Revision:Mathematical Modelling

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These notes are based on the requirements of the M1 A Level mathematics module.

Now, unlike me the textbook is a highly important and useful resource and I would advise you at least glance over it, since I could well miss something! However, I will just say that this chapter doesn’t require your full attention, as all the key points can be summarised as follows:

Models in M1

In mechanics we have to model things, particularly in the exam room since we cant start tossing projectiles around to investigate air resistance on a pen shaped object – though Edexcel probably encourages such disorder. Therefore we model real life things, like trees and balls, as rods and particles. This is so we can remove complexities from our calculations and gain a nice round and rough estimate that does not change based on the fact that the man standing on a slope has one foot larger than the other! Believe me, modelling makes all our lives easier. Below I've listed the key models used in M1 and an example of how they might be applied.

Particles

The most common model. Objects, such as tennis balls, books and even people can be modelled as particles. The particle model assumes that the object is SO small that it is COMPLETELY unaffected by air resistance (in the case of flying, or indeed falling). The particle is also seen to have such a SMALL surface area that its centre of mass can be assumed uniform and its mass always acts downwards from its centre due to this uniform mass distribution

Light strings

Light strings often crop up in questions involving pulleys, and the same always applies. Light strings are what they say on the tin, weightless (at least in our models). It may also be worth mentioning that they are considered so narrow and to have such a SMALL SURFACE area, that friction DOES NOT affect their motion over a pulley. Yet, they will often be declared frictionless (or the pulley will).

Inextensible string/rod

The strings occur in the pulley problems section and the rods are sometimes called inextensible during moments questions or those funny ones involving caravans. What this means is that the DIMENSIONS of the rod or string (length and width) CANNOT BE CHANGED. Even with extreme force (such as boats pulling each other). This models chief advantage is that when dealing with pulleys, since the string remains at a CONSTANT length, we can assume the ACCELERATION in all parts of the string to be UNIFORM.

Inelastic objects

These are objects or walls, which do not change shape or dimensions, even if a car is dropped on them, basically. They are modelled as inelastic so that we do not have the complications of restitution interfering with our calculations. What you need to know is that if the objects WERE elastic, collisions would cause ENERGY LOSSES, which would not allow us to use the idea of conserved momentum so literally. This model is commonly used during questions about things striking other things, such as those in the Collisions chapter.

Uniform rods

These are rods that have a UNIFORM distribution of MASS. The consequence of this model is that the total mass of the rod can always be assumed to act through its centre. So if the rod AB is 6m long and has a weight of 50g, the 50g will act 3m in from either of the ends A or B.

Planes

A plane is a COMPLETELY FLAT surface, that has no bumps or impeding objects place along it. Planes can be SMOOTH or ROUGH. When a plane is declared as smooth and you are drawing a diagram make sure you note this on the diagram, so you don’t start including friction calculations. Smooth planes has NO FRICTIONAL COEFFICIENT. Rough planes can be indicated (by your diagrams) with dashed lines along the plane to show its roughness. Also drawing a big mu symbol helps to remind you! If you are asked to criticise the model state that friction is always present – so the smooth model is unrealistic.

Gravity!

It is true, gravity is in M1 a model. This is because we assume gravitational acceleration to be constant. IT IS NOT. However, we assume that g is always 9.8 in M1 to simplify matters. This can be asked at any time, just mention that the model is assuming the gravitational force to be constant.

Example questions

1. How has the man been modelled while standing on the beam, what does this model imply? [1 mark]

   As a particle. The model suggests that the mans mass acts on a single point along the beam and so the distance of the mans acting mass can be calculated based on the model.

2. In the above question the string is said to be inextensible, is such an idea realistic, why? [1 mark]

   No it is not, due to the fact that the application of force to a string will cause slight extension, which whilst negligible in most cases will still have an effect. All materials are extensible and so the idea is unrelated to the real world.

3. Explain how the tennis ball has been modelled during its motion (bouncing) and how this simplifies the situation, [2 marks]

   The tennis ball has been modelled as a particle. This makes the situation more simple, due to the fact that air resistance, which may result in a decreased resultant force, and thus acceleration can be ignored.

Originally written by RobbieCc on TSR forums.