Assumptions: Definition, Examples & Regression, Maths

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What is mathematical modelling?

Mathematical modelling is used to describe a system using mathematical concepts and language. The resultant description is called a model. A model is often a simplified abstraction of reality.

Mechanics deals with the motion and the action of forces acting on objects. In mechanics, it is common practice to model the motion or impact of forces on objects. Let's look at an example.

Consider the motion of a ball being dropped from a height. In reality, there are multiple forces acting on the ball. This includes the force of gravity that is pulling the ball towards the Earth's surface, air resistance acting on the ball and also the gravitational pull of other objects and not just that of the Earth. In mechanics, the motion of the ball is often modelled in a simplified manner by considering only the acceleration of the ball due to gravity.

Now, you might be wondering, why does this simplification make sense? The answer is that it makes sense because the impact of all the other forces acting on the ball is likely negligible compared to the Earth's gravity pulling it downwards.

If we simplified the motion of the ball by considering only the effect of air resistance and ignored the acceleration due to gravity, that would not make sense and would not really help us accurately model the motion of the ball.

Modelling assumptions

To build the mathematical models that are used in mechanics, there are often assumptions that have to be made. For example, in the earlier example, the effect of air resistance being negligible was an assumption that had to be made.

Modelling assumptions allow us to simplify real-world problems and analyse them using known mathematical techniques.

It is necessary to note that modelling assumptions can affect the usability and validity of a model. For example, if we are modelling the motion of a falling feather instead of a falling ball, it would not be appropriate to ignore the effects of wind and air resistance.

Common assumptions in mechanics

Here are some commonly used models and their modelling assumptions in mechanics. It is a good idea to become familiar with each of these.

Model	Modelling assumptions
Particle – Dimensions of the object considered are negligible.	The mass of the particle is concentrated at a single point. Rotational forces and air resistance can be ignored
Rod – Dimensions of the object are negligible.	mass is concentrated along a line the rod has no thickness the rod is rigid (cannot be bent)
Lamina – an object with a wide area but negligible thickness, such as a sheet of paper.	mass is uniformly distributed across a completely flat surface
Uniform body – mass of the body is distributed evenly.	The mass of the body is concentrated at a single point that is the geometrical center of the body - the center of mass
Light object – the mass of the object is small compared to other masses, like a string or a pulley.	the object can be treated as having zero mass tension is the same at both ends of a light string
Inextensible string – a string that does not stretch under load.	acceleration is the same in objects connected by a taut inextensible string
Smooth surface – surface with no rugosity, so friction is negligible.	there is no friction between the surface and any object on it
Rough surface – if a surface is not a smooth surface, it is considered a rough surface.	objects in contact with the surface experience a frictional force if they are moving or are acted on by a force when at rest
Wire – rigid thin length of metal.	treated as one-dimensional object
Smooth and light pulley – all pulleys you consider will be smooth and light.	pulley has no mass tension is the same on either side of the pulley
Bead – a particle with a hole in it for threading through a wire or string.	moves freely along a wire or string tension is uniform on either side of the bead
Peg – support from which a body can be suspended or rested.	dimensionless and fixed can be considered rough or smooth as specified in the question
Air resistance – the resistance experienced by an object as it moves through air.	can be modeled as being negligible
Gravity – the force of attraction between all objects. Acceleration due to gravity is denoted by g, and g = 9.8 m/s² unless otherwise specified.	assume that all objects with mass are attracted towards the Earth Earth's gravity is uniform and acts vertically downwards g is constant and is taken as 9.8 m/s², unless otherwise stated

Assumptions - Key takeaways

Mathematical modelling is used to describe a system using mathematical concepts and language. The resultant description is called a model. A model is often a simplified abstraction of reality.
To build the mathematical models that are used in mechanics, there are often assumptions that have to be made. Modelling assumptions allow us to simplify real-world problems and analyse them using known mathematical techniques.
Modelling assumptions can affect the usability and validity of a model. Incorrect or spurious assumptions can render the model ineffective.

Flashcards in Assumptions 15

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An ice puck is hit and it slides across the ice. The ice is assumed to be a smooth surface. What will be the impact of this assumption on the calculation of the motion of the puck?

No effect of friction between the puck and the ice will be taken into account.

Why do we use modelling assumptions?

Modelling assumptions allow us to simplify real world problems and analyze them using known mathematical techniques.

State whether the following statement is true or false : A model is often a simplified abstraction of reality.

True

State whether the following statement is true or false : Modelling assumptions do not affect the validity of the model itself.

False

What are the common assumptions used when we model gravity in mechanics?

All objects with mass are attracted towards the Earth
Earth’s gravity is uniform and acts vertically downwards
g (gravitational constant) is constant and is taken as 9.8 m/s², unless otherwise stated

What are the common assumptions used when we model a particle in mechanics?

Mass of the particle is concentrated at a single point.
Rotational forces and air resistance can be ignored

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Frequently Asked Questions about Assumptions

What does assumption mean?

In mechanics, assumptions allow us to simplify real-world problems and analyse them using known mathematical techniques.

What is an example of an assumption?

A very common assumption used in mechanics is that a body falling from a height is not influenced significantly by air resistance.

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What is mathematical modelling?

Modelling assumptions

Common assumptions in mechanics