle

Right cable angle is an adjustable model parameter. Move the slider to change it from 1 to 89 °; the simulation should update its geometry or engineering response directly from this value.

35 °
Suspended traffic-signal equilibriumThe rendered unit vectors are exactly the coefficient vectors used in equilibrium.ABT₁ 7.57 kNT₂ 8.37 kNα = 25°β = 35°W 8.0 kN
Left tension
7.5670 kN
Right tension
8.3721 kN
Force residual
0.0000000000
Closed-form difference
0.0000000000

Closed-form difference

Maximum difference between matrix and analytical solutions.

equilibrium independently verified
Engineering model scope

Category

Particle equilibrium

Idealization

Concurrent forces act at an idealized particle; cables carry tension only.

Acceptance check

Require ΣF = 0 and reject negative cable tension or singular geometry.

Interpretation question

How does an unequal sag angle distribute the signal weight between the two cables?

Equivalent block-and-tackle

The pulley simulation uses one ideal continuous cable, equal tension in every rope leg, and an explicitly stated aggregate efficiency η\eta. If nn vertical rope segments support the moving block, the educational equilibrium model is

W=nηT,T=Wnη.W=n\eta T, \qquad T=\frac{W}{n\eta}.

The diagram traces a continuous equivalent reeving and contains exactly the selected number of supporting legs. This is not a detailed bearing-friction, rope-bending, or per-sheave efficiency model.

Pulley and Hanging-Load System

Concept and model scope

Trace one continuous equivalent rope and relate the selected supporting-leg count to required cable pull.

Simulation purpose: Continuous equivalent reeving, exact supporting-leg count, and aggregate-efficiency equilibrium.

Model scope: Ideal one-rope tackle with equal tension in every rope leg. The selected integer n is the number of vertical tension legs terminating at moving pulleys or the moving-block anchor. η is an explicitly stated aggregate system efficiency, so the educational model uses W = nηT rather than a detailed per-sheave loss model.

Verification: The continuous reeving contains exactly n supporting legs, every pulley has a visible fixed or moving attachment, and both the displayed pull and assessment answer satisfy nηT − W = 0 to numerical precision. Noninteger exact entries are normalized to the nearest permitted segment count.

Load

Load

Load is an adjustable model parameter. Move the slider to change it from 1 to 200 kN; the simulation should update its geometry or engineering response directly from this value.

80 kN
Supporting segments

Supporting segments

Supporting segments is an adjustable model parameter. Move the slider to change it from 1 to 8; the simulation should update its geometry or engineering response directly from this value.

4
Aggregate efficiency

Aggregate efficiency

Aggregate efficiency is an adjustable model parameter. Move the slider to change it from 50 to 100 %; the simulation should update its geometry or engineering response directly from this value.

90 %
Continuous equivalent block-and-tackle reevingEvery vertical blue leg terminates at a moving pulley or moving-block anchor.FIXED SUPPORTfree-end pull TT=22.22TTTfixed anchorMOVING BLOCK · n = 4 SUPPORTING LEGSAggregate efficiency η = 90%W = 80.0 kNEQUILIBRIUM MODELnηT = Wn = 4η = 0.9T = 22.22 kN
Required pull
22.2222 kN
Ideal pull
20.0000 kN
Actual mechanical advantage
3.6000
Equilibrium residual
0.0000000000 kN
reeving and equilibrium verified
Engineering model scope

Category

Particle equilibrium

Idealization

Concurrent forces act at an idealized particle; cables carry tension only.

Acceptance check

Require ΣF = 0 and reject negative cable tension or singular geometry.

Interpretation question

Why does the ideal pulling force equal the load divided by the number of supporting rope segments?

Three-dimensional guy-wire joint

Three-dimensional guy-wire joint

Concept and model scope

Solve three cable tensions for a joint carrying vertical and horizontal loads.

Governing relation: [uA uB uC] T = −P

Geometry: Anchor radius and rise use one fixed spatial scale and define the actual cable direction vectors. A cable solution requiring compression is physically invalid.

Physical dimensions use a fixed scene-units-per-metre scale. Force-arrow lengths use a fixed range-based readability scale; direction and application point remain mechanically correct.

Controls
Downward load

Downward load

Downward load belongs to the same geometry/equilibrium model used by the 3D scene and analytical solver.

45 kN
Horizontal load

Horizontal load

Horizontal load belongs to the same geometry/equilibrium model used by the 3D scene and analytical solver.

10 kN
Horizontal-load azimuth

Horizontal-load azimuth

Horizontal-load azimuth belongs to the same geometry/equilibrium model used by the 3D scene and analytical solver.

20 °
Anchor radius

Anchor radius

Anchor radius belongs to the same geometry/equilibrium model used by the 3D scene and analytical solver.

6.0 m
Anchor rise above joint

Anchor rise above joint

Anchor rise above joint belongs to the same geometry/equilibrium model used by the 3D scene and analytical solver.

8.0 m
Engineering model scope

Category

Particle equilibrium

Idealization

Concurrent forces act at an idealized particle; cables carry tension only.

Acceptance check

Require ΣF = 0 and reject negative cable tension or singular geometry.

Interpretation question

What geometric feature makes the three-dimensional direction matrix lose rank?

Equilibrium feasibility

Equilibrium feasibility tester

Concept and model scope

Test whether two selected cable directions can balance the applied load using nonnegative tensions.

Governing model: A T = −P; Ti ≥ 0

Every physical dimension shown by this studio is derived from the same state used by the solver. Readability-scaled force arrows preserve direction and application point.

Equilibrium feasibility. The load is compared with the tensile cone generated by the two cable directionsP 30 kNTensile equilibrium feasibleGeometry-faithful free-body diagram with automatic fit-to-content framing.
Controls
Load

Load

Load is part of the same engineering state used by the diagram and solver.

30 kN
Load direction

Load direction

Load direction is part of the same engineering state used by the diagram and solver.

-90 °
Left cable direction

Left cable direction

Left cable direction is part of the same engineering state used by the diagram and solver.

135 °
Right cable direction

Right cable direction

Right cable direction is part of the same engineering state used by the diagram and solver.

45 °
Engineering model scope

Category

Particle equilibrium

Idealization

Concurrent forces act at an idealized particle; cables carry tension only.

Acceptance check

Require ΣF = 0 and reject negative cable tension or singular geometry.

Interpretation question

Why must the applied load lie inside the positive cone generated by the cable directions?

Model limits

These simulations are rigid, static, small-connection models. They do not include cable self-weight, sag-induced geometric nonlinearity, elastic stretch, dynamic amplification, pulley rotational inertia, bearing friction, rope bending loss, or design-code factors unless explicitly stated.

Particle-Equilibrium Feasibility Workflow

Solve 2D or 3D concurrent-force equilibrium while checking equation independence, unilateral tension constraints, and force residuals.

Particle-Equilibrium Feasibility WorkflowSolve 2D or 3D concurrent-force equilibrium while checking equation independence, unilateral tension constraints, and force residuals.. Isolate the particle or joint → Draw every applied force; cable tensions act away from the particle; Draw every applied force; cable tensions act away from the particle → Convert force directions and cable geometry to unit vectors; Convert force directions and cable geometry to unit vectors → Assemble the independent 2D or 3D equations ΣF = 0; Assemble the independent 2D or 3D equations ΣF = 0 → Coefficient matrix has sufficient independent rank for all unknowns?; Coefficient matrix has sufficient independent rank for all unknowns? — No → Statics alone cannot uniquely determine the assumed unknowns; Coefficient matrix has sufficient independent rank for all unknowns? — Yes → Solve the force-equilibrium system; Solve the force-equilibrium system → Any tension-only element requires T < 0?; Any tension-only element requires T < 0? — Yes → Assumed active cable set is infeasible; revise slack members or geometry; Any tension-only element requires T < 0? — No → Force residual is within tolerance?; Assumed active cable set is infeasible; revise slack members or geometry → Draw every applied force; cable tensions act away from the particle; Force residual is within tolerance? — Yes → Feasible equilibrium: T ≥ 0; T = 0 is a boundary or slack state; Force residual is within tolerance? — No → Correct force directions, geometry, units, or algebra; Correct force directions, geometry, units, or algebra → Draw every applied force; cable tensions act away from the particle

Isolate the particle or joint → Draw every applied force; cable tensions act away from the particle; Draw every applied force; cable tensions act away from the particle → Convert force directions and cable geometry to unit vectors; Convert force directions and cable geometry to unit vectors → Assemble the independent 2D or 3D equations ΣF = 0; Assemble the independent 2D or 3D equations ΣF = 0 → Coefficient matrix has sufficient independent rank for all unknowns?; Coefficient matrix has sufficient independent rank for all unknowns? — No → Statics alone cannot uniquely determine the assumed unknowns; Coefficient matrix has sufficient independent rank for all unknowns? — Yes → Solve the force-equilibrium system; Solve the force-equilibrium system → Any tension-only element requires T < 0?; Any tension-only element requires T < 0? — Yes → Assumed active cable set is infeasible; revise slack members or geometry; Any tension-only element requires T < 0? — No → Force residual is within tolerance?; Assumed active cable set is infeasible; revise slack members or geometry → Draw every applied force; cable tensions act away from the particle; Force residual is within tolerance? — Yes → Feasible equilibrium: T ≥ 0; T = 0 is a boundary or slack state; Force residual is within tolerance? — No → Correct force directions, geometry, units, or algebra; Correct force directions, geometry, units, or algebra → Draw every applied force; cable tensions act away from the particle

  • Isolate the particle or joint: terminator
  • Draw every applied force; cable tensions act away from the particle: process
  • Convert force directions and cable geometry to unit vectors: process
  • Assemble the independent 2D or 3D equations ΣF = 0: process
  • Coefficient matrix has sufficient independent rank for all unknowns?: decision
  • Statics alone cannot uniquely determine the assumed unknowns: terminator
  • Solve the force-equilibrium system: process
  • Any tension-only element requires T < 0?: decision
  • Assumed active cable set is infeasible; revise slack members or geometry: process
  • Force residual is within tolerance?: decision
  • Correct force directions, geometry, units, or algebra: process
  • Feasible equilibrium: T ≥ 0; T = 0 is a boundary or slack state: terminator
Key Takeaways
  • A particle free-body diagram contains concurrent forces only.
  • Cable tensions are solved from the same unit direction vectors shown in the diagram and must remain nonnegative.
  • Matrix and closed-form solutions should agree for the two-cable cases.
  • Pulley mechanical advantage depends on supporting rope segments, not simply pulley count.
  • Equation count alone is insufficient; matrix rank controls uniqueness.
  • A small equilibrium residual verifies the numerical solution.
  • Singular, negative-tension, and impossible configurations must be reported explicitly.