ASLevel Maths Mechanics 1 for Edexcel M 1

AS-Level Maths: Mechanics 1 for Edexcel M 1. 1 Mathematical modelling These icons indicate that teacher’s notes or useful web addresses are available in the Notes Page. This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation. 1 of 16 © Boardworks Ltd 2005

Contents The modelling process Terminology Examples of models SI units 2 of 16 © Boardworks Ltd 2005

Introduction to mechanics Mechanics is a branch of applied maths that models physical situations involving forces and motion. At this level, it can be broadly divided into three sections: Kinematics – the study of how objects move. In kinematics, we consider the motion of objects ignoring any forces involved. Dynamics – the study of why objects move. In dynamics, we consider the forces that act to produce or change motion. Statics – the study of objects at rest. In statics, we examine how forces balance to prevent motion. 3 of 16 © Boardworks Ltd 2005

The modelling process In mechanics, modelling involves simplifying a real world problem to enable conclusions to be drawn. The modelling process can be illustrated as follows: Evaluate/improve the model Identify a real world problem make assumptions to allow simplification Arrive at a mathematical model 4 of 16 compare with the real world problem solve the problem Interpret the solution © Boardworks Ltd 2005

Common assumptions In almost all mechanics’ models a diagram should be drawn indicating any forces and motion. Assumptions need to be made when modelling to enable a problem to be simplified. Common assumptions include: 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 5 of 16 © Boardworks Ltd 2005

Contents Terminology The modelling process Terminology Examples of models SI units 6 of 16 © Boardworks Ltd 2005

Terminology Here are some terms that are often used when modelling problems in mechanics: 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. Particle – a body whose dimensions are so small compared with any other distances involved that it can be represented by a single point. Lamina – a flat object whose thickness is so small compared with its length and width that it can be ignored. 7 of 16 © Boardworks Ltd 2005

Terminology 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. Uniform rod – equal lengths have equal masses and so the mass can be assumed to act at the centre of the rod. Non-uniform rod – equal lengths do not have equal masses. Inextensible string – a string whose length does not change. It can also be referred to as inelastic. 8 of 16 © Boardworks Ltd 2005

Terminology Smooth surface – a surface on which friction can be ignored. Rough surface – a surface on which friction needs to be taken into account. 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. 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. 9 of 16 © Boardworks Ltd 2005

Contents Examples of models The modelling process Terminology Examples of models SI units 10 of 16 © Boardworks Ltd 2005

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 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. Air resistance as the ball is thrown into the air can also be ignored. 11 of 16 © Boardworks Ltd 2005

Examples of models 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. Also, as the masses move air resistance is ignored. 12 of 16 © Boardworks Ltd 2005

Examples of models A uniform rod of weight 30 N rests horizontally in equilibrium on two smooth supports at A and B as shown. RA 2 m RB 4 m 4 m Suppose we need to calculate the reaction forces at A and B. 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. 13 of 16 © Boardworks Ltd 2005

Examples of models 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. 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. 14 of 16 © Boardworks Ltd 2005

Contents SI units The modelling process Terminology Examples of models SI units 15 of 16 © Boardworks Ltd 2005

SI units Throughout mechanics the international system of units, or SI units, are used whenever possible. The following SI units are most commonly used in M 1. Quantity SI Unit Distance/displacement metre (m) Speed/velocity metres per second (ms– 1) Acceleration metres per second (ms– 2) Mass kilogram (kg) Weight/force Newton (N) Momentum/impulse Newton seconds (Ns) Moment of a force Newton-metre (Nm) 16 of 16 © Boardworks Ltd 2005