M1.1 - Mathematical models in mechanics
- Syllabus
- 2019
- Topic
- M1.1
- Level
- AS
Learning objectives
Turn a physical situation into a mechanics model
A mechanics model replaces a complicated real object or contact with a simpler idealisation. Each modelling word is an assumption, and the assumption tells you which features may be ignored and which mathematical consequences may be used.
| Model term | Assumption and useful consequence |
|---|---|
| Particle | Size and shape are ignored; the object's mass is treated as concentrated at one point. |
| Lamina | A flat object whose thickness is ignored; its mass is spread over an area. |
| Rigid body | The distance between any two points of the body stays fixed, so it does not deform. |
| Rod | A thin rigid body whose width is ignored; it remains straight. A uniform rod has constant mass per unit length, so its centre of mass is at its midpoint; a non-uniform rod need not. A light rod has negligible mass. |
| Bead on a wire | The bead is treated as a particle constrained to move along the wire. |
| Peg | A small fixed support or contact point; its size is ignored. |
| Model term | Assumption and useful consequence |
|---|---|
| Inextensible string | Its length is constant. While connected parts remain taut, their displacements, speeds and acceleration magnitudes are linked by that fixed length. |
| Light string | Its mass is negligible, so the tension is constant along one continuous section of string. |
| Smooth surface | Friction is ignored; the contact force is normal to the surface. |
| Rough surface | Friction may act parallel to the contact surface and oppose actual or impending relative motion. |
| Light smooth pulley | The pulley has negligible mass and axle friction is ignored; one continuous light string has the same tension on both sides. |
Translate every adjective before forming equations. For example, a block modelled as a particle may have its weight applied at one point; an inextensible string connecting two particles constrains their motions; and a smooth pulley lets the same tension be used on both sides of one continuous light string. These conclusions come from different assumptions and should not be interchanged.
A model is judged by purpose, not by whether it copies reality perfectly. Start with the stated idealisations, solve the simplified problem, then identify which omitted feature could materially change the result. Allowing string mass or extension, pulley friction, body dimensions, deformation, air resistance, measurement error, or changing acceleration can make a particular model more realistic, but only when that feature is relevant to the situation.
Do not infer more than the modelling word allows. Inextensible means constant length, not equal tension; equal tension additionally needs a light string and, across a pulley, a smooth light pulley. A rod remains straight, but it is not uniform or light unless stated. Smooth means frictionless, not that the normal reaction is zero.