Mathematics (9660) | Mathematical modelling

Welcome to Mathematical Modelling in Mechanics!

Hi there! This chapter might seem a bit different from calculating forces or using SUVAT equations, but it is incredibly important. Mathematical Modelling is the bridge between the messy, complicated real world and the clean, organized world of pure mathematics.

In Mechanics (M2), we use models to simplify complex physical situations so that we can apply mathematical rules (like Newton's Laws) to find solutions. You will learn how to make smart assumptions and how to comment on whether your simplified model is actually useful.

Why is this important? It teaches you critical thinking. In an exam, it's not enough just to calculate the answer; you must also understand why you were allowed to use that calculation in the first place!

Section 1: The Modelling Process

A Mathematical Model is a description of a real-world situation using mathematical concepts and language. Since the real world has infinite complexity (air resistance, bumpy surfaces, slight wind changes), we must make simplifying assumptions.

The Mathematical Modelling Cycle (The 4 Steps)

Think of modelling as creating a simplified map. You don't need every tree and pothole on the map, just the main roads and landmarks.

Step 1: Simplify and Formulate (Real World to Mathematics)

• Identify the core problem (e.g., how fast does this ball land?).
• Make simplifying assumptions (e.g., treat the ball as a particle, ignore air resistance).
• Translate the problem into mathematical equations (e.g., using \(F=ma\) or SUVAT equations).

Step 2: Analyse and Solve (Doing the Maths)

• Use mathematical techniques (like differentiation, integration, algebra, or SUVAT) to solve the equations derived in Step 1.
• This results in a mathematical solution (e.g., the time of flight is \(t=2.5\) seconds).

Step 3: Interpret (Mathematics to Real World)

• Translate the mathematical solution back into the context of the real-world problem.
• Your answer should make sense physically (e.g., "The ball will hit the ground after 2.5 seconds.")

Step 4: Refine and Validate (Checking the Model)

• Does the answer seem reasonable? Check the validity of your model.
• If the real-world observations don't match the model's predictions, the model is inaccurate.
• You must then refine the model by changing one or more initial assumptions (e.g., "The model was inaccurate, so we should now include air resistance").

Section 2: The Key Assumptions in Mechanics (M2.1)

When solving problems in Mechanics, you will frequently use specific vocabulary that tells you exactly what assumptions have been made. Knowing these terms is essential for both setting up the equations and commenting on the model's validity.

1. Simplifying the Object: Mass vs. Size

Assumption: Particle or Point Mass

Meaning: The object has mass, but its physical size and shape are ignored. Its mass is concentrated at a single point.

Implications (What you ignore):

• Air resistance (as size and shape are often key factors in drag).
• Rotation or tumbling of the object.
• The forces can all be treated as acting through the centre of mass.

Example: When studying the trajectory of a bullet, treating it as a particle simplifies the motion significantly.

Assumption: Light (or Massless)

Meaning: The object (usually a string, rod, or pulley) has negligible mass compared to the bodies it connects.

Implications (What you ignore):

• You don't need to include the weight (\(W = mg\)) of the object in force calculations.
• It simplifies connected particle problems greatly.

Assumption: Rigid Body

Meaning: The object has mass and physical size (unlike a particle), but it does not bend or change shape when forces are applied.

(This assumption is typically used in more advanced M2 topics like Moments and Centres of Mass when the size matters.)

2. Simplifying Connections: Strings and Rods

Assumption: Inextensible String

Meaning: The string connecting two objects cannot stretch.

Implications (The golden rule):

• Both objects connected by the string must have the same speed and the same acceleration.
• The Tension in the string is uniform throughout its entire length.

Assumption: Light/Smooth Pulley

Meaning: The pulley has negligible mass and there is no friction when the string passes over it.

Implications:

• The tension in the string remains constant on both sides of the pulley.

If the pulley were rough or heavy, the tensions on either side would be different!

3. Simplifying the Environment: Surfaces and Forces

Assumption: Smooth Surface or Smooth Contact

Meaning: There is no friction between the object and the surface.

Implications:

• The frictional force \(F\) is zero.
• Only the Normal Reaction force (\(R\)) acts perpendicular to the surface.

Assumption: Rough Surface

Meaning: Friction exists and resists motion.

Implications:

• The frictional force \(F\) must satisfy the inequality: \(F \le \mu R\).
• If the body is on the point of moving (Limiting Friction), then \(F = \mu R\). \(\mu\) is the coefficient of friction.

Assumption: Motion under Gravity Only

Meaning: The only force acting on the object (apart from contact forces) is gravity.

Implications:

• The acceleration is constant and equal to \(g\) (usually taken as \(9.8 \text{ ms}^{-2}\)).
• Crucially, we are ignoring Air Resistance (Drag), which is often a large simplification.

Section 3: Interpretation and Validation of Models (M2.1)

Once you have used your model (your simplified equations) to find an answer, the final, crucial stage is to assess its usefulness.

1. Why Do Models Fail? (Common Mistakes in Reality)

A real-world result often deviates from a mathematical result because reality does not follow our simple assumptions. When commenting on why a model might be inaccurate, you need to point out the assumptions you made that are most likely false.

Common source of error:

• If you assumed a particle, you ignored air resistance, which is always present in reality.
• If you assumed a smooth surface, you ignored friction.
• If you assumed an inextensible string, the string might have stretched slightly in reality.

Don't worry if this seems tricky at first! In the exam, usually, the question will guide you towards one or two specific assumptions to comment on.

2. Improving the Model (Refining Assumptions)

If your initial model proved invalid (i.e., its predictions were significantly wrong), you need to make the model more complex to better reflect reality. This is called Refining the Model.

• To account for friction: Replace the "smooth surface" assumption with "rough surface" and use \(F = \mu R\).
• To account for air resistance: Replace the "particle" assumption or "motion under gravity only" assumption by introducing a resistive force \(R\) (often modelled as proportional to speed or speed squared, \(R=kv\) or \(R=kv^2\)). (Note: This usually requires advanced calculus techniques beyond AS level).
• To account for stretching: Replace the "inextensible string" assumption with an "elastic string" assumption.

Did you know? The trajectory calculations for launching satellites or rockets must use extremely sophisticated models that account for variable gravity, atmospheric drag, and even the rotation of the Earth. A simple SUVAT model would be useless!

Key Takeaway Summary

Mathematical modelling in Mechanics is the skill of translating a complex physical problem into solvable equations by making sensible simplifications (assumptions), solving the mathematics, and then critically interpreting and validating the results against reality. Master the definitions of particle, light, smooth, and inextensible, as these are the tools you use to simplify the world.