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AS-Level Maths: Mechanics 1for Edexcel

M1.1 Mathematical modelling

For more detailed instructions, see the Getting Started presentation.

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The modelling process

The modelling process

Terminology

Examples of models

SI units

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Mechanics is a branch of applied maths that models physical situations involving forces and motion.

Introduction to mechanics

At this level, it can be broadly divided into three sections:

Kinematics – the study of how objects move.

Dynamics – the study of why objects move.

Statics – the study of objects at rest.

In kinematics, we consider the motion of objects ignoring any forces involved.

In dynamics, we consider the forces that act to produce or change motion.

In statics, we examine how forces balance to prevent motion.

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In mechanics, modelling involves simplifying a real world problem to enable conclusions to be drawn.

The modelling process

The modelling process can be illustrated as follows:

Arrive at a mathematical

model

Arrive at a mathematical

model

Interpret the solution

Interpret the solution

Evaluate/improve the model

Evaluate/improve the model

solve the problem

compare with the real world problem

make assumptions to allow simplification

Identify a real world problemIdentify a real world problem

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Assumptions need to be made when modelling to enable a problem to be simplified.

In almost all mechanics’ models a diagram should be drawn indicating any forces and motion.

Common assumptions include:

Common assumptions

ignoring friction

ignoring air resistance

treating a body as a particle

ignoring small dimensions (for example, in a lamina)

ignoring the stretching of an inelastic string

when a load is attached

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Terminology

The modelling process

Terminology

Examples of models

SI units

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Here are some terms that are often used when modelling problems in mechanics:

Lamina – a flat object whose thickness is so small compared with its length and width that it can be ignored.

Terminology

Particle – a body whose dimensions are so small compared with any other distances involved that it can be represented by a single point.

Rough – friction needs to be taken into account when a body moves on a rough surface.

Smooth – friction can be ignored on a smooth surface.

Thin – of negligible thickness.

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Terminology

Inextensible string – a string whose length does not change. It can also be referred to as inelastic.

Non-uniform rod – equal lengths do not have equal masses.

Uniform rod – equal lengths have equal masses and so the mass can be assumed to act at the centre of the rod.

Rigid body – a body that is assumed to not change shape when acted on by a force.

Light rod – a rod is assumed to have length only. A light rod has a negligible mass compared with other bodies.

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Peg – a support on which a body may rest or be hung. There is only one point of contact between a peg and a body.

Terminology

Light smooth pulley – a pulley whose mass is so small compared with other objects that it can be ignored and, as it is smooth, friction can also be ignored.

Bead – a particle with a hole drilled through so that it can be threaded onto a wire or string.

Wire – a rigid body in the form of a strip of metal.

Rough surface – a surface on which friction needs to be taken into account.

Smooth surface – a surface on which friction can be ignored.

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Examples of models

The modelling process

Terminology

Examples of models

SI units

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How can this situation be modelled – what assumptions can we make and what physical factors can we ignore?

The dimensions of the ball can be ignored and it can be modelled as a particle.

Examples of models

Suppose we need to calculate the greatest height reached by a ball that is thrown vertically upwards with an initial speed of 5 ms–1.

Air resistance as the ball is thrown into the air can also be ignored.

5 ms–1

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Two particles of mass 3 kg and 5 kg are attached to the ends of a light inextensible string. The string passes over a smooth pulley. The system is then released from rest.

What assumptions can we make in modelling the above situation and what physical factors can we ignore?

The two masses are modelled as particles.

The masses are attached to a light inextensible string. This assumes that the string has no weight and will not stretch. Thus tension in the string for each particle is equal.

The pulley being smooth assumes there is no friction.

Examples of models

Also, as the masses move air resistance is ignored.

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A uniform rod of weight 30 N rests horizontally in equilibrium on two smooth supports at A and B as shown.

How can the fact that the beam is uniform be used to model the situation?

As the beam is uniform the weight is taken to act at the centre.

Examples of models

Suppose we need to calculate the reaction forces at A and B.

RA RB

2 m 4 m 4 m

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A parachutist jumps from an aeroplane and falls vertically from rest towards the ground.

State any assumptions to be made when modelling the initial motion of the parachutist and any physical factors that have been ignored.

The parachutist can be modelled as a particle.

Examples of models

In this model the parachutist is taken to be travelling vertically with no rotational or sideways motion.

Any air resistance can be ignored in the time before the parachute opens.

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SI units

The modelling process

Terminology

Examples of models

SI units

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The following SI units are most commonly used in M1.

SI units

SI UnitQuantity

Throughout mechanics the international system of units, or SI units, are used whenever possible.

Newton-metre (Nm)Moment of a force

Newton seconds (Ns)Momentum/impulse

Newton (N)Weight/force

kilogram (kg)Mass

metres per second per second (ms–2)Acceleration

metres per second (ms–1)Speed/velocity

Distance/displacement metre (m)