Energy in Systems
4. Stability and Equilibrium
Learning outcomes
- I can define equilibrium and describe the conditions required for equilibrium.
- I can distinguish between stable, unstable, and neutral equilibrium.
- I can explain how the centre of mass affects stability.
- I can predict how changes in shape and mass distribution influence stability.
- I can apply equilibrium concepts to real-world structures and systems.
Stability and Equilibrium
Equilibrium describes a situation in which the forces and turning effects acting on an object are balanced.
An object in equilibrium has no acceleration. This means it can either:
- remain stationary, or
- move at constant velocity.
For an object to be in complete mechanical equilibrium, two conditions must be satisfied:
resultant force = 0 N
and
resultant moment = 0 N·m
The first condition prevents linear acceleration. The second prevents angular acceleration.
Translational Equilibrium
Translational equilibrium occurs when all the forces acting on an object balance.
In one dimension:
ΣF = 0
Consider a book resting on a table.
Two main forces act on the book:
- weight acts downward
- normal force acts upward
If:
normal force = weight
then:
resultant force = 0 N
The book does not accelerate.
Equilibrium Does Not Necessarily Mean Stationary
An object does not have to be stationary to be in translational equilibrium.
Suppose a car travels along a straight road at constant velocity.
If the driving force is:
800 N forward
and the total resistive force is:
800 N backward
then:
resultant force = 800 − 800
resultant force = 0 N
The car continues at constant velocity.
This is an example of dynamic equilibrium.
Rotational Equilibrium
An object can have balanced forces but still rotate if the forces produce an unbalanced turning effect.
The turning effect of a force is called a moment or torque.
For a force acting perpendicular to the distance from a pivot:
moment = force × perpendicular distance from pivot
or:
M = Fd
Units:
N·m
For rotational equilibrium:
total clockwise moments = total anticlockwise moments
or:
ΣM = 0
Worked Example: Balanced Beam
A child weighing 300 N sits:
2 m
from the pivot of a seesaw.
Anticlockwise moment:
M = Fd
M = 300 × 2
M = 600 N·m
For equilibrium, the other side must produce:
600 N·m
of clockwise moment.
If another child weighs 400 N:
600 = 400 × d
d = 1.5 m
The second child must sit 1.5 m from the pivot.
Complete Mechanical Equilibrium
For complete equilibrium:
ΣF = 0
and:
ΣM = 0
An object can therefore fail to be in equilibrium in two different ways.
It might:
- accelerate linearly because the forces are unbalanced
- rotate because the moments are unbalanced
Both conditions must be checked.
Centre of Mass
The centre of mass is the point at which the mass of an object can be considered to be concentrated when analyzing its motion.
For a uniform, symmetrical object, the centre of mass is usually near its geometric centre.
Examples include:
- uniform sphere → centre
- uniform rectangular block → geometric centre
- uniform disc → centre
For irregular objects, the centre of mass may not be located at the geometric centre.
It can even lie outside the physical material of an object.
Centre of Gravity
The centre of gravity is the point through which the object's weight can be considered to act.
Near Earth's surface, where gravitational field strength is approximately uniform:
centre of gravity ≈ centre of mass
For most everyday stability problems, the terms can therefore be treated as being at approximately the same position.
The Line of Action of Weight
Weight acts vertically downward through the centre of mass.
Imagine drawing a vertical line downward from the centre of mass.
This is the line of action of the weight.
Whether this line falls inside or outside the object's base of support is extremely important for stability.
Base of Support
The base of support is the area enclosed by the points where an object contacts its supporting surface.
For example:
- a person standing with feet apart has a wider base
- a table's legs define its base
- a vehicle's tyres define its base
- a building's foundation provides its base
Generally:
wider base → greater stability
When Does an Object Tip?
An object remains stable while the vertical line through its centre of mass falls inside its base of support.
As the object tilts, its centre of mass moves relative to the base.
At the tipping point, the line of action passes through the edge of the base.
If the line moves beyond the base:
weight produces a turning moment
and the object tips.
This provides a powerful rule:
line of action inside base → object can remain upright
line of action outside base → object tips
Stable Equilibrium
An object is in stable equilibrium if a small displacement causes it to tend to return to its original position.
Imagine a ball at the bottom of a bowl.
If the ball is moved slightly upward:
- its centre of mass rises
- its gravitational potential energy increases
- gravity causes it to roll back toward the bottom
The original position is therefore stable.
Stable Equilibrium and Energy
Stable equilibrium corresponds to a local minimum in gravitational potential energy.
Recall:
Eₚ = mgh
A small displacement from a stable position raises the centre of mass.
This increases gravitational potential energy.
The system therefore tends to return toward the lower-energy position.
Examples include:
- ball in a bowl
- pendulum hanging vertically downward
- rocking chair near its resting position
- weighted toy that rights itself
Unstable Equilibrium
An object is in unstable equilibrium if a small displacement causes it to move farther away from its original position.
Consider a ball balanced at the top of a hill.
If moved slightly:
- its centre of mass lowers
- gravitational potential energy decreases
- it continues moving away from its original position
The ball does not return to the top.
The original position is unstable.
Unstable Equilibrium and Energy
Unstable equilibrium corresponds to a local maximum in gravitational potential energy.
A small movement allows the system to move toward a lower-energy state.
Examples include:
- ball balanced on top of a hill
- pencil balanced vertically on its point
- object balanced precariously on an edge
- inverted pendulum
Neutral Equilibrium
An object is in neutral equilibrium if a small displacement does not cause it to return or move farther away.
A ball on a flat horizontal surface is a common example.
When the ball moves sideways:
- the height of its centre of mass remains approximately constant
- gravitational potential energy remains approximately constant
The ball can remain in its new position.
Examples include:
- ball on a level floor
- cylinder lying on its side on a flat surface
- wheel resting on level ground
Comparing the Three Types
| Type | After Small Displacement | Centre of Mass | Potential Energy |
|---|---|---|---|
| Stable | Returns toward original position | Rises when displaced | Minimum at equilibrium |
| Unstable | Moves farther away | Lowers when displaced | Maximum at equilibrium |
| Neutral | Remains in new position | Same height | Approximately unchanged |
Stability and Centre of Mass
The position of the centre of mass strongly influences stability.
Generally:
lower centre of mass → greater stability
A low centre of mass makes it more difficult for the line of action of the object's weight to move outside the base.
This is why many stable objects are designed to be:
- wide
- low
- heavier near the bottom
Worked Example: Two Objects
Consider two objects with identical bases.
Object A has its centre of mass:
0.5 m above the ground
Object B has its centre of mass:
1.5 m above the ground
Object B generally requires a smaller tilt before its line of action moves outside the base.
Therefore:
Object A is more stable.
Lowering the Centre of Mass
Athletes often lower their centre of mass to improve stability.
Examples include:
- wrestlers bending their knees
- rugby players crouching
- basketball defenders using a low stance
- skiers bending their legs
- martial artists using wide, low stances
Lowering the body makes it harder for external forces to tip the person.
Increasing the Base of Support
A wider base generally increases stability.
Compare standing with:
feet together
and:
feet far apart
With a wider stance, the centre of mass can move farther sideways before its line of action leaves the base of support.
This is why:
- cranes use outriggers
- ladders spread at the bottom
- tripods have widely spaced legs
- athletes widen their stance
- heavy equipment may use wide tracks
Shape and Stability
Changing an object's shape can alter:
- base width
- centre-of-mass position
- mass distribution
A tall, narrow object is generally less stable than a short, wide object of similar mass.
For example:
A tall narrow cabinet may tip relatively easily.
A low wide cabinet is usually harder to tip.
This principle is important in furniture and building design.
Mass Distribution
It is not only the total mass that matters.
Where the mass is located is extremely important.
Consider two objects with the same dimensions and total mass.
Object A has most of its mass near the bottom.
Object B has most of its mass near the top.
Object A has a lower centre of mass and is generally more stable.
Therefore:
lower mass distribution → lower centre of mass → greater stability
Stability of Vehicles
Vehicle designers must carefully control the centre of mass.
A low centre of mass improves resistance to rollover.
This is especially important for:
- cars
- buses
- trucks
- SUVs
- racing vehicles
Tall vehicles generally have a higher centre of mass and can be more vulnerable to tipping during sharp turns.
Vehicle Loading
Where cargo is placed affects vehicle stability.
Heavy cargo placed high above the ground raises the combined centre of mass.
Heavy cargo placed lower:
- lowers the centre of mass
- improves stability
- reduces rollover risk
This is why heavy objects should generally be loaded low whenever practical and secured properly.
Stability During Turning
When a vehicle turns, it requires a resultant force toward the centre of the curved path.
If the vehicle has:
- a high centre of mass
- a narrow wheelbase
- high speed
- a sharp turn
the risk of rollover can increase.
Designers therefore consider both:
centre-of-mass height
and:
base width
when designing vehicles.
Racing Cars
Racing cars are designed with extremely low centres of mass.
Features may include:
- engines mounted low
- drivers sitting close to the ground
- wide wheel spacing
- heavy components positioned low
- low vehicle bodies
These features improve stability when cornering at high speeds.
Cranes and Stability
Cranes must lift heavy loads far from their central support.
This produces large turning moments.
The moment produced by a load is:
M = Fd
where:
- F = load force
- d = perpendicular distance from pivot
As the load moves farther from the crane:
distance increases → moment increases
This can make the crane unstable.
Crane Counterweights
Cranes often use heavy counterweights.
Suppose a load produces:
200,000 N·m
of clockwise moment.
The counterweight must provide a sufficient anticlockwise moment to maintain equilibrium.
For equilibrium:
clockwise moments = anticlockwise moments
Counterweights help prevent the crane from rotating or tipping.
Crane Outriggers
Mobile cranes often extend outriggers before lifting.
Outriggers increase the effective base of support.
This allows the line of action of the combined weight to remain within the base over a greater range of loading conditions.
Therefore:
larger base → greater resistance to tipping
Worked Example: Crane
A crane lifts a load weighing:
20,000 N
at a horizontal distance of:
5 m
from the pivot.
Moment caused by the load:
M = Fd
M = 20,000 × 5
M = 100,000 N·m
A counterweight acts:
2 m
from the opposite side of the pivot.
For rotational equilibrium:
F × 2 = 100,000
F = 50,000 N
The counterweight must provide 50,000 N under this simplified model.
Buildings and Stability
Buildings must remain stable despite forces produced by:
- gravity
- wind
- earthquakes
- moving occupants
- equipment
Engineers can improve stability using:
- wide foundations
- carefully distributed mass
- structural bracing
- reinforced cores
- flexible structural elements
- tuned mass dampers
The exact approach depends on the building and environmental conditions.
Tall Buildings
Tall buildings naturally have high centres of mass.
Engineers cannot simply make every skyscraper short and wide, so other techniques are used.
These may include:
- deep foundations
- wide lower structures
- strong structural cores
- cross-bracing
- aerodynamic shaping
- damping systems
Stability therefore involves both static equilibrium and the response of structures to changing forces.
Ships and Stability
Ships must remain stable while floating.
The distribution of mass affects the position of the ship's centre of mass.
Placing heavy equipment too high can reduce stability.
Heavy components are therefore often positioned low.
Ballast may also be used to:
- lower the centre of mass
- improve stability
- control the ship's orientation
Ship stability also involves buoyancy, making the full analysis more complex than stability on a solid surface.
Stability of Furniture
Furniture design provides familiar examples.
A bookshelf is more likely to tip if:
- it is tall
- its base is narrow
- heavy objects are stored near the top
Stability can be improved by:
- widening the base
- placing heavy objects on lower shelves
- reducing height
- securing furniture to a wall
The physics is the same:
keep the centre of mass low and its line of action within the base.
Human Stability
Humans continuously adjust their bodies to maintain equilibrium.
When standing upright, the line of action of our weight must remain within our base of support.
If you lean forward, you can remain standing only until your centre of mass moves too far.
You may then:
- step forward
- move another body part backward
- fall forward
Taking a step creates a new base of support, allowing equilibrium to be restored.
Carrying a Load
Suppose you carry a heavy backpack.
The backpack moves the combined centre of mass of you and the backpack backward.
You may naturally lean forward.
Why?
Leaning forward shifts your body mass so the combined centre of mass remains over your base of support.
This is an example of the body automatically adjusting to maintain equilibrium.
Carrying a Heavy Object in Front
If you hold a heavy box in front of you, the combined centre of mass moves forward.
You may lean backward to compensate.
Again, the objective is to keep the line of action of the combined weight within the base formed by your feet.
Stability and Potential Energy
The three equilibrium types can be understood using gravitational potential energy.
Stable Equilibrium
Small displacement causes:
Eₚ to increase
The object tends to return.
Unstable Equilibrium
Small displacement causes:
Eₚ to decrease
The object moves farther away.
Neutral Equilibrium
Small displacement causes:
Eₚ to remain approximately constant
The object remains in its new position.
This provides an energy-based explanation for stability.
Worked Example: Tipping
A rectangular object has a base width of:
0.80 m
Its centre of mass is directly above the centre of the base.
The centre of mass is therefore initially:
0.40 m
horizontally from either edge.
As the object tilts, the line of action of its weight moves toward one edge.
At the critical tipping position, the line of action passes exactly through that edge.
Beyond this point, weight produces a moment that rotates the object farther toward the ground.
This is the tipping threshold.
Stability Versus Balance
An object can be balanced but still be extremely unstable.
A pencil standing perfectly upright on its point could theoretically have:
ΣF = 0
and:
ΣM = 0
at one instant.
However, an extremely small disturbance causes it to fall.
Therefore:
equilibrium describes whether forces and moments are balanced
while:
stability describes what happens after a small disturbance.
This distinction is important.
Stable Does Not Mean Impossible to Tip
A stable object can still be tipped if a sufficiently large force or moment is applied.
"Stable" means that after a small displacement, the object tends to return toward equilibrium.
It does not mean that the object can never fall.
Improving Stability
In many systems, stability can be increased by:
Lowering the Centre of Mass
Place heavy components closer to the ground.
Widening the Base
Increase the area over which the object is supported.
Redistributing Mass
Move heavy components downward or toward the centre.
Adding Counterweights
Balance large turning moments.
Using Outriggers
Increase the effective base of support.
Reducing Height
A shorter structure often has a lower centre of mass.
Securing the Structure
Anchors and supports can prevent tipping.
Stability and Engineering Trade-Offs
Maximum stability is not always the only design goal.
Engineers may also need to consider:
- mass
- cost
- mobility
- size
- speed
- aerodynamic drag
- available materials
- usability
For example, making a vehicle extremely wide could improve stability but make it impractical for ordinary roads.
Engineering therefore requires balancing competing requirements.
Multi-Step Example
A simplified crane has a 15,000 N load positioned:
4 m
to the right of the pivot.
Clockwise moment:
M = 15,000 × 4
M = 60,000 N·m
A counterweight of:
20,000 N
is positioned on the left.
For equilibrium:
20,000 × d = 60,000
Therefore:
d = 3 m
The counterweight must act 3 m from the pivot to balance the load under these simplified conditions.
If the load moves farther outward to 6 m:
New moment:
M = 15,000 × 6
M = 90,000 N·m
The existing counterweight at 3 m produces only:
20,000 × 3 = 60,000 N·m
The moments are no longer balanced.
This demonstrates why extending a crane's load farther outward can dramatically affect stability.
Common Mistakes
Thinking Equilibrium Means Stationary
An object moving at constant velocity can also be in equilibrium.
The requirement is:
resultant force = 0
not:
velocity = 0
Checking Forces but Ignoring Moments
Complete equilibrium requires:
ΣF = 0
and:
ΣM = 0
Balanced forces alone do not guarantee rotational equilibrium.
Assuming the Centre of Mass Must Be Inside the Material
The centre of mass can lie outside the physical material of some objects.
Thinking a Heavy Object Is Always More Stable
Total mass alone does not determine stability.
Mass distribution is extremely important.
Thinking a Lower Centre of Mass Makes Tipping Easier
The opposite is generally true.
A lower centre of mass usually increases stability.
Thinking a Narrow Base Is More Stable
A wider base generally provides greater stability.
Confusing Equilibrium and Stability
Equilibrium describes balanced forces and moments.
Stability describes what happens after a small displacement.
Assuming Stable Means Impossible to Tip
A stable object can still tip if subjected to a sufficiently large force or displacement.
Check Your Understanding
- Define equilibrium.
- What must the resultant force be for translational equilibrium?
- What must the resultant moment be for rotational equilibrium?
- State the two conditions required for complete mechanical equilibrium.
- Can a moving object be in equilibrium? Explain.
- A car experiences 900 N driving force and 900 N resistance. Describe its motion if it is already moving.
- Define moment.
- State the equation used to calculate a moment.
- A 300 N force acts 2 m from a pivot. Calculate the moment.
- A 500 N load produces a moment of 1000 N·m. Calculate its perpendicular distance from the pivot.
- Define centre of mass.
- Explain the relationship between centre of mass and centre of gravity near Earth's surface.
- What is the base of support?
- Explain how the line of action of weight determines whether an object tips.
- Define stable equilibrium.
- Define unstable equilibrium.
- Define neutral equilibrium.
- Give one example of each type of equilibrium.
- Explain stable equilibrium using gravitational potential energy.
- Explain unstable equilibrium using gravitational potential energy.
- Why does lowering the centre of mass generally increase stability?
- Why does widening the base generally increase stability?
- Explain why heavy cargo should generally be placed low in a vehicle.
- Explain why racing cars are designed with low centres of mass.
- Why do cranes use outriggers?
- Explain the purpose of a crane counterweight.
- A crane lifts a 10,000 N load 6 m from its pivot. Calculate the moment produced by the load.
- A counterweight acts 3 m from the opposite side of the pivot. Calculate the counterweight force required for equilibrium in Question 27.
- Explain why a person carrying a heavy backpack may lean forward.
- Explain the difference between equilibrium and stability.
Key Terms
Equilibrium — Condition in which the resultant force and resultant moment on an object are zero.
Translational equilibrium — Condition in which the resultant force is zero.
Rotational equilibrium — Condition in which the resultant moment is zero.
Static equilibrium — Equilibrium in which an object remains stationary.
Dynamic equilibrium — Equilibrium in which an object moves at constant velocity.
Resultant force — Single force representing the combined effect of all forces acting on an object.
Moment — Turning effect produced by a force about a pivot.
Torque — Another term for the turning effect of a force.
Pivot — Point around which an object can rotate.
Centre of mass — Point at which an object's mass can be considered concentrated for analysis.
Centre of gravity — Point through which the resultant gravitational force on an object acts.
Line of action — Imaginary line extending through the direction in which a force acts.
Base of support — Area enclosed by the contact points supporting an object.
Stable equilibrium — Equilibrium in which a small displacement produces a tendency to return to the original position.
Unstable equilibrium — Equilibrium in which a small displacement causes movement farther from the original position.
Neutral equilibrium — Equilibrium in which a displaced object can remain in its new position.
Counterweight — Mass positioned to produce a moment opposing another turning effect.
Outrigger — Extended support used to increase the effective base of a machine such as a crane.
Tipping point — Condition at which the line of action of weight reaches the edge of the base of support.
Mass distribution — Arrangement of mass throughout an object.
Key Takeaways
- Equilibrium occurs when forces and turning effects are balanced.
- Translational equilibrium requires ΣF = 0.
- Rotational equilibrium requires ΣM = 0.
- Complete mechanical equilibrium requires both conditions.
- An object in equilibrium does not accelerate.
- An object can be stationary or move at constant velocity while in equilibrium.
- Moments describe the turning effects of forces.
- For perpendicular forces, M = Fd.
- The centre of mass represents the average position of an object's mass.
- Near Earth's surface, the centre of gravity and centre of mass are usually approximately the same location.
- Weight acts vertically downward through the centre of mass.
- An object generally remains upright while the line of action of its weight stays within its base of support.
- An object reaches its tipping point when this line reaches the edge of the base.
- Stable equilibrium causes an object to return after a small displacement.
- Unstable equilibrium causes an object to move farther away after a small displacement.
- Neutral equilibrium allows an object to remain in its new position.
- Stable equilibrium corresponds to a minimum in gravitational potential energy.
- Unstable equilibrium corresponds to a maximum in gravitational potential energy.
- A lower centre of mass generally increases stability.
- A wider base of support generally increases stability.
- Mass distribution can be more important to stability than total mass.
- Vehicles, cranes, buildings, ships, furniture, and the human body all rely on the same fundamental stability principles.
- Counterweights balance moments.
- Outriggers increase the effective base of support.
- Equilibrium and stability are related but different concepts.
The central stability rule is:
Keep the line of action of the weight inside the base of support.
And the central equilibrium conditions are:
ΣF = 0
ΣM = 0