L.41/42. Pre-Leaving Certificate Examination, Applied Mathematics. Marking Scheme. Ordinary Pg. 2. Higher Pg. 21.

Transcription

1 L./ Pre-Leaving Certificate Examination, 0 Applied Mathematics Marking Scheme Ordinary Pg. Higher Pg. Page of 6

2 exams Pre-Leaving Certificate Examination, 0 Applied Mathematics Ordinary Level Marking Scheme (00 marks) General Instructions. Penalties of three types are applied to students work as follows: Slips - numerical slips S( ) Blunders - mathematical errors B( ) Misreading - if not serious M( ) Serious blunder or omission or misreading which oversimplifies: - award the attempt mark only. Attempt marks are awarded as follows: 5 (Att. ), 0 (Att. ). Mark all answers, including excess answers and repeated answers whether cancelled or not, and award the marks for the best answers.. Mark scripts in red unless a student uses red. If a student uses red, mark the script in blue or black.. Number the grid on each script to 9 in numerical order, not the order of answering. 5. Scrutinise all pages of the answer book. 6. The marking scheme shows one correct solution to each question. In many cases there are other equally valid methods. Six questions to be answered. All questions carry equal marks. (6 50m) 0 L./_MS /8 Page of 8 exams

3 . A car travels along a straight level road. It starts from rest at a point P and accelerates uniformly for 0 seconds to a speed of 0 ms. It then moves at a constant speed of 0 ms for 0 seconds. Finally the car decelerates uniformly from 0 ms to rest at Q in k seconds. The distance from P to Q is,00 m. (i) Draw a speed-time graph of the motion of the car from P to Q. (0) Velocity (ms ) Time (s) k t... (0m) (ii) Find the uniform acceleration of the car. (0) v u + at u 0 v 0 t a(0) 0a a 0 0 m/s (iii) Find the distance travelled while the car is accelerating. (5) s ut + ½at u 0 a t 0 s 0(0) + ()(0) m 0 L./_MS /8 Page of 8 exams

4 (iv) Find the distance travelled while the car is decelerating. (5) Velocity (ms ) k Time (s) t s (area ) Total distance area + area,00 (0)(0) + (0)(0) or s u + v t u 0 v 0 t 0 t 0 Distance accelerating s (0) (0)(0) 00 m, , m Distance at constant speed s (0) (0)(0),00 m Distance decelerating d (total distance) s + s + s s d s s,00 (0)(0) (0)(0),00 00, m 0 L./_MS /8 Page of 8 exams

5 (v) Find the value of k. (0) or s area 600 u 0 t k 0k 600 k s v u + as u 0 v 0 s (0) + a(600),00a,600 a, 600, 00 v u + at u 0 v 0 a 0 0 k k 0 (0)() k 0 s 0 L./_MS 5/8 Page 5 of 8 exams

6 . A river is 8 m wide and is flowing with a speed of. ms parallel to the straight banks. The speed of a boat in still water is ms. (i) Find the total time it takes the boat to travel up river (against the flow) for 800 m and then back the same distance (with the flow). (0) t up river d v boat v river t down river d v boat + v river d 800 v boat v river. t total (0m) , ,50 s... (0m) (ii) How long does it take the boat to cross the river by the shortest path? (5). v v (v boat ) (v river ) () (.)..56 v.6 t d v t s (iii) What is the shortest time it takes the boat to cross the river? (5) t d v boat 8... (0m) s 0 L./_MS 6/8 Page 6 of 8 exams

7 . A straight vertical cliff is 500 m high. A particle is fired horizontally directly out to sea from the top of the cliff with an initial speed of 00 ms. (i) 500 m 00 ms v ms Find the distance from the foot of the cliff that the particle strikes the sea. s y u y t + ½a y t s y 500 u y 0 a y g 0 (0) 500 0(t) + ( 0)t... (0) 500 5t 5t 500 t 00 t 0 s s x u x t u x 00 t 0 s x 00(0)... (0m),000 m A second particle is fired horizontally directly out to sea with an initial speed of v ms from a point 80 m above the foot of the cliff. This particle strikes the sea at the same point as the first particle. (ii) Find the value of v. s y u y t + ½a y t s y 80 u y 0 a y g 0 (0) 80 0(t) + ( 0)t 80 5t 5t 80 t 6 t s s x u x t s x,000 t,000 v() v, ms 0 L./_MS 7/8 Page 7 of 8 exams

8 . A particle, of mass kg, rests on a smooth horizontal table. It is connected by two taut, light, inextensible strings which pass over smooth light pulleys at the edges of the table to particles, of masses 5 kg and kg, which hang freely under gravity. kg kg 5 kg The system is released from rest. (i) Show on separate diagrams the forces acting on each particle. (5) T kg a g S 5kg a 5g a R T kg g S (ii) Write down the equation of motion for each particle. (5) g T a S 5g 5a T S a R g (0) 0 N (iii) Find the common acceleration of the particles. (0) g T a (0) T a 0 T a S 5g 5a S 5(0) 5a 50 + S 5a 0 T a 50 + S 5a T S a 70 9a a 70 ms 9 /.68 m/s... (0m) 0 L./_MS 8/8 Page 8 of 8 exams

9 (iv) Find the tension in each string. (0) g T a T g a 70 (0) ( ) ,0 N / N 9 S 5g 5a S 5a 5g 70 5( ) 5(0) N /.579 N 9 0 L./_MS 9/8 Page 9 of 8 exams

10 5. A smooth sphere A, of mass 5 kg, collides directly with another smooth sphere B, of mass kg, on a smooth horizontal table. Before impact A and B are moving in opposite directions with speeds of ms and 6 ms respectively. A ms 6ms 5 kg B kg B The coefficient of restitution for the collision is. Find (i) the speed of A and the speed of B after the collision (0) Sphere Mass Speed Speed before after A 5 kg p B kg 6 q COM: m A u A + m B u B m A v A + m A v A 5() + ( 6) 5(p) + (q) 8 5p + q NEL: v A vb u A u B p q ( 6) e (p q) ( + 6) 0 p q 0 p q 5 5p + q 8 ( ) p q 5 ( ) 7p p ms 7 5p + q 8 q 8 5p 8 5( ) 7 7 q ms 0 L./_MS 0/8 Page 0 of 8 exams

11 5(ii) the loss in kinetic energy due to the collision (5) K.E. before collision K.E. before m Au A + mbub (5)() + ()( 6) (80) + (7) J K.E. after collision K.E. after m AvA + mbvb (5)(7 ) + ()( ) 5, ,77 J / 9.06 J 96 K.E. loss K.E. before K.E. after, ,5 J / J 96 (iii) the magnitude of the impulse imparted to A due to the collision. (5) I A m A v A m A u A 5( ) 5() Ns / N 7 0 L./_MS /8 Page of 8 exams

12 6. (a) Particles of weights N, N and 5 N are placed at the points (, 0), (, 5) and (5, 6), respectively. Find the co-ordinates of the centre of gravity of the system. (0) (, 0) + (, 5) + 5 (5, 6) 0 (x, y) () + () + 5(5) 0 x x () + () + 5(5) ( 0) + (5) + 5( 6) 0 y y ( 0) + (5) + 5( 6) cog (6, 5) 0 L./_MS /8 Page of 8 exams

13 F 6(b) A uniform lamina ABCFE consists of a rectangle ABCE and two right-angled triangles EDF and DCF. E D C The co-ordinates of the points are A(0, 0), B(5, 0), C(5, 8), D(9, 8), E(0, 8) and F(9, ). Find the co-ordinates of the centre of gravity of the lamina. A B (0) Rectangle ABCE Area of rectangle ABCE (5 0)(8 0) (5(8) 0 units Centres of gravity of ΔECF g , cog ΔECF (7.5, ) Triangle ECF E(0, 8) (0, 0) C(5, 8) (5, 0) F(9, ) (9, 6) Area of ΔECF (5)(6) (9)(0) 90 5 units Centres of gravity of ΔECF g, cog ΔECF (8, 0) Co-ordinates of the centre of gravity of the lamina (7.5, ) + (8, 0) (0 + (x, y) 0(7.5) + 5(8) 65(x) x,60 65x x, / () + 5(0) 65(y) y 90 65y y / cog of lamina (, ) / (7.66, 5.66) 0 L./_MS /8 Page of 8 exams

14 7. A uniform ladder, of weight W, rests on rough horizontal ground and leans against a smooth vertical wall. The foot of the ladder is 8 m from the wall and the top of the ladder is 5 m above the ground. 5 m (i) Find the length of the ladder. (5) l m 89 l 7 m The coefficient of friction between the ladder and the ground is. (ii) Show on a diagram all of the forces acting on the ladder. (0) S 7 5 m R 7 l W R 8m... (0m) (iii) Show that the value of > 5. (0) R W S R S W Assume that the ladder is on the point of slipping Taking foot of the ladder (5)S ( 8 )W S (8) W (5)() W 5 0 L./_MS /8 Page of 8 exams

15 7(iii) (Continued) S W W 5 W W 5 5 if, then the ladder will not slip 5 (iv) A person, whose weight is the same as the ladder, starts to climb the ladder. If the coefficient of friction between the ladder and the ground is, how far can the person safely climb the ladder before it begins to slip? (5) S x W W 5 m R l R 8m R W + W W S R (W) W S(5) W() + W(x sin ) 5( W) W + Wx(7 8 ) Wx( 7 8 ) 0W W 6W x 7 ( )( )(6) m 0 L./_MS 5/8 Page 5 of 8 exams

16 8. A sphere of diameter 0 m is fixed to a horizontal surface. A smooth particle of mass kg describes a horizontal circle of radius r m on the smooth inside surface of a sphere. The plane of the circular motion is. m below the centre of the sphere. kg r. m (i) Find the value of r. (0) kg 5m r. m r + (.) (5) r r r.8 m (ii) Find the normal reaction between the particle and the surface of the sphere. (0) R R Rsin Rcos mg sin R sin g... (0m) 7 R( ) 5 (0) R (0)(5) N 0 L./_MS 6/8 Page 6 of 8 exams

17 8(iii) Calculate the angular velocity of the particle. Give your answer correct to one decimal place. (0) cos R cos m r 500( ) 5 ()( )(.8)... (0m) rad/s 0 L./_MS 7/8 Page 7 of 8 exams

18 9. (a) State the Principle of Archimedes. (0) when a body is wholly or partly immersed in a liquid, it suffers an upthrust or buoyancy equal in magnitude to the weight of the liquid displaced A solid piece of metal, of relative density 5, weighs 0 N in air. (i) How much does the metal weigh in water? (0) B w W s 0 5 Apparent weight 0 6 N (ii) How much does the metal weigh in a liquid of relative density 0.8? (0) B l s l W s 0.8(0) 5. Apparent weight N 0 L./_MS 8/8 Page 8 of 8 exams

19 9(b) A sphere has a radius of.5 m. The relative density of the sphere is 0.8 and it is completely immersed in a tank of liquid of relative density The sphere is held at rest by a light inextensible string which is attached to the base of the tank. (i) Find the weight of the sphere, correct to two decimal places. (0) V πr (...)(.5) m W l Vg (0.8,000)(8.8)(0) 65,0 N (ii) Find the tension in the string. [Density of water,000 kg m.] (0) B l Vg (0.85,000)(8.8)(0) 69,50 N B W + T 69,50 65,0 + T T 69,50 65,0,090 N 0 L./_MS 9/8 Page 9 of 8 exams

20 Notes: 0 L./_MS 0/8 Page 0 of 8 exams

21 exams Pre-Leaving Certificate Examination, 0 Applied Mathematics Higher Level Marking Scheme (00 marks) General Instructions. Penalties of three types are applied to students work as follows: Slips - numerical slips S( ) Blunders - mathematical errors B( ) Misreading - if not serious M( ) Serious blunder or omission or misreading which oversimplifies: - award the attempt mark only. Attempt marks are awarded as follows: 5 (Att. ), 0 (Att. ). Mark all answers, including excess answers and repeated answers whether cancelled or not, and award the marks for the best answers.. Mark scripts in red unless a student uses red. If a student uses red, mark the script in blue or black.. Number the grid on each script to 0 in numerical order, not the order of answering. 5. Scrutinise all pages of the answer book. 6. The marking scheme shows one correct solution to each question. In many cases there are other equally valid methods. Six questions to be answered. All questions carry equal marks. (6 50m) 0 L./_MS /8 Page of 8 exams

22 . (a) Two people are running straight towards each other. At a certain instant they are 57.5 m apart. The first person is moving at ms and constant acceleration of 0. ms and the other person is moving at ms and constant acceleration of 0.05 ms. (i) How long does it take for them to meet? (0) s ut + ½at s st person + s nd person 57.5 u u a 0. a 0.05 t t t t (t + (0.)t ) + (t + (0.05)t ) 57.5 t t + t t t t 57.5 t + 00t,00 0 (t + 0)(t 0) 0 t t 0 Solution not valid t 0 0 t 0 (ii) Find the speed of the two people when they meet. (0) v u + at u a 0. t 0 v st person + (0.)(0) ms v u + at u a 0.05 t 0 v nd person + (0.05)(0).5 ms 0 L./_MS /8 Page of 8 exams

23 (b) A cyclist, who is travelling at a constant speed of 8 ms on a straight road, observes a bus 50 m ahead. The bus starts to accelerate from rest, with a constant acceleration. If the cyclist continues at a constant speed of 8 ms, he can just catch up with the bus. (i) Find the constant acceleration of the bus. (5) v u + as Relative to the bus u cb u c u b ms a cb a c a b 0 a a s cb 50 v cb 0 0 (8) + ( a)(50) 00a 6 a ms (ii) How near would the cyclist get to the bus if he can only manage to cycle at 6 ms? Relative to the bus u cb u c u b ms Minimum gap when their speeds are equal v c v b v b u b + a b t v b 6 u b 0 a b t t s at t 9.75 s s c 6(9.75) 56.5 m s b ut + at 0 + (0.6)(9.75) 8.5 Rel. displacement Minimum gap m (5) 0 L./_MS /8 Page of 8 exams

24 . (a) Two straight roads intersect at right angles. A car is travelling north on one road at ms. A bus is travelling east on the other road at ms. Bus The bus is 50 m away from the intersection as the car passes through the intersection. (i) Find the velocity of the bus relative to the car. (0) v C j v B i v BC v B v C i j v BC () + () Car (ii) Find the shortest distance between the car and the bus and the time at which this occurs. (0) Direction tan tan tan (...) Relative path 50 x x 50 sin 50( ) 5 y y 0 m Time x t vbc s 0 L./_MS /8 Page of 8 exams

25 (a) (iii) Calculate the length of time for which the car and the bus are less than or equal to m apart. (0) y + x y + 0,68 y,68,600 8 y 9 Time y v BC (9) 5.6 s (b) The wind is blowing with a constant speed of 5 ms. To a man running at ms in a southerly direction, the wind appears to blow from the north-west. (i) Find the true velocity of the wind. (0) Let v w x i + y j v w 5 x + y 5 x + y 5 v m 0 i j v wm v w v m (x i + y j ) (0 i j ) x i + (y + ) j from NW x (y + ) By substitution: [ (y + )] + y 5 y + y + + y 5 y + y 0 y + y 0 (y )(y + ) 0 y 0 y + 0 y y x ( + ) x ( + ) If v w i + j v wm i + j to NW... (reject) If v w i j v wm i j from NW v w i j 0 L./_MS 5/8 Page 5 of 8 exams

26 (b) (ii) From what direction would the wind appear to blow if the man reversed direction but continued to run at the same speed? (0) v m 0 i + j v wm ( i j ) (0 i + j ) i 5 j tan 5 tan South of East 0 L./_MS 6/8 Page 6 of 8 exams

27 . (a) A particle is projected with a velocity of ms at an angle of 5 to the horizontal plane. Find (i) the maximum height of the particle above the plane (0) sin 5 cos 5 u cos 5 i + 56 sin 5 j ( ) i + ( ) j i + j Maximum height: u y j a y g v y 0 v () + ( g)h max 0,5 (9.8)h max 9.6h max,5 h max, m (ii) the velocity of the particle after seconds. (0) v x i v y (u y + a y t) j ( 9.8()) j ( 9.) j 7.8 j Speed x v y v + ( ) + (7.8),5 + 5, , m/s 0 L./_MS 7/8 Page 7 of 8 exams

28 (b) Two particles X and Y are fired at the same time and with the same velocity of 98 ms from a point O on horizontal ground. X is fired at an angle 5 to the horizontal. Y is fired at an angle tan to the horizontal. Both particles hit the same stationary target, but not at the same time. (i) Write down the position vector of X after time t and the position vector of Y after time t. (Take O as the origin.) (0) s x u cos t s y u sin t gt r 98 X t i + ( 98) t (9.8) t ti + t. 9t j j r 98 t Y i + 0 ( 98) t (9.8) t j ti + 0 t. 9t j 0 (ii) Show that t 5 t. (5) Y r r X 98 9 ti t. 9t j ti + 0 t. 9t t t 0 j t 0 t 5 t 5 t... (5) 0 L./_MS 8/8 Page 8 of 8 exams

29 (b)(iii) Show that the position of the target is 90 i + 5 j. (5) 98 9 t. 9t t. 9t 0 9 ( 5t ).9( 5t) 0 9 t. 5t t +.5t t t t t 96 t t (9.6) 0 t t t r t X i + t. 9t j 98 0 i j i + j i + j 90 i + 5 j 0 L./_MS 9/8 Page 9 of 8 exams

30 . (a) The diagram shows a light inextensible string having one end fixed, passing under a smooth movable pulley P of mass 8 kg and then over a fixed smooth light pulley Q. A particle of mass 0 kg is attached to the free end of the string. The system is released from rest. Find the acceleration of P and the particle, in terms of g. P 8kg Q 0 kg (0) Q T P 8kg T T 0 kg a T 8g 8a 0g T 0(a) T 8g 8a 0g T 0a ( ) T 8g 8a T + 0g 0a g 8a a g 8 g ms a P g ms a particle ( g ) g ms 0 L./_MS 0/8 Page 0 of 8 exams

31 (b) A light inextensible string passes over a small smooth fixed pulley. A particle A, of mass M, is attached to one end of the string and a light smooth movable pulley is attached to the other end. Two particles of masses of m and m are connected by a light inextensible string which passes over this pulley. The system is released from rest. A M (i) Find, in terms of g, M, m and m, m m the acceleration of the particle A. (5) Mg T Ma T S 0 S m g m (a + b) m g S m (b a) Mg T Ma T Mg Ma T S 0 S T Mg Ma S Mg Ma S m g m (a + b) Mg Ma m g m a + m b Mg Ma m g m a + m b m Mg m Ma m m g m m a + m m b 0 L./_MS /8 Page of 8 exams

32 (b) (Continued) m g S m (b a) m g Mg + Ma m b m a m g Mg + Ma m b m a m m g m Mg + m Ma m m b m m a m m g + m Mg m Ma m m b + m m a Adding these two equations: m Mg m Ma m m g m m a + m m b m Mg m Ma m m g m m a m m b m Mg m Ma + m Mg m Ma m m g m m a g(m M + m M m m ) a(m m + m M + m M) g(m(m + m ) m m ) a(m m + M(m + m )) a g( M ( m + m) mm ) m m + M ( m + m ) (ii) Show that the system will remain at rest mm if the mass of particle A is. (5) m + m Let a 0 g( M ( m + m) mm ) m m + M ( m + m ) 0 g(m(m + m ) m m ) 0 M(m + m ) m m 0 M(m + m ) m m M mm m + m 0 L./_MS /8 Page of 8 exams

33 5. (a) A smooth sphere P, of mass m, moving with speed u, collides directly with an identical smooth sphere Q, which is moving in the same direction with speed u. The coefficient of restitution for the collision is e. Find, in terms of m, u and e, the loss of kinetic energy due to the collision. (0) Sphere Mass Speed Speed before after P m u p Q m u q COM: m A u A + m B u B m A v A + m A v A m(u) + m(u) mp + mq mp + mq mu p + q u NEL: va vb u A ub e u u p q e p q ue p + q u p q ue p u( e) p u( e) q u p q u u( e) Before collision: K.E. After collision: K.E. u(6 + e) u( + e) (m)(u) + (m)(u) 5 mu (m)[u( e)] + (m)[u( + e)] (m) u ( e) + (m) u ( + e) (mu )(9 6e + e ) + (mu )(9 + 6e + e ) 8 8 (mu )(8 + e ) 8 (mu )(9 + e ) Loss in K.E. 5 mu (mu )(9 + e ) mu 0 9 ( e ) mu ( e ) 0 L./_MS /8 Page of 8 exams

34 5(b) A smooth sphere A, of mass m, moving with speed u, collides with a smooth sphere B, of mass m, which is at rest. The direction of motion of A, before impact, makes an angle of 5 with the lines of centres of the instant of impact. m 5 u A B m After the collision the two spheres move in perpendicular directions. The coefficient of restitution between the spheres is e. (i) Show that e. (0) Sphere A Mass m Speed before Speed after u u i + j p i + u j B m 0 i + 0 j q i + 0 j After collision, A and B perpendicular: (p u i + j )( q i + 0 j ) 0 pq 0 p 0 COM: m A u A + m B u B m A v A + m B v B u m( ) + m(0) m(0) + mq mq mu q u u NEL: v A vb u A u B 0 q u 0 q e e eu eu e u 0 L./_MS /8 Page of 8 exams

35 5(b) (ii) Find, in terms of u, the speed of each sphere after the collision. (5) v A 0 i + u j v B q i + 0 j eu i + 0 j (iii) Find the percentage loss in kinetic energy due to the collision. (5) K.E. before (m)(u ) + (m)(0 ) mu u K.E. after (m)( ) u + (m)( ) u u m( + ) () mu ( + ) mu K.E. loss mu mu mu % K.E. loss mu 00 mu 5% 0 L./_MS 5/8 Page 5 of 8 exams

36 6. (a) A light inextensible string, of length m, is attached at one end to a fixed point P which is m above a smooth horizontal table. The other end of the string is attached to a particle of mass 0 kg which moves uniformly in a horizontal circle, centre O. O is vertically below P. The reaction between the particle and the table is 8 N. (i) Find the tension in the string. (0) r + ( ) r + r mg R + T sin 0(9.8) 8 + T( ) T T 50 (ii) Find the angular velocity of the particle. (0) T cos m r 50 ( ) (0)( )() rad/s 0 L./_MS 6/8 Page 6 of 8 exams

37 6(b) A particle P, of mass m, is attached to the midpoint of an elastic string [ AB ], of natural length l and elastic constant k. A and B are attached to fixed points on a smooth horizontal table, a distance of l apart. Initially the particle is held at rest in a position such that AP l and PB l, and is then released. (i) Show that the motion of the particle is simple harmonic. (0) l l x x F r k(l l o ) k( l x l) ke kx F l k(l l o ) k( l + x l) k( l + x) kl + kx F F r F l ( kl kx)( lk + kx) kl kx lk kx kx ma kx a k x m a x (ii) Find, in terms of m and k, the period of the motion. (0) k a x m a x k m T π ω π m k 0 L./_MS 7/8 Page 7 of 8 exams

38 7. (a) One end of a uniform ladder, of weight W and length l, rests against a rough vertical wall, and the other end rests on rough horizontal ground. The coefficient of friction at each contact is. l The ladder makes an angle of tan with the horizontal and is in a vertical plane which is perpendicular to the wall. Find the distance that a person of weight W can safely climb before the ladder begins to slip. (0) R + S W + W R + S W S R (W) W R S R S (W S) S R W W S S 9 R 6W S 9S 6W 0S S W 5 x S W S Taking foot of the ladder (5)S 8 ( )W W(l cos ) + W(x cos ) S(l sin ) + S(l cos ) Wl + Wx ls tan + Sl tan Wl + Wx ls + Sl Wl + Wx l( W) + (5 W)l 5 Wl + Wx Wx x Wl + Wl 5 5 Wl 5 Wl Wl 5 9 Wl 5 9 l 5 0 L./_MS 8/8 Page 8 of 8 exams

39 7(b) Two uniform ladders, AC and BC, of equal length l and weights W and W respectively, are freely jointed at C. They stand in a vertical plane with A and B on rough horizontal ground. Both ladders make an angle with the ground. The coefficient of friction at both A and B is. (i) Find, in terms of W, the normal reactions at A and B. (5) C A C B R A W W F G S B R + S W + W W F G Taking A W(l cos ) + W(l cos ) S(l cos ) W + 9W S S 0W S 5 W R W 5 W W (ii) Show that slipping will occur first at A. (5) R < S R < S slipping will occur at A before C 0 L./_MS 9/8 Page 9 of 8 exams

40 7(b) (iii) Find the value of tan at which slipping will occur. (0) X Y R F W X μr ( W) W Y + W R W Y W W W Taking A W(l cos ) + Y(l cos ) X(l sin ) W(l cos ) + ( W)(l cos ) ( W)(l sin ) Wl cos + Wl cos Wl sin cos + cos sin cos sin cos sin sinθ cosθ tan 0 L./_MS 0/8 Page 0 of 8 exams

41 8. (a) Prove that the moment of inertia of a uniform square lamina, of mass m and side l, about an axis through its centre parallel to one of the sides is ml. (0) Bookwork m l ΔI set up integral and solve Deduce 8(b) A uniform circular disc, of mass m and radius r, has a particle, of mass m, attached to its centre. The system performs small oscillations in a vertical plane about a horizontal axis l through a point on its circumference, perpendicular to the plane of the disc. (i) Find, in terms of g and r, the period of small oscillations. (5) Disc: I centre Mr I P I c + Mr Mr + Mr Mr Point Mass: I point mass Md (m)(r) mr I total I P + I point mass mr + mr 7 mr I T π Mgh π π 7 mr (m) g( r) 7r 6g 0 L./_MS /8 Page of 8 exams

42 8(b) The disc is held with the particle vertically above l, and is then released from rest. (ii) Find the maximum angular velocity in the subsequent motion. (5) Mg h + Mg h + I Mg h + Mg h + I Disc Point Mass System Disc Point Mass System mg(r) + mg(r) + 0 mg(r) + mg(r) + ( 7 mr ) RHS LHS 9mgr 7 mgr + mr 7 mr 9mgr mgr 6mgr 7 r 6g ()(6g) 7r g 7r g 7r 0 L./_MS /8 Page of 8 exams

43 9. (a) A piece of metal weighs 5 N in air and N in water. The metal weighs N in a light oil. Find (i) the relative density of the metal (0) B w 5 B w W s 0 5 s0 s (ii) the relative density of the oil. (0) B l 5 B l s l s l W0 s0 s l (5) 5 ()(5) 5 0 L./_MS /8 Page of 8 exams

44 9(b) A solid hemisphere of radius r is held immerged in a tank of liquid of density. Its plane face is in a horizontal position at a distance r below the surface of the liquid. Find the magnitude and direction of (i) the downward thrust exerted on the plane face by the liquid (0) Pressure hg rg Thrust Pressure area (rg)(πr ) πr g (ii) the buoyancy force (5) Buoyancy Vg ( πr )g πr g (iii) the total thrust upwards on the curved surface of the hemisphere. (5) T downwards T upwards B T upwards T downwards πr g T upwards πr g T upwards πr g + πr g 5 πr g 0 L./_MS /8 Page of 8 exams

45 0. (a) Solve the differential equation ( + x dy ) x y dx given that y when x. (0) ( + x dy ) dx dy y dy y ln y, x ln x y x dx + x x dx + x ln ( + x ) + c ln ( + ) + c c ln ln ln ln y ln ( + x ) + ln ln( + x ) () ln( + x ) () ln( + x ) y ( + x ) / + x 0 L./_MS 5/8 Page 5 of 8 exams

46 m 0(b) A force of magnitude 5 thrusts a particle, of mass m, directed away from x a fixed point o, where x is the distance of the particle from O. The particle starts from rest at x d. (i) Show that the velocity of the particle is when x d. (5) d F m x ma m x a x x 5 dv v dx x 5 v 0 v dv v v x d x 5 dx x 0 d v (0) + x d v d x when x d v d ( d) d 9d 8 9d v 8 9d d x (ii) Determine the limiting speed, v, of the particle. (that is, v v as t ). (5) v d x as t, x (as the particle always accelerates to the right) as x, x 0 v d 0 v d d 0 L./_MS 6/8 Page 6 of 8 exams

47 0 L./_MS 7/8 Page 7 of 8 exams

48 0 L./_MS 8/8 Page 8 of 8 exams