Mathematical Models — Complete Topic
LESSON 01
What Is Mathematical Modelling?
Mechanics is concerned with describing and predicting the behaviour of physical objects. Real objects, however, contain enormous amounts of detail that may be irrelevant to the mathematical problem.
A mathematical model removes selected details and replaces the real situation with an idealised version that can be analysed mathematically.
A mathematical representation of a real physical situation, based on a set of simplifying assumptions.
The modelling cycle
- Identify the real situation. Decide what objects, forces and movements matter.
- Choose an appropriate model. For example, represent a car as a particle.
- State or interpret assumptions. For example, a smooth surface means friction is ignored.
- Apply mathematics. Use equations, vectors, Newton's laws, moments or other mechanics methods.
- Interpret the result physically. Decide what the mathematical answer means for the original physical system.
- Evaluate the model. Ask whether the assumptions are reasonable.
The model is not the physical situation itself. It is an approximation of that situation designed to make the problem mathematically manageable.
Why are assumptions necessary?
A real car has a length, width, height, four tyres, suspension, rotating wheels, air resistance, engine forces, friction and many other properties. A first mechanics model might simply represent the car by one particle moving along a straight road.
This does not mean the car literally has no size. It means that its size is not important for the particular calculation being performed.
LESSON 02
The Particle Model
A particle is an idealised object whose size and shape are ignored. Its mass is treated as being concentrated at a single point.
An object whose dimensions and shape are ignored and whose mass may be treated as concentrated at a point.
What does the particle assumption remove?
- The physical dimensions of the object.
- Its shape.
- Rotational effects caused by its size, when those effects are outside the model.
- The exact location of different parts of the object.
What does the particle model NOT mean?
A particle can still have mass. It can still experience forces, accelerate, move along a surface, be connected to a string and interact with other particles.
"Particle" does not mean "massless". The particle has mass; only its size and shape are ignored.
When is a particle model useful?
Car on a road
If only the car's translational motion matters, its dimensions can often be ignored.
Ball in flight
If rotation and air flow are irrelevant, the ball can be represented by a particle.
Hanging mass
A small mass attached to a string can be treated as a particle when only its motion matters.
When it fails
If the size, shape, rotation or distribution of mass is important, a particle model may no longer be appropriate.
WORKED MODELLING DECISION
A 1500 kg car travels along a straight road. We want to calculate its acceleration under a known resultant force.
Model: Treat the car as a particle.
Reason: Its size and shape are not required when applying Newton's second law to its translational motion.
What is retained? Its mass and the forces acting on it.
LESSON 03
Lamina
A lamina is a thin, flat object. In mechanics it is commonly treated as a two-dimensional body.
A thin, flat body whose thickness is negligible compared with its other dimensions.
The lamina model is useful when the shape and distribution of mass across a flat region matter, but the thickness does not.
Uniform lamina
A uniform lamina has mass distributed evenly over its area. This allows its centre of mass to be located using the symmetry of the shape in appropriate situations.
MODELLING EXAMPLE
A thin rectangular metal plate is modelled as a uniform lamina.
The thickness is ignored, and the mass is assumed to be distributed uniformly across the rectangular area.
Do not confuse a lamina with a particle. A lamina retains its two-dimensional shape; a particle does not.
LESSON 04
Rigid Bodies
A body whose shape is assumed not to change.
A rigid body can still translate and rotate. The important assumption is that distances between points in the body remain fixed.
Why is rigidity useful?
Real materials can bend, stretch, compress or deform. A rigid model ignores these deformations when they are sufficiently small or irrelevant to the problem.
Retained
Shape, geometry, relative distances between points and the overall dimensions of the body.
Ignored
Elastic deformation, bending and stretching of the body.
A rigid body is not the same as a particle. A rigid body retains its shape and dimensions; a particle does not.
LESSON 05
Rods
A rod is an idealised long, thin body. The specification uses the terms light, uniform and non-uniform rod.
Light rod
The mass of the rod is considered negligible.
Therefore its weight is ignored in the mathematical model.
Uniform rod
The mass is distributed uniformly along the length of the rod.
For a uniform rod, the centre of mass is at its midpoint.
Non-uniform rod
The mass is not distributed uniformly along its length.
Therefore the centre of mass need not be at the midpoint.
| Model | Mass distribution | Weight | Centre of mass |
|---|---|---|---|
| Light rod | Mass negligible | Ignored | Usually irrelevant |
| Uniform rod | Even | Included if mass is given | Midpoint |
| Non-uniform rod | Uneven | Included if mass is given | Not necessarily midpoint |
"Uniform" and "light" mean different things. Uniform tells you how mass is distributed. Light tells you that the mass itself is negligible.
LESSON 06
Strings
Strings appear constantly in Mechanics 1 problems involving connected particles.
Light string
The mass of the string is considered negligible.
Because its mass is neglected, the string does not require its own equation of motion based on its weight.
Inextensible string
The string does not stretch, so its length remains constant while the string is taut.
Why does inextensibility matter?
If two particles are connected by a taut inextensible string, movement of one particle constrains the movement of the other. In standard connected-particle models, their speeds have the same magnitude when the string arrangement is fixed.
If one end moves a certain distance, the other end must move by the corresponding amount required to preserve the total string length.
Tension
A string under tension pulls on the objects attached to it. Tension acts along the string and away from the object.
A light string does NOT mean that its tension is zero. "Light" refers to mass, not tension.
When the string becomes slack
The standard relationship between connected particles depends on the string remaining taut. If the string becomes slack, the tension cannot continue acting as though the string were pulling the particles together.
Never blindly apply connected-particle equations after an event that makes the string slack. The physical model has changed.
LESSON 07
Smooth and Rough Surfaces
Smooth surface
Friction is ignored.
A smooth surface can still exert a normal reaction force. Smooth does not mean "no contact force".
Rough surface
Friction is allowed to act between the object and the surface.
A rough surface does not automatically mean that friction has a particular fixed direction. Friction opposes the relative motion or tendency of relative motion between the surfaces.
| Surface | Friction? | Normal reaction? |
|---|---|---|
| Smooth | Ignored | Yes, if there is contact |
| Rough | May act | Yes, if there is contact |
"Smooth" does not mean that the reaction force disappears. It means the frictional component is ignored.
LESSON 08
Light Smooth Pulley
A pulley is used to change the direction of a string and can connect the motion of two particles.
The pulley is modelled as having negligible mass and negligible friction at its contact with the string.
Consequence for tension
For a light, smooth pulley with a light inextensible string, the tension is treated as the same throughout the continuous string in the standard model.
This does not mean the particles have the same mass or the same forces. It means the ideal string-pulley model gives one common tension magnitude.
Why can't we normally treat the whole pulley system as one particle?
The connected particles can move in different directions. Their individual equations of motion must therefore normally be considered separately.
MODELLING EXAMPLE
Particles P and Q are attached to opposite ends of a light inextensible string passing over a small smooth pulley.
The string has negligible mass, does not stretch, and the pulley is treated as frictionless.
If the string remains taut, the particles have accelerations of equal magnitude in the corresponding directions.
The tension magnitude is common to both particles.
LESSON 09
Bead, Wire and Peg
Bead
A small particle-like object that can move along a wire.
In a typical model, the bead is treated as a particle while the wire determines the path available to it.
Wire
A thin structure that constrains the movement of a bead or other particle to a particular path.
Peg
A fixed point around which a string can change direction.
When a string passes around a smooth peg in the standard ideal model, the peg changes the direction of the string while the idealised tension remains governed by the assumptions of the string and contact.
If a question says a bead moves along a wire, the wire is usually acting as a constraint. If a string passes over or around a peg, pay attention to the direction of the tension on each attached object.
LESSON 10
Gravity and Weight in Models
Near the Earth's surface, the weight of a particle of mass m is modelled as acting vertically downward with magnitude mg.
In Edexcel mechanics questions, unless another value is given, use the specified value of g from the question or the convention required by the examination.
What does a model neglect?
Many introductory mechanics models neglect air resistance. This means that, for an object moving vertically under gravity, the resultant vertical force may be taken to be its weight alone.
MODELLING EXAMPLE
A ball is projected vertically upwards and is modelled as a particle. Air resistance is neglected.
The ball continues to have downward acceleration due to gravity throughout its flight.
At the highest point, its instantaneous velocity is zero, but its acceleration is still downward.
At maximum height, velocity is zero. Acceleration is NOT zero if gravity is still acting.
LESSON 11
Reading Modelling Assumptions Like an Examiner
In mechanics, words such as "light", "smooth", "uniform" and "inextensible" are mathematical instructions. Each word changes the model.
| Wording | Mathematical interpretation | What you ignore / assume |
|---|---|---|
| Particle | Point-like object | Size and shape |
| Lamina | Thin flat body | Thickness |
| Rigid body | Fixed shape | Deformation |
| Light rod | Negligible mass | Rod's weight |
| Uniform rod | Even mass distribution | Uneven density |
| Inextensible string | Fixed length | Stretching |
| Smooth surface | No friction | Frictional force |
| Rough surface | Friction can act | Nothing automatically |
| Light smooth pulley | Ideal pulley | Pulley mass and friction |
| Bead | Particle constrained to a wire | Bead dimensions |
A useful distinction
Geometry assumptions
Particle, lamina, rigid body, rod and wire determine how the physical geometry is represented.
Mass assumptions
Light, uniform and non-uniform determine how mass is treated or distributed.
Contact assumptions
Smooth and rough determine whether friction is included.
Constraint assumptions
Inextensible strings, wires and pegs constrain motion.
LESSON 12
Model Limitations
A model is useful only when its assumptions are reasonable for the question being asked.
Example: car as a particle
For calculating acceleration along a straight road, ignoring the car's dimensions may be reasonable.
For calculating whether the car can fit through a narrow gap, the particle model would be inappropriate because the dimensions suddenly matter.
Example: light string
Ignoring the mass of a string may be reasonable if its mass is tiny compared with the attached particles.
If the string itself has substantial mass, treating it as light can produce an unrealistic tension distribution.
Example: smooth surface
Ignoring friction can simplify the problem dramatically. However, a real surface may exert friction, so the model is an approximation.
When evaluating a model, explain what assumption has been made and what physical effect has been ignored. Do not simply say "the model is unrealistic."
LESSON 13
Exam Technique: Decode the Model First
Before writing an equation, translate the wording of the question into mathematical consequences.
- Identify every object. Decide whether it is a particle, rod, lamina, bead or another model.
- Identify mass assumptions. Look for light, uniform or non-uniform.
- Identify connection assumptions. Look for strings, wires, pegs and pulleys.
- Identify surface assumptions. Decide whether friction is included or ignored.
- Draw the forces that the model requires. Do not add forces that the model explicitly ignores.
- Check the geometry. A constraint such as an inextensible string may connect the motions of different particles.
- Only then calculate.
Do not add friction simply because two objects are touching. If the surface is stated to be smooth, friction is excluded from the model.
LESSON 14
Worked Modelling Examples
EXAMPLE 01 · PARTICLE
A 900 kg van moves along a straight road. It is modelled as a particle. Explain the meaning of this model.
The van's size and shape are ignored. Its mass is retained because mass is required when considering its dynamics.
EXAMPLE 02 · LIGHT STRING
A particle is connected to another particle by a light string. What property of the string is being neglected?
The mass of the string is considered negligible.
Its tension is not zero merely because it is light.
EXAMPLE 03 · INEXTENSIBLE
Two particles are connected by an inextensible string. Particle P moves 0.8 m toward a pulley. What does the inextensibility condition tell us?
The total string length remains constant. Therefore the movement of P must be matched by the corresponding movement of the other end, provided the string remains taut and the geometry of the system is unchanged.
EXAMPLE 04 · SMOOTH SURFACE
A 5 kg particle rests on a smooth horizontal table. What horizontal contact force does the table provide?
There is no horizontal frictional force because the table is smooth.
The normal reaction is vertical, so it does not provide a horizontal component.
EXAMPLE 05 · UNIFORM ROD
A uniform rod has mass 6 kg and length 2 m. Where is its centre of mass?
Because the rod is uniform, its centre of mass is at its midpoint.
Therefore it is 1 m from either end.
EXAMPLE 06 · LIGHT ROD
A rod is described as light. What happens to its weight in the model?
The rod's mass is considered negligible, so its weight is ignored.
EXAMPLE 07 · LIGHT SMOOTH PULLEY
Two particles are connected by a light inextensible string over a small smooth pulley. What standard tension assumption is made?
The tension is treated as having the same magnitude throughout the light string.
EXAMPLE 08 · RIGID BODY
A metal beam is modelled as a rigid body. What physical behaviour is ignored?
Deformation of the beam is ignored. Its shape and distances between its points are assumed to remain fixed.
EXAMPLE 09 · MODEL EVALUATION
A tennis ball is modelled as a particle while its flight through the air is studied. Give one limitation of this model.
The model ignores the ball's size and rotational behaviour. It may also be combined with an assumption that air resistance is negligible.
EXAMPLE 10 · COMPLETE MODEL
A particle P is attached to a particle Q by a light inextensible string passing over a small smooth pulley. P is on a smooth horizontal table and Q hangs vertically. List the modelling consequences.
Particles: sizes and shapes are ignored.
Light string: string mass is negligible.
Inextensible: string length is constant while taut, linking the particles' motion.
Smooth pulley: friction at the pulley is ignored and the ideal tension is common through the string.
Smooth table: friction on P is ignored.
Hanging Q: its weight acts vertically downward.
LESSON 15
Targeted Practice
SHOW ANSWER
The size and shape of the object.
SHOW ANSWER
Yes. The particle model ignores dimensions and shape, not mass.
SHOW ANSWER
A thin, flat body whose thickness is negligible.
SHOW ANSWER
The body does not deform; its shape remains fixed.
SHOW ANSWER
The mass of the string is negligible.
SHOW ANSWER
The string does not stretch, so its length remains constant while taut.
SHOW ANSWER
Friction is ignored.
SHOW ANSWER
Yes. Smooth means no friction, not no contact force.
SHOW ANSWER
At its midpoint.
SHOW ANSWER
No. Light means negligible mass. A light string can have significant tension.
If you can explain every word in the following phrase, you have understood the core modelling language: "two particles are connected by a light, inextensible string passing over a small smooth pulley."
MASTER SUMMARY
The Modelling Atlas
Never memorise modelling words as vocabulary alone. Memorise their mathematical consequences.
100-Question Modelling Exam
A complete mixed assessment covering definitions, assumptions, consequences, modelling decisions, exam traps and applied scenarios.