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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. |
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Rod – Dimensions of the object are negligible. |
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Lamina – an object with a wide area but negligible thickness, such as a sheet of paper. |
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Uniform body – mass of the body is distributed evenly. |
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Light object – the mass of the object is small compared to other masses, like a string or a pulley. |
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Inextensible string – a string that does not stretch under load. |
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Smooth surface – surface with no rugosity, so friction is negligible. |
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Rough surface – if a surface is not a smooth surface, it is considered a rough surface. |
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Wire – rigid thin length of metal. |
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Smooth and light pulley – all pulleys you consider will be smooth and light. |
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Bead – a particle with a hole in it for threading through a wire or string. |
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Peg – support from which a body can be suspended or rested. |
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Air resistance – the resistance experienced by an object as it moves through air. |
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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. |
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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.
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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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