KAHIBARO
Table of Contents
Balance of Forces
Equilibrium is the state in which an object has no acceleration. In the context of Newton's first law, this means the object either remains at rest or continues moving with constant velocity in a straight line. The key idea is not that there are no forces, but that the forces cancel each other.
If the total force on an object is zero, the object is in translational equilibrium. Mathematically,
$$
\sum \vec{F} = 0
$$
This vector equation means that the forces balance in every direction. In one dimension, this becomes
$$
\sum F_x = 0
$$
In two dimensions, both component conditions must be true:
$$
\sum F_x = 0, \qquad \sum F_y = 0
$$
An object is in equilibrium when the net force is zero:
$$
\sum \vec{F} = 0
$$
This does not mean that each force is zero. It means all the forces together add up to zero.
Rest and Constant Velocity
There are two common situations of equilibrium. A book resting on a table is in equilibrium because its weight downward is balanced by the table's normal force upward. A car moving at constant speed in a straight line can also be in equilibrium if the driving force forward is balanced by resistive forces backward.
This shows an important point. Equilibrium is not the same as being at rest. Rest is only one special case. Constant velocity motion also satisfies equilibrium because acceleration is zero.
Balanced Forces in One Dimension
In one dimension, equilibrium is often simple to check. Suppose two opposite forces act on a box, one to the right and one to the left. If their magnitudes are equal, the net force is zero.
$$
F_{\text{right}} - F_{\text{left}} = 0
$$
So,
$$
F_{\text{right}} = F_{\text{left}}
$$
If they are not equal, the object is not in equilibrium and it accelerates in the direction of the larger net force.
Balanced Forces in Two Dimensions
In two dimensions, equilibrium requires balance along each axis separately. A hanging sign supported by two ropes is a common example. The upward components of the rope tensions must balance the weight, and the horizontal components must cancel each other.
This can be written as
$$
\sum F_x = 0
$$
and
$$
\sum F_y = 0
$$
An object may fail to be in equilibrium even if forces balance in one direction, if they do not balance in another direction.
For equilibrium in two dimensions, both conditions must hold:
$$
\sum F_x = 0
$$
$$
\sum F_y = 0
$$
Balancing forces in only one direction is not enough.
Common Examples
A lamp hanging from the ceiling is in equilibrium when the upward tension equals the downward weight.
$$
T = mg
$$
A box on a floor is in equilibrium when the normal force balances the weight.
$$
N = mg
$$
A sled pulled horizontally at constant velocity is in equilibrium when the pulling force equals the friction force.
$$
F_{\text{pull}} = f
$$
These examples show that equilibrium often means identifying opposite effects and seeing how they cancel.
Visualizing Equilibrium
A simple force diagram helps make equilibrium clear. Equal and opposite forces produce zero net force.
In this drawing, the upward normal force and downward weight have equal magnitude, so the box does not accelerate vertically.
Equilibrium as a Test
Equilibrium is often used as a condition for solving problems. If a problem states that an object is at rest, suspended, balanced, or moving with constant velocity, then you can immediately use
$$
\sum \vec{F} = 0
$$
This allows unknown forces to be found from known ones.
The following table summarizes common equilibrium situations.
| Situation | Condition | Meaning |
|---|---|---|
| Object at rest | $\sum \vec{F} = 0$ | No acceleration |
| Object moving at constant velocity | $\sum \vec{F} = 0$ | No change in velocity |
| Vertical balance | $\sum F_y = 0$ | Upward and downward forces cancel |
| Horizontal balance | $\sum F_x = 0$ | Leftward and rightward forces cancel |
A Simple Worked Idea
Imagine a 5 kg object resting on a table. Its weight is
$$
W = mg = 5 \times 9.8 = 49 \text{ N}
$$
Since the object is at rest, it is in equilibrium, so the upward normal force must be 49 N as well.
$$
N = 49 \text{ N}
$$
The net force is then
$$
\sum F_y = N - mg = 49 - 49 = 0
$$
So the object remains at rest.
Final Rule
Equilibrium is one of the most important ideas in mechanics because it connects motion and force in a very direct way. Whenever acceleration is zero, the net force must be zero. Whenever the net force is zero, the object stays at rest or keeps moving with constant velocity.
Equilibrium means zero acceleration, not necessarily zero motion.
$$
\sum \vec{F} = 0 \iff \vec{a} = 0
$$
An object in equilibrium can be resting, or it can be moving in a straight line with constant velocity.