Questions on modelling, quantities and units

Question 1

A car is modelled as a particle. State what the model assumes, and one consequence for the mathematics.

Accepted 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. B1 for the assumption, B1 for a consequence.

Question 2

A car travels at 90 km h⁻¹ and a metal has density 2.5 g cm⁻³. Convert each to SI units.

Accepted 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.

Question 3

A particle moves along a straight east-west line, with east taken as the positive direction. At one instant its velocity is −4 m s⁻¹. State the speed of the particle and the direction in which it is moving.

Accepted Answer

Speed is the magnitude of the velocity, 4 m s⁻¹, and the negative sign means the particle is moving in the negative direction, so it is moving west. B1 for the speed, B1 for the direction.

Question 4

Find the weight, in newtons, of a mass of 250 g, taking g = 9.8 m s⁻². State the SI unit of the moment of a force.

Accepted Answer

Convert first: 250 g = 0.25 kg, so W = mg = 0.25 × 9.8 = 2.45 N. A moment is a force times a distance, so its SI unit is the newton metre, N m. B1 for 2.45 N, B1 for N m. Substituting 250 into W = mg gives 2450 N, the weight of a quarter of a tonne; the gram is not an SI base unit and has to go before the calculation, not after it.

Question 5

A load hangs from a crane by a cable modelled as a light inextensible string. State what each modelling word contributes.

Accepted 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.

Question 6

Explain the modelling words smooth, rough and uniform.

Accepted 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.

Question 7

Write the SI units of velocity, acceleration and force, and show that the equation s = ut + ½at² is consistent in units.

Accepted 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.

Question 8

Sort into scalars and vectors: mass, weight, speed, velocity, distance, displacement.

Accepted 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. 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.

Question 9

Explain what is meant by each of the modelling words rod, lamina and bead, and in each case state one thing the word allows the working to ignore.

Accepted Answer

A rod is a rigid body with length but no thickness, so every force on it acts at a point of one line and it does not bend. A lamina is a flat body with no thickness, so it is treated as a two-dimensional shape and its weight acts at a point of that shape. A bead is a particle threaded on a wire or string, free to slide along it, so the only contact force from a smooth wire is the reaction perpendicular to it. B1 B1 B1, one for each of rod, lamina and bead with its consequence.

Question 10

A car travelling at 72 km h⁻¹ brakes with constant deceleration 2 m s⁻² until it stops. A student substitutes into v² = u² + 2as and obtains a stopping distance of 1296 m. Identify the error, and find the correct stopping distance.

Accepted Answer

The student substituted u = 72 without converting: 72 km h⁻¹ is 72 × 1000/3600 = 20 m s⁻¹, and 72²/(2 × 2) = 1296 mixes kilometres per hour with metres per second squared. With u = 20, v = 0 and a = −2: 0 = 400 − 4s, so s = 100 m. B1 for identifying the unconverted speed as the error, M1 for converting to 20 m s⁻¹, M1 for v² = u² + 2as with v = 0, A1 for 100 m. The size of the wrong answer is itself a check: a car does not need well over a kilometre to stop from 72 km h⁻¹ under firm braking.

Question 11

A walker goes 12 m due east and then 5 m due north. Find the distance travelled and the displacement from the starting point, giving the magnitude of the displacement and its direction.

Accepted Answer

Distance is the length of the path, 12 + 5 = 17 m. Displacement is the straight line from start to finish: magnitude √(12² + 5²) = 13 m, in the direction tan⁻¹(5/12) = 22.6 degrees north of east. B1 for 17 m, M1 for Pythagoras on the two legs, A1 for 13 m with the direction. The two answers differ because one is a scalar and the other a vector; a displacement given as a bare 13 m is incomplete, since the same magnitude in another direction is a different displacement.

Question 12

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.

Accepted 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. 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.

Question 13

Extension. Using dimensions M, L and T, test each of the following for dimensional consistency, where v is a speed, g an acceleration, h and r lengths, t a time, m a mass and F a force: (i) v² = 2gh + at, where a is an acceleration; (ii) F = mv²/r. State what a failed test and a passed test each establish about an equation.

Accepted Answer

(i) [v²] = (L T⁻¹)² = L² T⁻², and [2gh] = L T⁻² × L = L² T⁻², which agrees. But [at] = L T⁻² × T = L T⁻¹, a velocity, which does not agree with L² T⁻². The equation is inconsistent, and the term at is the one at fault. (ii) [mv²/r] = M × L² T⁻² ÷ L = M L T⁻², which is the dimension of a force, so the equation is consistent. M1 for writing the dimensions of each term of (i), A1 for the conclusion naming at as the failing term, M1 for the dimensions of mv²/r, A1 for the agreement with [F]. B1 for the meaning of each outcome: a fail is decisive, the equation is wrong; a pass only shows the equation is not ruled out, because a wrong dimensionless factor such as a missing ½ passes the test unnoticed.

Question 14

Extension. The speed v of waves on deep water is assumed to depend only on the wavelength w, the acceleration g and the density D of the water, so that v = k w^p g^q D^r with k a dimensionless constant. The dimensions of a density are M L⁻³. Find p, q and r, and state what the argument has established about the wave speed and what it has left open.

Accepted Answer

Dimensions: [v] = L T⁻¹, [w] = L, [g] = L T⁻², [D] = M L⁻³. The right-hand side has dimensions L^p (L T⁻²)^q (M L⁻³)^r = M^r L^p + q − 3r T^−2q. Matching M: r = 0. Matching T: −2q = −1, so q = ½. Matching L: p + q − 3r = 1, so p = ½. Hence p = ½, q = ½, r = 0 and v = k√(gw). B1 for the dimensions of g and D, M1 for equating the indices of M, L and T, A1 for r = 0, A1 for q = ½, A1 for p = ½. B1 for the interpretation: the density drops out, so in this model the wave speed does not depend on the density, and the speed grows as the square root of the wavelength; the constant k is not determined, so the method gives the shape of the formula and not the numbers in it.

Question 15

A crate of mass 40 kg hangs at rest from a helicopter on a vertical steel cable of mass 9 kg. A student models the cable as a light inextensible string. Find the tension the model gives, then find the actual tension at the top of the cable and at the crate, explain which of the model's tensions is affected, and state when the light-cable assumption is reasonable.

Accepted Answer

In the model the cable has no mass, so the tension is the same throughout and, with the crate at rest, T = 40g = 392 N. In reality the tension varies along the cable because each part of it supports the cable below it as well as the crate. At the crate the cable supports only the crate, so the tension there is still 392 N. At the top it supports the crate and the whole cable, 49 kg, so the tension is 49g = 480.2 N. The model is exact at the bottom and about 18% low at the top. The assumption is reasonable when the cable's mass is small compared with the load it carries, for example a short cable with a heavy crate, and becomes poor for a long cable or a light load. B1 for one tension throughout in the model with T = 392 N, M1 for treating the crate and cable together at the top, A1 for 480.2 N, B1 for explaining that the tension at the crate is unchanged while the tension at the top is underestimated, B1 for a reasoned condition on when the assumption holds.