Assumptions for models

Topic summary

Terms and Assumptions in Mathematical Models

In mechanics and mathematical modelling, various idealised objects are used to simplify real-world scenarios. These terms describe the objects and their properties, allowing complex systems to be analysed more easily.

Particle

A particle is an object with mass but no size or shape. This assumption simplifies problems by neglecting rotational effects and treating all forces as acting on a single point.

• No dimensions — considered a point mass.
• Forces such as friction and air resistance are often ignored.

Rod

A rod is an object with length but no thickness, and it is usually assumed to be rigid.

• One-dimensional — has length but no width or depth.
• Cannot bend or stretch (assumed to be perfectly rigid).
• Examples include beams, poles, or thin bars.

Lamina

A lamina is a two-dimensional flat object with mass but negligible thickness.

• Has area but no thickness.
• Mass is distributed evenly across its surface.
• Examples include thin sheets or plates.

Uniform Body

A uniform body has mass distributed evenly throughout its volume or area.

• Mass is assumed to be consistent in all parts.
• The centre of mass is at the geometric centre of the object.

Non-Uniform Body

A non-uniform body has mass that is not distributed evenly, meaning certain parts are heavier or lighter.

• The centre of mass is not necessarily at the geometric centre.
• Mass distribution can vary across the object.

Light Object

A light object is an object with negligible mass compared to other elements of the system, allowing it to be ignored in calculations involving forces.

• Mass is considered negligible (approximated as zero).
• Often used for strings, pulleys, or wires in problems.

Inextensible String

An inextensible string is a string that does not stretch under tension.

• Constant length, regardless of forces applied.
• Used in problems involving tension, such as pulley systems.

Smooth Surface

A smooth surface assumes no friction between the object and the surface.

• Negligible resistance to motion across the surface.
• Friction is ignored in calculations.

Rough Surface

A rough surface includes friction between the object and the surface.

• Opposes motion and slows down moving objects.
• Frictional forces must be considered in calculations.

Wire

A wire is a thin, rigid length of material that can be modelled like a rod or a string, depending on the context.

• Thin, one-dimensional object.
• Can be smooth or rough depending on the problem.

Bead

A bead is a particle that can move freely along a wire or string without friction.

• Moves along a wire or string.
• Often assumed to have negligible friction with the wire.

Peg

A peg is a fixed point used in problems where an object, like a string or rod, is hung or pivoted.

• Can be smooth (no friction) or rough (with friction).
• Used to analyse rotational motion or tension in strings.

Gravity

In most models, gravity is treated as a constant force acting vertically downward with an acceleration of:

\[g \approx 9.8 \, \text{ms}^{-2}\]

• Force due to gravity acts on the mass of an object.
• In many problems, air resistance is neglected, simplifying the motion under gravity.