M1: Mathematical models in mechanicsEdexcel International A Level Maths: Topic test

20 questions, 54 marks

Edexcel International A Level Maths

M1: Mathematical models in mechanics topic test

Total 54 marks

Name

Class

Date

1
A long, thin, uniform wooden plank of mass 30 kg rests horizontally on two supports. The plank is modelled as a uniform rod.
(a)
Which statement is a consequence of modelling the plank as a rod?
[1 mark]
AThe mass of the plank is ignored.
BThe plank is treated as having length but negligible thickness.
CThe weight of the plank acts at one end.
DThe plank is able to stretch when loaded.
(b)
Which statement is a consequence of modelling the plank as uniform?
[1 mark]
AThe weight of the plank acts at its midpoint.
BThe mass of the plank is negligible.
CFriction between the plank and the supports is zero.
DThe plank cannot bend.
(c)
Give two reasons why modelling the plank as a rod may be unrealistic.
[2 marks]
Total for question 1: 4 marks
2
A footballer kicks a ball across a flat pitch. The ball is modelled as a particle, and air resistance is ignored.
(a)
Which statement follows from modelling the ball as a particle?
[1 mark]
AThe ball has no mass.
BThe ball moves in a straight line.
CThe dimensions of the ball are ignored, so any spin of the ball is ignored.
DThe ball experiences no force from gravity.
(b)
Compared with the real motion, the model will most likely predict a horizontal range that is
[1 mark]
Atoo small, because in reality the ball would move faster.
Bexactly correct, because a football is small.
Cunaffected, because the weight of the ball is the only force acting.
Dtoo large, because in reality air resistance slows the ball.
(c)
State two further modelling assumptions that are needed to treat the motion of the ball as that of a particle moving under gravity alone.
[2 marks]
Total for question 2: 4 marks
3
A sprinter completes a 100 m race in 12.5 s. A coach models the sprinter as a particle moving in a straight line at constant speed.
(a)
Find the speed of the sprinter according to the model, and state one reason why the model is unrealistic.
[3 marks]
(b)
In fact the sprinter takes 3.2 s to cover the first 20 m. Find the time that the model predicts for the first 20 m, and find the percentage by which this prediction is less than the actual time.
[4 marks]
Total for question 3: 7 marks
4
A builder lowers a bucket of mass 15 kg to the ground using a rope that passes over a horizontal metal bar fixed to scaffolding. The rope is modelled as a light inextensible string, the bar as a smooth peg and the bucket as a particle.
(a)
Explain what each of the following modelling assumptions means, and state one consequence of each for the forces or motion in the situation: (i) the rope is light, (ii) the rope is inextensible, (iii) the bar is a smooth peg.
[6 marks]
(b)
The model predicts that the bucket takes 2.0 s to reach the ground, but the builder measures a time of 2.6 s. (i) Find the percentage by which the predicted time is less than the measured time. (ii) Identify two features of the real situation that could explain why the bucket takes longer than the model predicts, and justify each.
[6 marks]
Total for question 4: 12 marks
5
A hockey puck of mass 0.17 kg slides across a horizontal ice rink. The puck is modelled as a particle, the ice is modelled as smooth and air resistance is ignored.
(a)
What does it mean to model the ice as smooth?
[1 mark]
AThe puck has no mass.
BThe puck cannot rotate as it slides.
CThere is no friction between the puck and the ice.
DThe ice has no thickness.
(b)
According to the model, what happens to the horizontal speed of the puck as it slides?
[1 mark]
AIt stays constant.
BIt decreases steadily until the puck stops.
CIt increases steadily.
DIt decreases at first, then stays constant.
(c)
The model predicts that the puck travels at a constant speed of 12 m s $^{-1}$ . In a test, the puck covers 30 m in 3.0 s. Find the percentage by which the time predicted by the model is less than the measured time.
[2 marks]
Total for question 5: 4 marks
6
A lamp of mass 2 kg hangs at rest from a ceiling hook by a light inextensible string. The lamp is modelled as a particle.
(a)
Which statement follows from modelling the string as light?
[1 mark]
AThe string does not stretch.
BThe tension in the string is zero.
CThe lamp has no size.
DThe mass of the string is ignored.
(b)
Taking $g=9.8$ m s $^{-2}$ , what is the weight of the lamp?
[1 mark]
A $2$ N
B $19.6$ N
C $0.204$ N
D $39.2$ N
(c)
Give one reason why modelling the lamp as a particle is reasonable in this situation, and one situation in which it would not be reasonable.
[2 marks]
Total for question 6: 4 marks
7
A small bead of mass 0.02 kg is threaded on a rigid straight wire that is fixed in a horizontal position. The bead is modelled as a particle and the wire as smooth.
(a)
Explain what each of the following modelling assumptions means: (i) the bead is a particle, (ii) the wire is smooth, (iii) the wire is rigid.
[3 marks]
(b)
Taking $g=9.8$ m s $^{-2}$ , find the weight of the bead according to the model. The actual mass of the bead is 0.0215 kg. Find the percentage by which the weight in the model is less than the actual weight.
[4 marks]
Total for question 7: 7 marks
8
A uniform square steel plate of side 0.5 m and mass 8 kg is hung from a smooth horizontal peg that passes through a small hole near one corner. The plate is modelled as a uniform lamina.
(a)
Explain what each of the terms (i) uniform, (ii) lamina and (iii) smooth peg means in this model, and state one consequence of each.
[6 marks]
(b)
(i) Taking $g=9.8$ m s $^{-2}$ , find the weight of the plate.
(ii) The thickness of the plate is 4 mm. Find the thickness as a percentage of the side length, and hence comment on whether modelling the plate as a lamina is reasonable.
(iii) State one feature of a real peg that could make the behaviour of the plate differ from the model.
[6 marks]
Total for question 8: 12 marks

End of questions

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).