Maths › Mechanics › Modelling, quantities and units
Modelling, quantities and units
Mechanics starts by deciding what to ignore. A crate becomes a particle, a rope becomes a light inextensible string, and what survives the pruning is exactly the physics the maths can handle.
Builds on Vectors in two dimensions.
IN THIS TOPIC
- Use the standard modelling words, particle, light, smooth, inextensible, uniform, rigid, and say what each assumption removes.
- Work in SI units and convert into them before calculating.
- Distinguish scalar quantities from vector ones, and signed values from magnitudes.
WHAT YOU PROBABLY THINK
The tension in a rope is always equal to the weight hanging from it.
The art of leaving things out
A model trades realism for solvability. Treating a crate as a particle means its size and shape stop mattering: all its mass acts at one point, every force acts at that point, and rotation is off the table. That single move turns most of this course into force arrows meeting at a dot.
The rest of the vocabulary works the same way. A light string has no mass, so tension is the same all along it. A smooth surface or pulley has no friction. An inextensible string cannot stretch, so the objects it joins share one acceleration. A uniform beam has its weight at its midpoint; a rigid one does not bend. Exam questions ask what these buy you, and the answer is always the simplification.
Units and quantities
Mechanics runs on SI units: metres, seconds, kilograms, and everything built from them, newtons for force, m s⁻¹ and m s⁻² for motion. Convert before calculating: a speed in km h⁻¹ fed into a suvat equation poisons every number after it. Weight is a force, W = mg, with g = 9.8 m s⁻² unless the question says otherwise.
Quantities split into scalars, size only, and vectors, size with direction: distance against displacement, speed against velocity. In one dimension direction is carried by sign, and choosing a positive direction at the start of every question, then believing the signs that follow, prevents most mechanics errors before they happen.
THE EXAM BIT
- “State a modelling assumption and its effect”: name the word, then the simplification, in one sentence each.
- Criticising a model earns marks the same way: air resistance ignored, string not really light, g taken constant.
- Convert to SI before any suvat or F = ma line.
- Fix a positive direction, write it down, and let signs carry direction from then on.
CHECK YOURSELF
A load hangs from a crane by a steel cable. The cable is modelled as a light inextensible string. State what each of the two words contributes to the model.
Show a hint
One word is about mass, the other about stretching.
Show the answer
Light: the cable's own weight is ignored, so the tension is the same throughout it.
Inextensible: the cable cannot stretch, so the load moves exactly with the cable end and shares its acceleration.
Model first: every standard word deletes one complication, and you should know which.
SI units in, signed one-dimensional vectors throughout, magnitudes only at the end.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Use the standard modelling words, particle, light, smooth, inextensible, uniform, rigid, and say what each assumption removes.
- Work in SI units and convert into them before calculating.
- Distinguish scalar quantities from vector ones, and signed values from magnitudes.
No animated video for this topic yet; these notes stand alone.