Unit 3 · Topic 01

Particles in equilibrium

Translational equilibrium does not mean that «nothing happens»: it means that the velocity vector does not change. Solving an equilibrium problem requires choosing the system, representing every external force, and checking that their vector sum is zero.

Unit 3EquilibriumOpen navigation

Concept 01

Equilibrium condition

Essential The minimum you should retain

An object is in translational equilibrium when its acceleration is zero. In an inertial frame this requires zero net force. The object may be at rest or move in a straight line at constant velocity.

UnderstandInterpret and connect

Zero net force does not mean that no forces exist. Several forces may act and balance one another. The condition is vectorial: if any component of net force is nonzero, there is acceleration along that component.

DeepenFormulation and conditions

The condition is ΣF→ext=0 \sum\vec F_{ext}=0 . In Cartesian coordinates this means ΣFx=0 \sum F_x=0 , ΣFy=0 \sum F_y=0 and, when needed, ΣF_z = 0. These are component equations of one vector balance.

ExploreConnections for further study

Equilibrium depends on the chosen system and reference frame. The same scene can be analysed with different systems, but each choice changes which forces are external and which unknowns appear. The goal is not to memorize equations but to build a consistent balance.

Mathematical relation

Vector condition for translational equilibrium

ΣF→ext=0→ \sum\vec F_{ext}=0
Represents

The zero-translational-acceleration condition in an inertial frame.

Physical interpretation

Rest and straight-line motion at constant velocity are both compatible.

DeepenVariables, conditions, and checks

Variables

ΣF⃗_ext
sum of external forces on the system; usual unit: N

Conditions of application

  • The frame is inertial.
  • The system boundary is declared.

Dimensional check

Every term has force dimensions.

Errors it helps prevent

  • Confusing equilibrium with rest.
  • Concluding that no force acts.

Mathematical relation

Cartesian component equilibrium

ΣFx=0,ΣFy=0,ΣFz=0 \sum F_x=0,\quad\sum F_y=0,\quad\sum F_z=0
Represents

Cartesian projections of one vector equilibrium condition.

Physical interpretation

Every independent component must vanish.

DeepenVariables, conditions, and checks

Variables

ΣF_i
component i of net force; usual unit: N

Conditions of application

  • All forces use the same Cartesian basis.
  • Signs are declared.

Dimensional check

Each equation sums forces in newtons.

Errors it helps prevent

  • Checking only one component.
Three force vectors form a closed triangle representing zero vector sum.F₁F₂F₃closed polygon: ΣF=0

The three forces are placed head to tail and close the polygon. Closure represents zero resultant; there is no fourth interaction.

Concept 02

A strategy for equilibrium

Essential The minimum you should retain

Before calculating, isolate the body or system, draw its FBD, choose axes, and name each force by its agent. Then write one equilibrium equation for each necessary direction.

UnderstandInterpret and connect

Choose axes that simplify components. On an incline, for example, axes parallel and perpendicular to the surface can avoid decomposing the normal force.

DeepenFormulation and conditions

Compare the number of useful equations with the number of unknowns. Missing equations often indicate a geometric constraint or an additional physical relation. Extra «forces» in an FBD often reveal that a non-force quantity has been included.

ExploreConnections for further study

A strong solution also checks signs, dimensions, and limiting cases. If an expression predicts a negative tension or a normal force incompatible with contact, the assumed physical regime may have changed.

Concept 03

Equilibrium with angled forces

Essential The minimum you should retain

When a force is angled relative to the axes, its effect along each direction is described by components. Equilibrium requires positive and negative components to balance separately.

UnderstandInterpret and connect

The correct sine or cosine depends on the axis from which the angle is measured. Before writing a component, identify geometrically which projection lies along the axis and what sign it has.

DeepenFormulation and conditions

A force of magnitude F at angle θ from +x has components F_x = F cosθ and F_y = F sinθ. Equilibrium equations can then determine unknown tensions, applied forces, or reactions.

ExploreConnections for further study

Vector diagrams can predict the structure of the answer before algebra. For two symmetric ropes holding a load, horizontal components must cancel. Using that symmetry reduces unnecessary calculation.

Worked example

Lamp supported by two symmetric cables

A 120 N lamp hangs in equilibrium from two identical cables, each 35° above the horizontal.

Given
  • Lamp weight: 120 N
  • Angle of each cable: 35° above horizontal
Target

Find the tension in each cable.

  1. System and FBD

    The lamp is the system: weight acts downward and the two tensions follow the cables.

  2. Symmetry

    The horizontal components of tension cancel.

  3. Vertical balance

    The equilibrium condition is 2T sin35° - 120 = 0.

  4. Calculation

    T = 120/(2 sin35°) ≈ 104.6 N.

Conclusion

Each cable carries about 105 N of tension. The tension is less than 120 N because the two cables share the vertical load.

A central lamp has two symmetric angled tensions and its downward weight.left Tright Tweight

Tensions follow the cables. Their horizontal components cancel and their vertical components support the weight, without revealing T numerically.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Confusing equilibrium with rest.

Translational equilibrium means a=0; velocity may be constant and nonzero.

Assuming ΣF=0 means no forces act.

Nonzero forces can balance vectorially.

Checking only one component and declaring equilibrium.

Every independent net-force component must be zero.