Further Mathematics | Mathematical models in mechanics

Welcome to Mathematical Models in Mechanics!

Hello and welcome to the first essential chapter of M1: Mechanics 1! Don't worry if 'Further Mathematics' sounds intimidating; this chapter is all about common sense and setting the stage for the calculations to come.

In real life, calculating the path of a bouncing tennis ball involves millions of variables (wind speed, ball spin, humidity, air resistance, etc.). That's too complicated! This chapter teaches you how to simplify the real world into a mathematical model so we can actually solve problems using standard equations like Newton's Laws.

Our goal here is to understand the assumptions we make and why they are necessary.

What is a Mathematical Model?

A Mathematical Model is a description of a real-world situation using mathematical concepts and language. We use models in mechanics to predict motion, forces, and equilibrium.

The Trade-Off: Simplicity vs. Accuracy

When we create a model, we are constantly making a trade-off:

• Simplification: This makes the math easier. We ignore minor forces (like air resistance).
• Accuracy: The more detail we include, the closer our model is to reality, but the math becomes much harder (often impossible without powerful computers).

In M1, we favor simplicity. We use a set of standard assumptions that allow us to use the standard formulas you will learn (like SUVAT and \(F = ma\)).

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Section 1: Modeling Objects

How we treat an object mathematically depends entirely on what we need to calculate. Do we care about its size, shape, or rotation?

1. The Particle Model

This is the most common and powerful simplification in M1.

Key Assumption: The object's mass is concentrated at a single point, and its size and rotational effects are ignored.

Think of it this way: when you look at a plane flying high in the sky, you don't see its wings or windows; you just see a dot. That dot is how we model a particle.

When is the Particle Model Appropriate?

• When the object's size is negligible compared to the distance it travels. (e.g., modeling Earth orbiting the Sun).
• When all forces acting on the object act through a single point.
• When we are not interested in rotation, air resistance, or the stability/shape of the object.

Example: Modeling a cricket ball thrown across a field. We treat it as a particle because its size relative to the field length is small, and we usually don't care about its spin (rotation) until we get to a more advanced level.

2. The Rigid Body Model

When an object's size does matter (e.g., it is resting on an inclined plane, or we are calculating the effect of forces applied at different points), we cannot use the particle model.

Key Assumption: The object has size and shape, but it maintains that shape perfectly. It is non-deformable (it cannot be squashed, stretched, or bent).

What does this allow us to do?

If an object is modeled as a rigid body, we can consider forces acting at different points and the effect of gravity acting at the centre of mass. However, we avoid the complexity of elasticity and internal forces.

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Section 2: Modeling Connections and Forces

In mechanics problems, objects are often connected (by strings) or moving on surfaces. We make crucial assumptions about these connections and the environment.

3. Modeling Strings and Rods

a) The String is Light

Key Assumption: The string or rod has zero mass (or its mass is so small compared to the objects it connects that it can be ignored).

Implication (VERY important): If the string is light, the Tension (\(T\)) is constant throughout its entire length. This drastically simplifies calculations involving pulleys.

If the string had mass, the tension would be different at different points, making the problem much harder!

b) The String is Inextensible

Key Assumption: The string cannot be stretched; its length is fixed.

Think of using a steel cable instead of a rubber band.

Implication (VERY important): If two particles are connected by an inextensible string, they must move together. They have the same speed and the same magnitude of acceleration (\(a\)).

4. Modeling Surfaces and Pulleys

a) The Surface is Smooth

When you hear the word "smooth" in mechanics, your mind should immediately scream "NO FRICTION!"

Key Assumption: There is no resistive force acting parallel to the surface (i.e., the frictional force \(F\) is zero).

Did you know? In real life, no surface is truly smooth. We assume this because calculating friction often requires knowing the coefficient of friction, which complicates M1 problems. We will deal with rough (frictional) surfaces later in the curriculum.

b) The Pulley is Small and Smooth

In problems involving strings passing over a pulley, we often make these assumptions:

• Small/Light: We ignore the pulley's size and mass.
• Smooth: We ignore any friction between the string and the pulley.

Implication: The tension \(T\) on both sides of the string passing over the pulley is the same.

5. Air Resistance and Other Environmental Forces

Air Resistance is Negligible

Key Assumption: The force exerted by the air (drag) opposing motion is zero.

Implication: This simplifies forces greatly. For example, a falling object is only subjected to gravity. If we didn't ignore air resistance, the resistive force would usually depend on the velocity (\(v\)), making the equations much harder to solve.

Common Mistake to Avoid: Students sometimes confuse air resistance (drag) with friction (on surfaces). They are both resistive forces, but they apply in different contexts.

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Section 3: Environmental Modeling (Gravity and Earth)

We also make standard assumptions about the fundamental forces acting on our objects.

6. Gravity and the Earth

a) Constant Acceleration due to Gravity (\(g\))

Key Assumption: We assume the acceleration due to gravity, \(g\), is constant and acts vertically downwards.

Standard Value: Unless otherwise stated, we usually use \(g = 9.8 \text{ m/s}^2\). Sometimes you might be asked to use \(g = 10 \text{ m/s}^2\) for approximation purposes, so always check the question!

In reality, gravity slightly changes depending on altitude and location on Earth, but we ignore this variation completely in M1.

b) The Earth is Flat

Key Assumption: We treat the Earth as a flat plane over the distances considered in M1 problems.

Implication: The vertical direction (perpendicular to the plane) is consistent everywhere in the problem, and all weights act parallel to one another. We do not worry about the curvature of the Earth.

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Summary and Review

Understanding these modeling assumptions is crucial. If a question asks you to "State the assumptions used in the model," you need to recall the relevant points below.

The Essential M1 Assumptions Checklist

When simplifying a real scenario into a mechanics problem, remember these key idealizations:

For Objects:

• Particle: Size and rotation ignored.
• Rigid Body: Size matters, but the object cannot deform (stretch or squash).

For Connections and Surfaces:

• Light String/Rod: Mass is zero, so Tension (\(T\)) is constant.
• Inextensible String: Length is fixed, so connected objects share the same acceleration (\(a\)).
• Smooth Surface/Pulley: No friction (\(F=0\)).

For Environment:

• Air Resistance: Negligible (set to zero).
• Gravity (\(g\)): Constant (\(9.8 \text{ m/s}^2\)) and acts vertically downwards.

You are now equipped with the essential tools to tackle any M1 problem. Every calculation you do from now on relies on these fundamental simplifications!