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Modelling, quantities and units questions
Mechanics begins by deciding what to leave out. A crate is modelled as a particle, a rope as a light inextensible string, a surface as smooth or rough, and each assumption is what makes the problem tractable. This lesson covers those standard modelling assumptions, the S.I. units, and what each simplification costs in accuracy.
7 original questions · 21 marks · the modelling, quantities and units notes · Mechanics
Every question here is written for this library rather than taken from a past paper. Write your answer out before opening the worked one: the answers award marks point by point, and the marks are easier to see when you have something of your own to compare against.
A car is modelled as a particle. State what the model assumes, and one consequence for the mathematics.
Worked answer
All the mass is treated as concentrated at a single point, so size, shape and rotation are ignored. Every force then acts at that one point, which means there are no turning effects and no moments to take. The whole of Newton's second law becomes available with nothing to resolve about. B1 for the assumption, B1 for a consequence.A car travels at 90 km h⁻¹ and a metal has density 2.5 g cm⁻³. Convert each to SI units.
Worked answer
Speed: 90 km h⁻¹ = 90 × 1000 m per 3600 s = 25 m s⁻¹. Density: 1 g is 10⁻³ kg and 1 cm³ is 10⁻⁶ m³, so 1 g cm⁻³ is 1000 kg m⁻³ and 2.5 g cm⁻³ = 2500 kg m⁻³. B1 for the speed, B1 for the density. Convert before substituting, never after. A speed left in km h⁻¹ inside a suvat equation alongside g in m s⁻² is wrong by a factor of 3.6.A load hangs from a crane by a cable modelled as a light inextensible string. State what each modelling word contributes.
Worked answer
Light means the cable's own weight is ignored, so the tension is the same at every point along it. Inextensible means the cable cannot stretch, so the load moves exactly as the top end does, with the same speed and the same acceleration. B1 B1, one for each word, and each has to be tied to its consequence.Explain the modelling words smooth, rough and uniform.
Worked answer
A smooth surface or pulley exerts no friction, so it provides only a normal reaction, and a smooth pulley leaves the tension unchanged across it. Rough means friction acts and has to enter the equations. Uniform means the mass is evenly spread, so a uniform rod's weight acts at its midpoint. B1 B1 B1 for the three words. Each word is a licence to leave something out of the working, which is why they are worth reading twice.Write the SI units of velocity, acceleration and force, and show that the equation s = ut + ½at2 is consistent in units.
Worked answer
Velocity m s⁻¹, acceleration m s⁻², force N = kg m s⁻². In the equation, ut has units (m s⁻¹)(s) = m, and at2 has (m s⁻²)(s²) = m. Every term is a length, which is what an equation for displacement demands. B1 for the units of velocity and acceleration, B1 for the newton in base units, M1 for checking the units of each term, A1 for the conclusion. The ½ is a pure number and carries no units. A check like this catches a formula misremembered as s = ut + ½at, where the last term would be a velocity.Sort into scalars and vectors: mass, weight, speed, velocity, distance, displacement.
Worked answer
Scalars: mass, speed and distance, each a bare size. Vectors: weight, velocity and displacement, each carrying a direction as well. B1 B1 for the two lists. The pairings are the point. Speed is the magnitude of velocity, distance is the path-length companion of displacement, and mass is a scalar in kilograms while weight is a force in newtons pointing down.A stone and a feather are released together from the same height and a student's model predicts they land together. Identify the modelling assumption responsible, discuss for which object it is reasonable, and describe what a modeller does when a prediction disagrees with experiment.
Worked answer
The model neglects air resistance and treats each object as a particle acted on by its weight alone, so both fall with acceleration g and the masses cancel out of ma = mg. For the stone the assumption holds well over a short drop. It is dense and compact, so the air resistance it meets stays small compared with its weight. For the feather it fails almost immediately, because a large surface area and a tiny weight mean air resistance rises to match the weight within a fraction of a second, after which the feather drifts at near-constant speed. Assumptions are judged object by object, not once per problem. When a prediction misses, the modeller identifies the weakest assumption, replaces it with something more realistic such as a resistance force that grows with speed, solves again and compares again. B1 for identifying the neglect of air resistance, B1 for the masses cancelling, B1 for the stone with a reason, B1 for the feather with a reason, B1 for identifying the weakest assumption, B1 for the refine and compare loop. That loop of model, compare, refine is the whole discipline, and a model is kept when it is simple enough to solve and close enough to be useful, not when it is true.
The same practice on paper: the printable workbook for this topic, questions and a worked answer book.
Practise modelling, quantities and units one question at a time
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