Centre Number Candidate Number Edexcel GCE Mechanics M3 · 2016. 5. 18. · One end of a light...

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Turn over Centre Number Candidate Number Write your name here Surname Other names Total Marks Paper Reference Pearson Edexcel GCE Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. Instructions Use black ink or ball-point pen. If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used. Fill in the boxes at the top of this page with your name, centre number and candidate number. Answer all questions and ensure that your answers to parts of questions are clearly labelled. Answer the questions in the spaces provided there may be more space than you need. You should show sufficient working to make your methods clear. Answers without working may not gain full credit. Whenever a numerical value of g is required, take g = 9.8 m s –2 , and give your answer to either two significant figures or three significant figures. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Information The total mark for this paper is 75. The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question. Advice Read each question carefully before you start to answer it. Try to answer every question. Check your answers if you have time at the end. You must have: Mathematical Formulae and Statistical Tables (Pink) *P46709A0124* P46709A ©2016 Pearson Education Ltd. 1/1/1/1/ Wednesday 18 May 2016 – Morning Time: 1 hour 30 minutes 6679/01 Mechanics M3 Advanced/Advanced Subsidiary

Transcript of Centre Number Candidate Number Edexcel GCE Mechanics M3 · 2016. 5. 18. · One end of a light...

Page 1: Centre Number Candidate Number Edexcel GCE Mechanics M3 · 2016. 5. 18. · One end of a light elastic string, of natural length 1.5m and modulus of elasticity 14.7N, is attached

Turn over

Centre Number Candidate Number

Write your name hereSurname Other names

Total Marks

Paper Reference

Pearson

Edexcel GCE

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions• Use black ink or ball-point pen.• If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Coloured pencils and highlighter pens must not be used.• Fill in the boxes at the top of this page with your name,

centre number and candidate number.• Answer all questions and ensure that your answers to parts of questions are clearly labelled.• Answer the questions in the spaces provided – there may be more space than you need.• You should show sufficient working to make your methods clear. Answers without working may not gain full credit.• Whenever a numerical value of g is required, take g = 9.8 m s–2, and give your answer to either two significant figures or three significant figures.• When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information• The total mark for this paper is 75.• The marks for each question are shown in brackets – use this as a guide as to how much time to spend on each question.

Advice• Read each question carefully before you start to answer it.• Try to answer every question.• Check your answers if you have time at the end.

You must have:

Mathematical Formulae and Statistical Tables (Pink)

*P46709A0124*P46709A©2016 Pearson Education Ltd.

1/1/1/1/

Wednesday 18 May 2016 – MorningTime: 1 hour 30 minutes 6679/01

Mechanics M3Advanced/Advanced Subsidiary

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1. A particle P of mass 0.5kg is moving along the positive x-axis under the action of a resultant force. The force acts along the x-axis. At time t seconds, P is x metres from the

origin O and is moving away from O in the positive x direction with speed 12

31

x +−m s

(a) Find the magnitude of the force acting on P when x = 3 (4)

Given that x = 4 when t = 2

(b) find the value of t when x = 10 (5)

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(Total 9 marks)

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2.

Figure 1

Figure 1 shows a uniform triangular lamina ABC in which AB = 6cm, BC = 9cm and angle ABC = 90°. The centre of mass of the lamina is G. Use algebraic integration to find the distance of G from AB.

(6)

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9cm

6cm

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(Total 6 marks)

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3. One end of a light elastic string, of natural length 1.5m and modulus of elasticity 14.7N, is attached to a fixed point O on a ceiling. A particle P of mass 0.6kg is attached to the free end of the string. The particle is held at O and released from rest. The particle comes to instantaneous rest for the first time at the point A.

Find

(a) the distance OA,(6)

(b) the magnitude of the instantaneous acceleration of P at A.(3)

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(Total 9 marks)

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4.

Figure 2

A uniform solid S consists of two right circular cones of base radius r. The smaller cone has height 2h and the centre of the plane face of this cone is O. The larger cone has height kh where k > 2. The two cones are joined so that their plane faces coincide, as shown in Figure 2.

(a) Show that the distance of the centre of mass of S from O is

h k4

2−( )(5)

The point A lies on the circumference of the base of one of the cones. The solid is suspended by a string attached at A and hangs freely in equilibrium.

Given that r = 3h and k = 6

(b) find the size of the angle between AO and the vertical.(3)

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r

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(Total 8 marks)

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5.

Figure 3

A particle P of mass m is attached to the ends of two light inextensible strings. The other ends of the strings are attached to fixed points A and B, where B is vertically below A and AB = l. The particle is moving with constant angular speed in a horizontal circle.Both strings are taut and inclined at 30° to AB, as shown in Figure 3.

(a) (i) Show that the tension in AP is 23 (2 )6m g lω+

(ii) Find the tension in BP.(9)

(b) Show that the time taken by P to complete one revolution is less

than 2lg (4)

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B

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P (m)l

30°

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(Total 13 marks)

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6. One end of a light inextensible string of length l is attached to a particle P of mass 2m. The other end of the string is attached to a fixed point A. The particle is hanging freely at

rest with the string vertical. The particle is then projected horizontally with speed 72gl

(a) Find the speed of P at the instant when the string is horizontal.(4)

When the string is horizontal and P is moving upwards, the string comes into contact with a small smooth peg which is fixed at the point B, where AB is horizontal and AB < l. The particle then describes a complete semicircle with centre B.

(b) Show that AB 12

l(9)

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(Total 13 marks)

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7. A particle P of mass 0.5kg is attached to one end of a light elastic spring, of natural length 1.2m and modulus of elasticity 15N. The other end of the spring is attached to a fixed point A on a smooth horizontal table. The particle is placed on the table at the point B where AB = 1.2m. The particle is pulled away from B to the point C, where ABC is a straight line and BC = 0.8m, and is then released from rest.

(a) (i) Show that P moves with simple harmonic motion with centre B.

(ii) Find the period of this motion.(5)

(b) Find the speed of P when it reaches B.(2)

The point D is the midpoint of AB.

(c) Find the time taken for P to move directly from C to D.(3)

When P first comes to instantaneous rest a particle Q of mass 0.3kg is placed at B. When P reaches B again, P strikes and adheres to Q to form a single particle R.

(d) Show that R also moves with simple harmonic motion.(3)

(e) Find the amplitude of this motion.(4)

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Question 7 continued

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Question 7 continued

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TOTAL FOR PAPER: 75 MARKS

END

Q7

(Total 17 marks)