AcademyApplying Force Models
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Equilibrium with Newton's First Law

Level 1 - Physics topic page in Applying Force Models.

Principle

Equilibrium means the vector sum of external forces on the chosen particle is zero.

Notation

\(\sum\vec F\)
resultant external force
\(\mathrm{N}\)
\(F_x,F_y\)
force components
\(\mathrm{N}\)
\(T\)
tension magnitude
\(\mathrm{N}\)
\(N\)
normal reaction magnitude
\(\mathrm{N}\)
\(W\)
weight magnitude
\(\mathrm{N}\)
\(\theta\)
angle from the chosen horizontal axis
rad or deg

Method

Derivation 1: Start from Newton's second law

Equilibrium is the zero-acceleration case of the particle model. The forces may be nonzero, but their resultant must vanish.

Second law
\[\sum\vec F=m\vec a\]
Equilibrium condition
\[\vec a=\vec 0\Rightarrow\sum\vec F=\vec 0\]
Component form
\[\sum F_x=0,\qquad \sum F_y=0\]
In three dimensions, add \(\sum F_z=0\).

Derivation 2: Resolve angled forces

Once axes are chosen, each angled force contributes a signed component to each balance equation.

Resolve a force
\[F_x=F\cos\theta,\qquad F_y=F\sin\theta\]
Insert signed components
\[\sum F_x=F_{1x}+F_{2x}+\cdots=0\]
Interpret signs
\[F_{\mathrm{solved}}<0\Rightarrow\text{assumed direction was opposite}\]

The free-body diagram should contain only forces acting on the particle. The two tensions pull along their strings, and weight acts downward.

PT1T2W
The force arrows do not vanish; their vector sum does.

Use the diagram to decide signs before solving. A force pointing left contributes a negative x-component if right is positive; a force pointing downward contributes a negative y-component if upward is positive.

Rules

These are the compact results from the method above.

Vector equilibrium
\[\sum \vec F=\vec 0\]
Horizontal balance
\[\sum F_x=0\]
Vertical balance
\[\sum F_y=0\]
Weight
\[W=mg\]

Examples

Question
A
\[120\,\mathrm{N}\]
sign hangs from two identical ropes, each
\[40^\circ\]
above horizontal. Find the rope tension.
Answer
The horizontal components cancel by symmetry. Vertical balance gives
\[2T\sin40^\circ=120\]
so
\[T=\frac{120}{2\sin40^\circ}=93.3\,\mathrm{N}\]

Checks

  • Equilibrium is a vector statement, not just vertical balance.
  • Components must use one sign convention throughout.
  • A normal reaction is perpendicular to the contact surface.
  • Tension pulls away from the object along the rope.
  • A negative solved force means the assumed direction was reversed.

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Free-Body Diagrams

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Particle Dynamics