2.1 Equilibrium | Vectors | Physics 11

In physics, equilibrium is the state of an object in which all the forces and torques acting upon it are balanced. This results in no change in the object's motion — it will either remain at rest or continue to move at a constant velocity.

Types of Equilibrium

There are two primary types of mechanical equilibrium:

Static Equilibrium: An object is in static equilibrium when it is completely at rest ( $v = 0$ , $ω = 0$ ). Both conditions of equilibrium are satisfied. For example, a book resting on a table is in static equilibrium because the downward gravitational force is perfectly balanced by the upward normal force.
Dynamic Equilibrium: An object is in dynamic equilibrium when it moves at a constant velocity (zero acceleration, $a = 0$ , $α = 0$ ). A classic example is a parachutist who has reached terminal velocity — the upward air resistance equals the downward gravitational force, so they fall at a constant speed.

Type	State of Motion	Net Force	Example
Static	At Rest ( $v = 0$ )	$\sum F = 0$	A book lying on a table
Dynamic	Constant Velocity ( $a = 0$ )	$\sum F = 0$	A car at steady speed on a straight road

Conditions for Equilibrium

For an object to be in complete equilibrium, two conditions must be satisfied simultaneously.

First Condition of Equilibrium (Translational Equilibrium)

The vector sum of all external forces acting on the object must be zero. This ensures the object has no linear acceleration.

$\sum F = 0$

In component form:

$\sum F_{x} = 0 \sum F_{y} = 0$

Meeting this condition prevents linear acceleration but does not prevent rotation. See also Rectangular Components Of A Vector→ for resolving forces into components.

Second Condition of Equilibrium (Rotational Equilibrium)

The net external torque acting on the object about any pivot point must be zero. This ensures the object has no angular acceleration.

Torque ( $τ$ ) is the rotational equivalent of force — it represents the turning effect of a force about an axis: $τ = r \times F$ .

$\sum τ = 0$

Sign Convention: Counter-clockwise (CCW) torques are positive; clockwise (CW) torques are negative.

When $\sum τ = 0$ , the total clockwise torque equals the total counter-clockwise torque, and the object does not change its rate of rotation.

Couple

A couple consists of two forces that are:

Equal in magnitude
Opposite in direction
Parallel but with different lines of action

A couple produces a pure torque (rotation) without any net translational force: $\sum F = 0 but \sum τ \neq = 0$

Complete Equilibrium

An object is in complete equilibrium only when both conditions are satisfied simultaneously:

Condition	Expression	Effect
First (Translational)	$\sum F = 0$	No linear acceleration
Second (Rotational)	$\sum τ = 0$	No angular acceleration

Types of Equilibrium

There are two primary types of mechanical equilibrium:

Static Equilibrium: An object is in static equilibrium when it is completely at rest ( $v = 0$ , $ω = 0$ ). Both conditions of equilibrium are satisfied. For example, a book resting on a table is in static equilibrium because the downward gravitational force is perfectly balanced by the upward normal force.
Dynamic Equilibrium: An object is in dynamic equilibrium when it moves at a constant velocity (zero acceleration, $a = 0$ , $α = 0$ ). A classic example is a parachutist who has reached terminal velocity — the upward air resistance equals the downward gravitational force, so they fall at a constant speed.

Type	State of Motion	Net Force	Example
Static	At Rest ( $v = 0$ )	$\sum F = 0$	A book lying on a table
Dynamic	Constant Velocity ( $a = 0$ )	$\sum F = 0$	A car at steady speed on a straight road

Conditions for Equilibrium

For an object to be in complete equilibrium, two conditions must be satisfied simultaneously.

First Condition of Equilibrium (Translational Equilibrium)

The vector sum of all external forces acting on the object must be zero. This ensures the object has no linear acceleration.

$\sum F = 0$

In component form:

$\sum F_{x} = 0 \sum F_{y} = 0$

Meeting this condition prevents linear acceleration but does not prevent rotation. See also Rectangular Components Of A Vector→ for resolving forces into components.

Second Condition of Equilibrium (Rotational Equilibrium)

The net external torque acting on the object about any pivot point must be zero. This ensures the object has no angular acceleration.

Torque ( $τ$ ) is the rotational equivalent of force — it represents the turning effect of a force about an axis: $τ = r \times F$ .

$\sum τ = 0$

Sign Convention: Counter-clockwise (CCW) torques are positive; clockwise (CW) torques are negative.

When $\sum τ = 0$ , the total clockwise torque equals the total counter-clockwise torque, and the object does not change its rate of rotation.

Couple

A couple consists of two forces that are:

Equal in magnitude
Opposite in direction
Parallel but with different lines of action

A couple produces a pure torque (rotation) without any net translational force: $\sum F = 0 but \sum τ \neq = 0$

Complete Equilibrium

An object is in complete equilibrium only when both conditions are satisfied simultaneously:

Condition	Expression	Effect
First (Translational)	$\sum F = 0$	No linear acceleration
Second (Rotational)	$\sum τ = 0$	No angular acceleration