Oxford Cambridge and RSA Level 3 Cambridge …...2 OC 2018 Answer all the questions.1 Fig. 1 shows part of a circular pipe. The diameter of the pipe varies. At a part of the pipe where

Oxford Cambridge and RSA

Level 3 Cambridge Technical in Engineering05823/05824/05825/05873

Unit 23: Applied mathematics for engineeringWednesday 17 January 2018 – AfternoonTime allowed: 2 hours

You must have:• the formula booklet for Level 3 Cambridge

Technical in Engineering (inserted)• a ruler (cm/mm)• a scientific calculator

First Name Last Name

CentreNumber

CandidateNumber

Date of Birth D D M M Y Y Y Y

INSTRUCTIONS• Use black ink. You may use an HB pencil for graphs and diagrams.• Complete the boxes above with your name, centre number, candidate number and date of birth.• Answer all the questions.• Write your answer to each question in the space provided. • Additional paper may be used if necessary but you must clearly show your candidate number,

centre number and question number(s).• The acceleration due to gravity is denoted by g m s−2. Unless otherwise instructed, when a

numerical value is needed, use g = 9.8.

INFORMATION• The total mark for this paper is 80.• The marks for each question are shown in brackets [ ].• Where appropriate, your answers should be supported with working.

Marks may be given for a correct method even if the answer is incorrect.• An answer may receive no marks unless you show sufficient detail of

the working to indicate that a correct method is being used.• Final answers should be given to a degree of accuracy appropriate to the

context.• This document consists of 20 pages.

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FOR EXAMINER USE ONLY

Question No Mark

1 /102 /133 /114 /115 /126 /117 /12

Total /80

2

© OCR 2018

Answer all the questions.

1 Fig. 1 shows part of a circular pipe. The diameter of the pipe varies. At a part of the pipe where the diameter is d1 m the water is flowing at a rate of v1 m s–1. At another part, where the diameter is d2 m, the water is flowing at a rate of v2 m s–1.

v2 d2

v1d1

Fig. 1

The volume flow rate, V m3 s–1, of water flowing at a rate of v m s–1 through a section of the pipe with diameter d m is given by

(i) When the water volume flow rate remains constant along the pipe shown in Fig. 1 show that

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Fig. 2 shows a dam constructed at the end of a reservoir. Water flows from the dam through a circular downpipe of diameter 1 m. The downpipe is connected to a circular feedpipe of diameter 0.5 m which passes water into the turbine. The surface of the water in the reservoir is h m above the turbine.

damwater surface

15 m s downpipe

1 m

0.5 m

h m

feedpipe turbine

reservoir

Fig. 2

2d .V v2

r= b l

.v v dd

2 11

2

2

= d n

© OCR 2018 Turn over

3

(ii) Water flows from the dam through the downpipe at the rate of 15 m s–1. Calculate the rate, v m s–1, at which the water flows through the feedpipe and enters the turbine.

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(iii) For this system the rate at which water enters the turbine, v m s–1, is also given by the formula

Using the value of v found in part (ii) calculate the value of h.

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(iv) When the volume flow rate of water entering the turbine is V m3 s–1 the corresponding mass

flow rate is Q kg s–1, where Q = 1000V.

Calculate the values of V and Q.

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(v) The power, P W, produced by the turbine is given by

P = , where is the efficiency of the turbine.

Calculate the efficiency of the turbine when the power produced is 20 MW.

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Qghh h

2 .v gh=

4

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Question 2 begins on page 5


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2 Fig. 3 shows a mass supported by an energised coil spring which is fixed securely to an overhead support.

mass

y

support

Fig. 3

As the spring expands and contracts the distance y mm from the support to the centre of the mass oscillates according to the formula

,

where t is time measured in seconds.

(i) State the technical name for oscillating movement of this type.

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(ii) Calculate

(A) the frequency of the oscillation in cycles per second,

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(B) the period of the oscillation in seconds.

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60 20 sin (4 )y tr= +

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(iii) On the grid below, sketch the graph of

Label the axes.

[3]

(iv) In practice the spring loses energy as it expands and contracts. In this case the distance

y mm is given by

(A) The table below shows some values for t and y. Work out the missing y-values and complete the table.

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t 0.125 0.375 0.625 0.875y 77.65

[3]

60 20 e (4 ) .y tsint r= + -

60 20 sin (4 )for 0 1.y t t# #r= +


7

(B) Plot these values on the grid below and draw the graph of

Label the axes.

[2]

(C) Use your graph in part (iv)(B) to estimate the values of t when y = 50.

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60 20 e sin (4 )for 0 1.y t tt # #r= + -

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3 Fig. 4 shows the outside profile of a circular steel tube which has three sections and a uniform wall thickness throughout. The section on the left is 150 mm long and has an outside diameter of 40 mm. The section on the right is 50 mm long and has an outside diameter of 60 mm. The central section is 40 mm long and joins the two sections on either side as shown. A picture of the tube is shown in Fig. 5.

You might find it useful to consider the central section of the tube as part of a complete cone.

40

40

60

50150

Fig. 4 Fig. 5

(i) Calculate the outside surface area of the tube, giving your answer correct to the

nearest mm2.

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(ii) An engineer has estimated the volume of steel in the tube by multiplying the outside surface area by the wall thickness. Explain why this calculation does not produce an accurate result and, without performing any additional calculations, describe a method that will produce an accurate value.

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4 Two vertical poles, each of height 10 m, are sited on the side of a hill. The horizontal distance between the poles is 30 m and a cable is suspended between them. Fig. 6 shows the two poles, the cable and the side of the hill beneath the cable aligned within a Cartesian coordinate system (x, y). The base of the pole on the left is at the origin (0, 0).

The cable may be modelled by y = 0.01x2 − 0.4x + 10.

The side of the hill may be modelled by y = − 0.001x2 − 0.06x.

30 m

xNot to scale

y

h

Fig. 6

(i) Find an equation, h = F(x), where F(x) defines the vertical height of the cable above the side of the hill for 0 x 30.

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(ii) Calculate the minimum height of the cable above the side of the hill. You must show that your answer is a minimum.

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# #


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(iii) Calculate the vertical area between the cable and the side of the hill.

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5 Fig. 7 shows a schematic diagram of a crank and piston mechanism.

AC

B

Connecting rod

Piston Cylinder

l

r

Crank

axis ofcylinder

ω rad s -1

θa

Fig. 7

The crank has radius r mm and rotates anticlockwise at a speed of ω radians per second about the crankshaft which is indicated by point A. The connecting rod has length l mm and connects the crank to the piston at the pivot points B and C. As the crank rotates the piston moves backwards and forwards along the cylinder. When the crank makes an angle of θ with the axis of the cylinder the connecting rod makes an angle of α with the axis of the cylinder.

(i) Show that

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Fig. 8 shows velocity vectors p, q and s associated with the mechanism shown in Fig. 7.

Fig. 8

.sin sinlr1a i= - b l

C

q

s

B

A

p

ϴα


13

Vector p represents the linear velocity of point B relative to point A; this acts perpendicularly to AB with a magnitude of ωr mm s–1. Vector q represents the velocity of point C relative to point B; this acts perpendicularly to BC. Vector s represents the velocity of point C relative to point B; this acts along the axis of the cylinder. Since point A is fixed, the magnitude of s is the absolute linear speed of the piston along the cylinder.

(ii) Fig. 9 is a diagram showing a closed triangle of the vectors p, q, and s associated with Fig. 8. Find the three internal angles of the triangle in terms of α and θ, and mark them on Fig. 9.

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s

pq

Fig. 9

[3]

(iii) The magnitudes of vectors p, q and s are , and respectively.

Derive a formula for in terms of , and θ.

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sp q

s p a

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(iv) Given that r = 50, l = 175 and ω = 200 calculate the speed of the piston along the cylinder when θ = 35°.

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Question 6 begins on page 16

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© OCR 2018

6 Water in a tank is heated by a thermostatically controlled immersion element. When the element is switched on the temperature T of the water in the tank, measured in oC, satisfies the equation

where t represents the time in minutes after the element is switched on.

When the element is switched on the following conditions apply.

(i) Starting from show that, while the water is being heated,

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(ii) Using the formula for T given in part (i), calculate the maximum theoretical temperature that the water can reach.

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dd 0.7 e ,tT2

2=-

dd 0.7 etT2

2=-

0, 10 and dd 7t T tT

= = =

10 70 1 e .T = + -] g10t

-

10t

-

10t

-


17

(iii) Calculate the time taken after the element is switched on for the temperature of the water to reach 70

oC.

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7 (i) A parachutist jumps from a plane and falls under the influence of gravity until her speed reaches 40 m s–1 vertically downward. Assuming that aerodynamic drag can be neglected while she is falling, calculate the time taken to reach this speed after jumping from the plane. You may assume that her initial vertical speed is zero.

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(ii) When her speed is 40 m s−1 she opens her parachute and continues downward for 100 seconds before landing on the ground. From the time when the parachute is first opened to the time when she lands on the ground her vertical speed, v m s−1, is modelled by

where t is the time in seconds after the parachute was opened.

Show that

where A and B are constants to be determined.

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30 3 17 105 12 4,v t tt

2=+ ++

+c m

3 17 105 12

5 3 2,t tt

tA

tB

2 + ++

= + + +c m


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(iii) For this part of the question you are given the following result.

where a, b and C are constants.

Use the formula for v given in part (ii) to find the vertical distance, d dv t0

100= # ,

travelled by the parachutist in the 100 seconds that the parachute is open.

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(iv) Calculate the height above ground at which the parachutist jumped from the plane.

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END OF QUESTION PAPER

( ) ,lndat b t a

at bC1

+ =++#

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Documents

Oxford Cambridge and RSA Level 3 Cambridge …...2 OC 2018 Answer all the questions.1 Fig. 1 shows part of a circular pipe. The diameter of the pipe varies. At a part of the pipe where