The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.

Size: px

Start display at page:

Download "The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72."

Rodney Blankenship
5 years ago
Views:

1 ADVANCED GCE UNIT 76/ MATHEMATICS (MEI Mechanics MONDAY MAY 7 Additional materials: Answer booklet (8 pages Graph paper MEI Examination Formulae and Tables (MF Morning Time: hour minutes INSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces proided on the answer booklet. Answer all the questions. You are permitted to use a graphical calculator in this paper. Final answers should be gien to a degree of accuracy appropriate to the context. The acceleration due to graity is denoted by g ms. Unless otherwise instructed, when a numerical alue is needed, use g 9.8. INFORMATION FOR CANDIDATES The number of marks is gien in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 7. ADVICE TO CANDIDATES Read each question carefully and make sure you know what you hae to do before starting your answer. You are adised that an answer may receie no marks unless you show sufficient detail of the working to indicate that a correct method is being used. This document consists of 6 printed pages and blank pages. HN/ OCR 7 [K//6] OCR is an exempt Charity [Turn oer

2 (a (i Write down the dimensions of the following quantities. Velocity Acceleration Force Density (which is mass per unit olume Pressure (which is force per unit area [] For a fluid with constant density r, the elocity, pressure P and height h at points on a streamline are related by Bernoulli s equation where g is the acceleration due to graity. P r rgh constant, (ii Show that the left-hand side of Bernoulli s equation is dimensionally consistent. [] (b In a wae tank, a float is performing simple harmonic motion with period.9 s in a ertical line. The height of the float aboe the bottom of the tank is h m at a time t s. When t, the height has its maximum alue. The alue of h aries between.6 and.. (i Sketch a graph showing how h aries with t. [] (ii Express h in terms of t. [] (iii Find the magnitude and direction of the acceleration of the float when h.7. [] OCR 7 76/ June 7

3 A fixed hollow sphere with centre O has an inside radius of.7 m. A particle P of mass. kg moes on the smooth inside surface of the sphere. At first, P is moing in a horizontal circle with constant speed, and OP makes a constant angle of 6 with the ertical (see Fig... O 6.7m P Fig.. (i Find the normal reaction acting on P. [] (ii Find the speed of P. [] The particle P is now placed at the lowest point of the sphere and is gien an initial horizontal speed of 9 m s. It then moes in part of a ertical circle. When OP makes an angle q with the upward ertical and P is still in contact with the sphere, the speed of P is ms and the normal reaction acting on P is R N (see Fig... O q RN P ms - 9ms - Fig.. (iii Find in terms of q. [] (i Show that R.6.76 cos q. [] ( Find the speed of P at the instant when it leaes the surface of the sphere. [] OCR 7 76/ June 7 [Turn oer

4 A light elastic string has natural length. m and stiffness 67 N m. (i The string is stretched to a length of. m. Find the tension in the string and the elastic energy stored in the string. [] One end of this string is attached to a fixed point A. The other end is attached to a heay ring R which is free to moe along a smooth ertical wire. The shortest distance from A to the wire is. m (see Fig...m A R Fig. The ring is in equilibrium when the length of the string AR is. m. (ii Show that the mass of the ring is. kg. [] The ring is gien an initial speed u ms ertically downwards from its equilibrium position. It first comes to rest, instantaneously, in the position where the length of AR is. m. (iii Find u. [7] (i Determine whether the ring will rise aboe the leel of A. [] OCR 7 76/ June 7

5 (a The region bounded by the cure y x for x, the x-axis and the line x, is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina. [8] (b The region bounded by the circular arc y - x for x, the x-axis and the line x, is rotated through p radians about the x-axis to form a uniform solid of reolution, as shown in Fig... y x Fig.. (i Show that the x-coordinate of the centre of mass of this solid of reolution is.. [6] This solid is placed on a rough horizontal surface, with its flat face in a ertical plane. It is held in equilibrium by a light horizontal string attached to its highest point and perpendicular to its flat face, as shown in Fig... Fig.. (ii Find the least possible coefficient of friction between the solid and the horizontal surface. [] OCR 7 76/ June 7

6 76 Mark Scheme June 7 (a(i [ Velocity ] L T [ Acceleration ] [ Force ] [ Density ] [ Pressure ] LT M L T M L (ii [ ] M L P T (b(i [ ρ ] (M L M L M L T (L T T [ ρ g h ] (M L (LT (L M L T All terms hae the same dimensions B B B B B M A A E (Deduct mark if answers gien as ms, ms, kgms etc Finding dimensions of nd or rd term Allow e.g. Equation is dimensionally consistent following correct work M A For a cos cure (starting at the highest point Approx correct alues marked on both axes (ii π Period. 9 ω ω.8 h.9 +.cos. 8t (iii When h.7, float is. m below centre Acceleration is ω x ms upwards OR When h.7, cos.8t (.8t., t. 8 Acceleration h..8 cos. 8t M..8 ( A.68 ms upwards A cao M A M F MA A cao π Accept.9 For h c+ acos/ sin with either c (.6 +. or a (..6 Award M if there is at most one error 6

7 76 Mark Scheme June 7 (i R cos Normal reaction is 7.8 N M A Resoling ertically (e.g. R sin 6 mg is MA R mgcos6 is M (ii R sin 6..7sin 6 Speed is 6. ms M M A A cao Horizontal equation of motion Acceleration (M for r.7 OR R sin 6. (.7sin 6 ω M A ω.69 (.7sin 6ω M Speed is 6. ms A cao Horizontal equation of motion or R..7 ω For rω (M for.7ω (iii By conseration of energy, (i. ( ( cosθ cosθ cosθ R cosθ..7. R +.9 cosθ (8.8.9 cosθ.7 R +.9 cosθ cosθ R.6.76 cosθ ( Leaes surface when R.6 cos θ Speed is.6 ms ( 9.6 M A A M A M A E M A M A cao Equation inoling KE and PE Any (reasonable correct form e.g. 8.9( + cos θ Radial equation with terms Substituting expression for SR If θ is taken to the downward ertical, maximum marks are: MAA in (iii MAMAE in (i Dependent on preious M or using m mg cosθ r 6

8 76 Mark Scheme June 7 (i Tension is N Energy is J (ii Let θ be angle between RA and ertical (iii cosθ ( θ 67. T cosθ mg 6.7 m 9.8 Mass of ring is. kg Loss of PE is. 9.8 (.9. EE at lowest point is 67. ( 8.66 By conseration of energy, u u u.8. u. 67. (i From lowest point to leel of A, Loss of EE is 8.66 Gain in PE is Since 8.66 >., Ring will rise aboe leel of A.8 B M A B M A E M A M A M F A cao M M M A cao 7 Resoling ertically Considering PE or PE at start and finish Award M if not more than one error Equation inoling KE, PE and EE EE at start and at leel of A PE at start and at leel of A (For M it must be the same start Comparing EE and PE (or equialent, e.g. mu +.8 mg. + m Fully correct deriation SR If 67 is used as modulus, maximum marks are: (i BMA (ii BMAE (iii MAMAMFA (i MMMA 6

9 76 Mark Scheme June 7 (a Area is x [ x ] x y x B M 6. x.6 y [ x ] 6. y y y 7 [ x ] 6 x 6 7 A A M A M Condone omission of A 8 Accept. from correct working (b(i Volume is π y π x y π x y x π y π ( x π [ x ] π x π x( x 9 π [ x x ] π M A M A M π may be omitted throughout For For 9 9 π 7. π E 6 Must be fully correct (ii Height of solid is h Th mg. F T.mg, R mg F Least coefficient of friction is. R B M F A Taking moments Must be fully correct (e.g. A if m π is used 66

MEI STRUCTURED MATHEMATICS 4763

MEI STRUCTURED MATHEMATICS 4763 OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MEI STRUCTURED MATHEMATICS 76 Mechanics Monday MAY 006 Morning hour