8.1 & 8.2 - Modelling Assumptions

• Mathematical models can be constructed to simulate real-life situations, but in many cases, it is necessary to simplify the problem by making assumptions so that it can be described using equations or graphs in order to solve it.
• The solution to a mathematical model needs to be interpreted in the context of the original problem. At times, your model may need to be refined and your assumptions reconsidered.
• Modelling assumptions can simplify a problem and allow you to analyse a real-life situation using known mathematical techniques.
• However, modelling assumptions can affect the validity of a model.

  For example, when modelling the landing of a commercial flight, it would not be appropriate to ignore the effects of wind and air resistance.
• There are 14 common models and modelling assumptions that you need to know.
• The 1st model is the particle, where the dimensions of the object are negligible.

  The 2 modelling assumptions for this model are:

  • The mass of the object is concentrated at a single point.
  • Rotational forces and air resistance can be ignored.
• The 2nd model is the rod, where all dimensions are negligible, except either its horizontal length or its vertical height. The choice between horizontal length or vertical height depends on the object's orientation and the problem context.

  The 3 modelling assumptions for this model are:

  • The mass of the object is concentrated along a line.
  • The rod has no thickness.
  • The rod is rigid so it does not bend or buckle.
• The 3rd model is the lamina, which is an object with area but its thickness is negligible, like a sheet of paper.

  The 1 modelling assumption for this model is:

  • The mass of the object is distributed across a flat surface.
• The 4th model is the uniform body, which is an object which has its mass distributed evenly throughout its volume.

  The 1 modelling assumption for this model is:

  • Though its mass is distributed evenly throughout its volume, we can treat the entire body as if all its mass is concentrated at a single point at the geometrical centre of the body (which is the centre of mass).
  • The centre of mass is a mathematical representation that simplifies calculations, but it does not mean that the mass is physically concentrated there.
• There are several differences between a particle and a uniform body:

  1. A particle is considered to have negligible dimensions, while a uniform body has finite dimensions.
  2. Particles don't experience rotational motion or torque, but uniform bodies can rotate and are affected by torques.
  3. Particles cannot deform, but uniform bodies can undergo elastic or plastic deformation.
• The 5th model is the light object, which has a mass that is small compared to other masses. Examples of light objects include strings/ropes in pulley systems or frictionless pulleys.

  The 2 modelling assumptions for this model are:

  1. The object is treated as having a mass of 0, in order to simplify calculations.
  2. The tension is the same at both ends of a light string.
• The 6th model is the inextensible string, which is a string that does not stretch under load.

  The 1 modelling assumption for this model is:

  1. The acceleration is the same in objects connected by a taut inextensible string.
• The acceleration is the same in objects connected by a taut, inextensible string.

  The string is taut, which means there is no slack (which is a condition where there is excess material in the string which allows for some free movement before tension is applied) in the string.

  This ensures immediate force transmission, which allows the objects to move together as a single system. This is also due to how the inextensible nature of the string maintains a constant distance between the objects.
• The taut string transmits forces instantly between connected objects.
• The 7th model is the smooth surface.

  You should assume that there is no friction between the surface and any object on it.
• The 8th model is the rough surface - if a surface is not smooth, it is therefore rough.

  You should assume that objects in contact with the surface experience a frictional force if they are moving or acted on by a force.
• The 9th model is the wire, which is a rigid thin length of metal that can transmit forces along its length.

  It is treated as one-dimensional.

  A one-dimensional object can have its position or motion described by a single coordinate eg a train on a straight track as we just need to know how far along the track it is.

  In terms of the wire, only its length is relevant (1 dimension) as it has a high length-to-width ratio - as in, how far a force has been transmitted as the wire can transmit forces along its length.
• The 10th model is the smooth and light pulley.

  All pulleys you consider will be smooth and light.

  You assume here that the pulley has no mass and that the tension is the same on either side of the pulley.
• The 12th model is the peg, which is a support from which a body can be suspended or rested.

  You assume here that the peg is fixed and dimensionless.

  As specified in the question, the peg can be rough or smooth.
• There are 2 types of pegs:

  1. Rough pegs have an uneven surface, which increases friction when in contact with other surfaces, making it less likely to slip or rotate freely.
  2. Smooth pegs have a uniform surface, which reduces friction, allowing it to move more easily against other surfaces.
• Rough pegs are often used when you want to prevent movement, while smooth pegs are used when you want to allow for easy rotation or sliding.
• The 13th model is air resistance, which is the resistance experienced as an object moves through the air.

  It is usually modelled as being negligible.
• The 14th model is gravity, which is the force of attraction between all objects. Acceleration due to gravity is denoted by g.

  You make several assumptions here, including:

  1. All objects with mass are attracted towards the Earth.
  2. Earth's gravity is uniform and acts vertically downwards.
  3. g is constant and is taken as 9.8m/s2, unless stated otherwise in the question.