When Is a Particle in Equilibrium?

In physics, a particle is a simplified model of a body - it has mass, but no size or shape. To tell if a particle is in equilibrium, all we need to do is ask: does it stay completely still over time? If it doesn't move or shift in space, it's in equilibrium.

The rule is simple: all the forces must cancel out

A particle is in static equilibrium when the total force acting on it is zero. That’s both a necessary and sufficient condition. Mathematically, it looks like this:

$$ \vec{R} = \sum \vec{F}_i = 0 $$

In other words, every force acting on the particle must be balanced out by one or more opposing forces, so that the resultant vector is zero.

No matter how many forces are at play, they must all add up vectorially to zero:

$$ \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \dots + \vec{F}_n = 0 $$

Example

Think of a cup of milk sitting on a table. It doesn't move - so it's in equilibrium.

Diagram showing a cup in static equilibrium on a table: weight and normal force cancel out

Two main forces act on the cup:

The gravitational force $ \vec{P} $, pulling it downward toward the Earth, with magnitude $ P = mg $.
The normal force $ \vec{N} $, from the table pushing upward with equal strength.

These two forces balance each other out perfectly:

$$ \vec{P} + \vec{N} = 0 $$

That’s why the cup doesn’t move - it's in equilibrium.

Which forces matter? When analyzing equilibrium, focus only on the actual forces acting on the object. The most common are the weight force ($ \vec{P} $, always downward) and any reaction forces ($ \vec{R}_v $), which depend on the type of surface or support (table, wall, string...) keeping the object in place.

Here’s another example.

Picture a chandelier hanging from the ceiling.

Diagram showing a chandelier in vertical equilibrium: tension and weight are balanced

In this case, two forces act:

The weight force $ \vec{P} $ pulling downward,
The tension $ \vec{T} $ in the cable pulling upward.

If those two forces are equal in magnitude and opposite in direction, the chandelier stays still - it’s in equilibrium.

What are constraints? A constraint is anything that limits an object’s motion - keeping it from falling, sliding, or rotating. Each constraint generates a reaction force that pushes back when the object tries to move in a forbidden direction.

Now, what if the surface is sloped?

When an object is on an inclined plane, the reaction from the surface splits into two parts:

A normal force $ \vec{N} $ perpendicular to the plane, which balances the part of the weight acting into the surface ($ \vec{P_y} $).
A parallel component $ \vec{P_x} $ pulling down the slope, which can be resisted by static friction $ \vec{S} $.

The object won’t slide if friction $ S $ is strong enough to cancel out $ P_x $, and the normal force $ N $ supports the perpendicular weight $ P_y $.

Diagram showing a box in equilibrium on a slope: weight components, friction, and normal force

So, the object is in equilibrium if the net force is zero in both directions: along the slope ($ P_x + S = 0 $) and perpendicular to it ($ N + P_y = 0 $).