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Assumptions

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Assumptions

To understand the concept of assumptions in mechanics, we must first understand the concept of mathematical modelling, in particular as applied to the field of mechanics.

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.

Frequently Asked Questions about Assumptions

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

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

Final Assumptions Quiz

Question

Why do we use modelling assumptions?

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Answer

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

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Question

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

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Answer

True

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Question

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

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Answer

False

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Question

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

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Answer

  • 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

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Question

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

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Answer

  • Mass of the particle is concentrated at a single point.

  • Rotational forces and air resistance can be ignored

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Question

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

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Answer

The wire is treated as a one-dimensional object

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Question

What is/are the common assumption(s) used when we model an inextensible string in mechanics?

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Answer

Acceleration is the same in objects connected by a taut inextensible string

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Question

In mechanics, what is a bead?

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Answer

A particle with a hole in it for threading on a wire or string

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Question

In mechanics, what is a lamina?

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Answer

Object with area but negligible thickness, like a sheet of paper.

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Question

Which of the following is a common assumption when we mention a light object in mechanics?

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Answer

Mass of the object is zero or negligible.

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Question

Which of the following is a common assumption when we mention a smooth surface in mechanics?

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Answer

There is no friction between the surface and any object on it.

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Question

In mechanics, what is a peg?

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Answer

A support from which a body can be suspended or rested.

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Question

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

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Answer

  • mass is concentrated along a line

  • no thickness

  • rigid (does not bend or buckle)

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Question

An ant falls off the top of a building. Is the following an appropriate assumption to model the motion of the ant: the effect of air resistance is negligible.

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Answer

No.

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Question

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?

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Answer

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

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