What Is a Rigid Body?

When physicists study the motion of real objects, they often simplify them using models. One particularly useful model is the rigid body.

A rigid body is an idealized object whose shape and size do not change, even when forces act on it. Real objects can always deform at least slightly, but many objects are stiff enough that we can treat them as rigid for a particular problem.

This approximation allows us to study not only how an object moves from place to place, but also how it rotates, balances, tips, and responds to forces applied at different positions.

Learning Targets

• I can define a rigid body and explain the assumptions of rigid body mechanics.
• I can distinguish between particle models and rigid body models.
• I can identify examples of rigid bodies in everyday life.
• I can explain why rigid body approximations are useful in physics.
• I can recognize situations where objects cannot be treated as rigid bodies.

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What Is a Rigid Body?

A rigid body is an idealized object in which the distance between any two points in the object remains constant.

In simpler terms:

A rigid body keeps the same shape and size while it moves or rotates.

Imagine marking two points, A and B, on a solid metal bar.

If the bar is treated as a rigid body, the distance between A and B never changes.

The bar might:

• Move left or right.
• Move upward or downward.
• Rotate.
• Accelerate.
• Fall.
• Balance.
• Experience several forces.

But its shape and dimensions are assumed to remain unchanged.

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The Rigid Body Assumption

A perfectly rigid object does not actually exist.

All real materials can:

• Stretch
• Compress
• Bend
• Twist
• Vibrate
• Break

However, these changes may be so small that they do not significantly affect the problem we are studying.

For example, a steel beam bends slightly when supporting a load. If that bending is extremely small compared with the dimensions of the beam, we may still model the beam as a rigid body.

This is called a rigid body approximation.

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Why Physicists Use Models

Real objects are extremely complicated.

Consider a car moving along a road.

The car contains:

• Thousands of individual parts.
• Flexible tyres.
• A suspension system.
• Vibrating engine components.
• Moving wheels.
• Deforming materials.
• Air moving around the vehicle.

Trying to include every detail would make even a simple calculation extremely difficult.

Instead, physicists choose a model that includes the features important to the problem and ignores features that have little effect on the answer.

A rigid body is one such model.

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The Main Assumptions of Rigid Body Mechanics

When an object is treated as a rigid body, we make several assumptions.

Shape remains constant

The object does not significantly bend, stretch or compress.

Size remains constant

Its dimensions do not change.

Distances within the object remain constant

The distance between any two points within the object remains fixed.

The object can translate

The entire object can move from one location to another.

The object can rotate

The object can turn about an axis or point.

These assumptions make it possible to study the overall motion of an object without analyzing the motion of every atom within it.

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Translational Motion

Translation occurs when an object moves from one position to another.

For example, imagine a box sliding across a floor.

Every part of the box moves in the same general direction.

The box may move:

Left → Right

without changing its orientation.

This is translational motion.

Other examples include:

• An elevator moving upward.
• A train travelling along a straight track.
• A book sliding across a table.
• A crate being pushed across the floor.

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Rotational Motion

Rigid bodies can also rotate.

Rotation occurs when an object turns around an axis.

Examples include:

• A door rotating around its hinges.
• A wheel rotating around its axle.
• A wrench turning a bolt.
• A seesaw rotating around its pivot.
• A fan blade rotating around its centre.

During rotation, different points on the object move along different paths.

This is one reason a rigid body model contains more information than a simple particle model.

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Translation and Rotation Together

Many real objects experience translation and rotation at the same time.

Consider a bicycle wheel rolling along a road.

The centre of the wheel moves forward.

At the same time, the wheel rotates around its centre.

Therefore:

Wheel motion = Translation + Rotation

The rigid body model allows us to analyze both types of motion.

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What Is a Particle Model?

Another common simplification is the particle model.

In a particle model, we imagine that all of an object's mass is concentrated at a single point.

The object's:

• Size
• Shape
• Orientation
• Rotation

are ignored.

This can make motion problems much simpler.

For example, when calculating the path of a ball thrown through the air, we might treat the entire ball as a single particle located at its centre of mass.

We do not need to consider the exact size or orientation of the ball if these features do not affect the problem.

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Particle Model vs Rigid Body Model

The choice of model depends on what we need to investigate.

Neither model is automatically "better."

The correct model is the simplest model that still includes the physics needed for the problem.

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Example — A Moving Car

Suppose we want to calculate how long a car takes to travel 100 m.

The exact shape of the car probably does not matter.

We can model the car as a particle.

But suppose we want to determine whether the car will tip while turning sharply.

Now the:

• Width of the car
• Height of its centre of mass
• Positions of the wheels
• Forces acting on the tyres

become important.

A particle model is no longer sufficient.

We need a rigid body model.

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Why the Position of a Force Matters

Imagine pushing a door.

If you push close to the hinges, the door is difficult to rotate.

If you push near the handle, the door rotates much more easily.

The force might be the same in both cases.

What changed?

The position where the force was applied.

A particle model cannot properly represent this difference because it treats the object as a single point.

A rigid body model preserves the object's dimensions, allowing us to study the turning effect of forces.

This turning effect is called a moment or torque.

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Centre of Mass

Although a rigid body has size and shape, it is often useful to identify a special point called its centre of mass.

The centre of mass is the point at which we can consider the object's mass to be concentrated when analyzing its overall translational motion.

For a uniform, symmetrical object, the centre of mass is often near its geometric centre.

Examples include:

• Centre of a uniform ball
• Centre of a uniform rectangular block
• Centre of a uniform disc

For an irregular object, the centre of mass may not be at its geometric centre.

It can even lie outside the physical material of some objects, such as a ring.

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Everyday Examples of Approximate Rigid Bodies

Many objects can be treated as rigid bodies under normal conditions.

Examples include:

A book

A book resting on a desk does not significantly change shape.

A door

A door rotates about its hinges while approximately maintaining its shape.

A bicycle wheel

The wheel rotates while its overall dimensions remain approximately constant.

A wrench

A wrench transfers force to a bolt while remaining nearly rigid.

A seesaw

A seesaw rotates around a pivot without significant deformation.

A ladder

A ladder leaning against a wall can often be analyzed as a rigid body.

A table

A table supporting ordinary loads can usually be treated as rigid.

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Worked Example — A Ladder

Imagine a ladder leaning against a wall.

We want to determine whether the ladder will remain in equilibrium.

Several forces may act:

• Weight of the ladder
• Normal force from the ground
• Normal force from the wall
• Friction at the ground

The positions of these forces matter.

The ladder's length also matters because forces acting at different distances can produce different turning effects.

Therefore, treating the ladder as a particle would remove important information.

A rigid body model is much more appropriate.

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Worked Example — A Falling Ball

Now consider a ball falling vertically.

Suppose we only want to calculate its:

• Acceleration
• Velocity
• Distance fallen
• Time of fall

The size and rotation of the ball may not matter.

We can therefore treat the ball as a particle.

However, if we want to analyze how the ball rolls, spins or rotates after hitting the ground, the rigid body model becomes more useful.

The appropriate model depends on the question being asked.

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When Is the Rigid Body Approximation Useful?

The rigid body approximation works well when deformation is small enough to ignore.

For example:

A wooden ruler lying on a table can often be treated as rigid.

A steel beam under a relatively small load may be treated as rigid.

A wheel rotating around an axle can often be treated as rigid.

A door rotating around its hinges can usually be treated as rigid.

The approximation simplifies the problem while preserving important information about:

• Size
• Shape
• Rotation
• Centre of mass
• Force positions
• Torque
• Equilibrium

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When Is an Object Not Rigid?

Some objects deform so much that the rigid body approximation becomes inappropriate.

Examples include:

• A stretched elastic band
• A compressed spring
• A bending diving board
• A bouncing rubber ball during impact
• A flexible rope
• A soft sponge being compressed
• A trampoline
• A car body during a collision

In these situations, changes in shape are an important part of the physics.

Ignoring them would remove information needed to understand what is happening.

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Example — A Spring

A spring is a particularly clear example.

If we are simply moving a spring from one location to another without changing its shape, we might sometimes approximate it as rigid.

But if we are studying:

• Stretching
• Compression
• Hooke's Law
• Elastic potential energy

then treating the spring as rigid would make no sense.

The deformation itself is what we are studying.

Therefore, the correct model depends not only on the object but also on the situation.

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Example — Car Collision

During normal driving, a car body can often be approximated as rigid for basic motion calculations.

During a collision, however, the car may deform significantly.

Crumple zones are deliberately designed to deform.

If we were studying the collision and passenger safety, treating the entire car as perfectly rigid would ignore an essential part of the physics.

So:

Normal car motion → rigid body approximation may be useful

Car deformation during collision → rigid body approximation may be inappropriate

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Example — Bridge

A bridge may appear completely rigid when cars travel across it.

In reality, bridges:

• Bend slightly
• Expand when heated
• Contract when cooled
• Vibrate
• Respond to wind and traffic

For some calculations, these changes are so small that the bridge can be approximated as rigid.

For structural engineering, however, deformation can be extremely important.

Again, the model depends on what we are trying to understand.

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Rigid Bodies and Equilibrium

Rigid body mechanics becomes especially important when studying equilibrium.

An object is in equilibrium when it has no overall acceleration.

For a rigid body to remain in equilibrium, we must consider both:

• The overall forces acting on it
• The turning effects of those forces

For example, a balanced seesaw may have downward forces on both sides and an upward force at the pivot.

Simply knowing the forces is not enough.

We must also know where the forces act.

This is one of the central ideas of rigid body mechanics.

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Rigid Bodies and Free-Body Diagrams

A free-body diagram represents the external forces acting on an object.

For a particle model, these forces can often be drawn as if they all act at one point.

For a rigid body, the positions of the forces may matter.

Consider a beam supported at both ends.

Its weight acts through its centre of mass, while support forces act at different positions.

Those different positions determine whether the forces produce clockwise or anticlockwise turning effects.

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Choosing the Correct Model

A useful question is:

Does the object's size, shape, rotation or deformation matter?

If none of these matter:

Particle model may be sufficient.

If size, shape or rotation matters but deformation does not:

Rigid body model may be appropriate.

If deformation itself matters:

A deformable-body model is needed.

We can summarize this as:

Position only → Particle

Position + size + rotation → Rigid body

Position + size + rotation + significant deformation → Deformable body

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Worked Example — Choosing a Model

Consider a football in several situations.

Football flying through the air

If we only need its trajectory:

Particle model

Football spinning through the air

If rotation matters:

Rigid body model

Football being compressed when kicked

If we are studying the deformation:

Deformable body model

The same physical object can therefore require different models for different problems.

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Why Rigid Body Models Are Useful

Rigid body mechanics provides a useful balance between simplicity and realism.

A particle model may be too simple because it removes size and shape.

A complete model of a real object may be far too complicated.

The rigid body model sits between these extremes.

It allows us to analyze:

• Translation
• Rotation
• Torque
• Equilibrium
• Centre of mass
• Stability
• Tipping
• Rolling

without needing to model every tiny deformation within the object.

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Common Mistakes

Mistake: Thinking rigid means completely stationary

A rigid body can move and rotate.

"Rigid" means its shape and dimensions remain approximately constant, not that it cannot move.

Mistake: Thinking rigid bodies actually exist

Perfect rigidity is an idealization.

Every real object deforms at least slightly.

Mistake: Assuming an object is always either rigid or non-rigid

The appropriate model depends on the situation.

A car can be treated as rigid during ordinary motion but not necessarily during a collision.

Mistake: Treating a rigid body as a particle

A particle has no meaningful size or orientation.

A rigid body retains its size and shape.

Mistake: Ignoring where a force acts

For rigid bodies, the location of a force can determine its turning effect.

Mistake: Treating a spring as rigid while studying its extension

If deformation is central to the problem, a rigid body model is inappropriate.

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Key Terms

Rigid body – an idealized object whose shape and dimensions remain constant.

Rigid body approximation – treating a real object as rigid when its deformation is small enough to ignore.

Particle model – representing an object as a single point with mass.

Translation – movement of an object from one position to another.

Rotation – turning of an object around an axis.

Deformation – a change in an object's shape or dimensions.

Centre of mass – the point representing the average position of an object's mass.

Torque – the turning effect of a force.

Equilibrium – a condition in which an object has no overall linear or angular acceleration.

Deformable body – an object whose changes in shape or dimensions must be considered.

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Check Your Understanding

1. What is meant by a rigid body?

2. Does a rigid body have to remain stationary? Explain.

3. Why is a perfectly rigid object an idealization?

4. Give three everyday objects that could often be approximated as rigid bodies.

5. Explain one important difference between a particle model and a rigid body model.

6. Why might a falling ball be treated as a particle in one problem but as a rigid body in another?

7. Why is the rigid body model useful when studying a door opening?

8. Explain why a spring should not be treated as rigid when studying Hooke's Law.

9. A car is travelling along a straight road. When might a particle model be sufficient?

10. The same car crashes into a barrier. Why might the rigid body approximation now be inappropriate?

11. Why does the position where a force acts matter when studying a rigid body?

12. Decide whether a particle, rigid body or deformable-body model would be most suitable for studying a diving board bending under a diver. Explain your choice.

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Key Takeaways

• A rigid body is an idealized object whose shape and dimensions remain constant.
• The distance between any two points in a rigid body is assumed to remain constant.
• A rigid body can translate, rotate, or do both at the same time.
• Perfectly rigid objects do not exist, but many real objects can be approximated as rigid.
• A particle model ignores an object's size, shape and rotation.
• A rigid body model preserves size and shape, allowing rotation and torque to be analyzed.
• The position where a force acts can be important when studying rigid bodies.
• Doors, wheels, beams, ladders and tools can often be approximated as rigid bodies.
• Springs, elastic bands, flexible ropes and objects undergoing collisions may require deformation to be considered.
• The best model depends on what aspect of the object's behaviour we are trying to study.
• Rigid body mechanics provides the foundation for studying torque, equilibrium, centre of mass, stability and rotational motion.

Did You Know?

An object does not have to be made from a particularly hard material to be treated as a rigid body. Even Earth can sometimes be approximated as a rigid body when studying its overall rotation. In other situations—such as studying earthquakes, tides, or seismic waves—Earth's deformation becomes important, and the rigid body approximation is no longer sufficient.