Mechanics of particles and systems Definition

Mechanical equilibrium

Builds on Rigid body Torque Couple Generalized coordinates

Definition

Let a rigid body, or a particle, be acted on by forces $F_1, \dots, F_k \in \R^3$ at points $p_1, \dots, p_k \in \R^3$ . The body is in mechanical equilibrium if both conditions hold:

the net force vanishes, $\sum_{i=1}^{k} F_i = 0$ ;
the net torque about a reference point $O$ vanishes, $\sum_{i=1}^{k} (p_i - O) \times F_i = 0$ .

For a particle all $p_i$ are equal, and condition 2 follows from condition 1.

Remarks

When condition 1 holds, the net torque is the same about every point, so condition 2 does not depend on the choice of $O$ .

Example: a weightless beam along the $x$ -axis carries the downward force $(0, 0, -300)$ at $x = -2$ and $(0, 0, -200)$ at $x = 3$ , and rests on a support at $x = 0$ that pushes up with $(0, 0, 500)$ . The net force is $0$ , and the net torque about $0$ is $(0, -600, 0)$ from the first load plus $(0, 600, 0)$ from the second, which is $0$ .

Non-example: take the couple formed by $(0, 5, 0)$ at $(0.2, 0, 0)$ and $(0, -5, 0)$ at $(-0.2, 0, 0)$ . It satisfies condition 1 but has net torque $(0, 0, 2)$ , so a body subject only to it is not in equilibrium.

For a system with generalized coordinates $q$ and potential energy $V(q)$ , a configuration $q_0$ is an equilibrium configuration when $\nabla V(q_0) = 0$ , that is, all generalized forces vanish there.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Couple concept
Holonomic constraint concept
Rigid body concept
Smooth manifold concept
Torque concept
Configuration space concept
Generalized coordinates concept
Mechanical equilibrium definition

Tag	`5C4JYQZKXF`
URI	`m2://5C4JYQZKXF`
Kind	definition
Writing	W0 stub
MSC	70C20
Defines	Mechanical equilibrium
Uses	Rigid body, Torque, Couple, Generalized coordinates
Changed	3 November 2026

Mechanical equilibrium

Definition

Remarks

Reading path

Further reading

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