Equilibrium: The State of Perfect Balance
The Two Pillars of Equilibrium
For an object to be in a state of equilibrium, two separate conditions must be satisfied simultaneously. Think of it as a two-part security system; if either part fails, the object will start to move or spin.
1. Translational Equilibrium: The sum of all forces acting on the object is zero. $ \Sigma F = 0 $.
2. Rotational Equilibrium: The sum of all moments (turning effects) acting on the object is zero. $ \Sigma M = 0 $.
Let's explore each of these conditions in more detail.
Translational Equilibrium: No Net Force
When we say the resultant force is zero ($ \Sigma F = 0 $), it means that all the forces pushing and pulling on the object cancel each other out. Imagine a game of tug-of-war where both teams are pulling with exactly the same force. The rope doesn't move in either direction. The forces are balanced.
This condition ensures that the object does not accelerate linearly. If it was at rest, it will remain at rest. If it was moving, it will continue to move in a straight line at a constant speed[1]. For example, a book lying on a table is in translational equilibrium because the gravitational force pulling it down is exactly balanced by the normal force[2] pushing up from the table.
Rotational Equilibrium: No Net Moment
The second condition involves the resultant moment being zero ($ \Sigma M = 0 $). A moment, also called a torque, is the turning effect of a force. It depends on two things:
- The size of the force.
- The perpendicular distance from the pivot point[3] (the point around which rotation happens) to the line of action of the force.
The formula for a moment is $ M = F \times d $, where $ M $ is the moment, $ F $ is the force, and $ d $ is the perpendicular distance.
For an object to be in rotational equilibrium, the sum of all clockwise moments about a pivot must equal the sum of all anticlockwise moments about that same pivot. This means the object will not start to spin or rotate. A seesaw is a perfect example. If two people of different weights want to balance, the heavier person must sit closer to the pivot, and the lighter person must sit farther away, so that their moments are equal and opposite.
Types of Equilibrium in Action
While we often think of equilibrium as a stationary object, it can be classified into different types based on how an object behaves when slightly disturbed.
| Type of Equilibrium | Description | Everyday Example |
|---|---|---|
| Stable | If the object is slightly displaced, it returns to its original position. Its center of gravity[4] is raised when displaced. | A rocking chair. When you rock it, it comes back to its flat, resting position. |
| Unstable | If the object is slightly displaced, it moves farther away from its original position. Its center of gravity is lowered when displaced. | A pencil balanced on its tip. The slightest breeze will make it fall over. |
| Neutral | If the object is displaced, it comes to rest in its new position. Its center of gravity remains at the same height. | A ball resting on a flat, level floor. You can push it to a new spot, and it will just stay there. |
Solving an Equilibrium Problem Step-by-Step
Let's apply what we've learned to solve a classic physics problem: the balanced seesaw.
Scenario: A 3 meter long seesaw is pivoted at its center. A 60 kg person sits 1 meter to the left of the pivot. Where should a 40 kg person sit on the right side to balance the seesaw?
The pivot is at the center of the seesaw.
Force (F) = weight = mass $\times$ gravity = $60 \times 10 = 600$ N. (Using $g \approx 10$ m/s$^2$ for simplicity).
Distance (d) = 1 m.
Moment (M) = $ F \times d = 600 \times 1 = 600$ Nm. This moment acts in the clockwise direction.
For balance, Sum of Clockwise Moments = Sum of Anticlockwise Moments.
So, the moment from the right side must also be 600 Nm, but in the anticlockwise direction.
Force (F) = $40 \times 10 = 400$ N.
Moment (M) = $600$ Nm.
$ M = F \times d $
$ 600 = 400 \times d $
$ d = 600 / 400 = 1.5$ m.
Answer: The 40 kg person must sit 1.5 meters to the right of the pivot for the seesaw to be in equilibrium. Notice how the lighter person has to sit farther away to create the same turning effect as the heavier person.
Common Mistakes and Important Questions
Q: If an object is not moving, does that automatically mean it is in equilibrium?
A: Yes, but with a crucial clarification. If an object is completely at rest (static), it is in static equilibrium, meaning both $ \Sigma F = 0 $ and $ \Sigma M = 0 $. However, an object moving at a constant velocity in a straight line is also in equilibrium (dynamic equilibrium), as there is no net force or net moment causing acceleration.
Q: Can an object be in translational equilibrium but not rotational equilibrium?
A: Absolutely! This is a common point of confusion. Imagine pushing on the top and bottom of a steering wheel with equal and opposite forces. The net force is zero (translational equilibrium), but the wheel will spin because there is a net moment (rotational disequilibrium). Both conditions must be met for complete equilibrium.
Q: When calculating moments, does the pivot point choice matter?
A: For an object to be in complete equilibrium, the condition $ \Sigma M = 0 $ must hold true about any pivot point you choose. It is often smart to choose the pivot at a point where an unknown force acts, as the moment of that force about that pivot will be zero (since the distance $d=0$), simplifying your calculations.
Footnote
[1] Constant Speed: A rate of motion that does not change over time. The object covers equal distances in equal time intervals.
[2] Normal Force (N): The support force exerted upon an object that is in contact with another stable object. It is always perpendicular to the surface.
[3] Pivot Point (Fulcrum): The fixed point about which a lever rotates.
[4] Center of Gravity (CG): The unique point in an object where the total weight of the object is considered to act.