Examination Paper: Physics Problems on Motion, Forces, and Circles, Exams of Applied Mathematics

Download Examination Paper: Physics Problems on Motion, Forces, and Circles and more Exams Applied Mathematics in PDF only on Docsity! Page 1 Coimisiún na Scrúduithe Stáit State Examinations Commission LEAVING CERTIFICATE 2010 MARKING SCHEME APPLIED MATHEMATICS ORDINARY LEVEL Page 2 Page 3 2. A river is 100 metres wide and has parallel banks. Boat B departs from point P on its western bank and lands at point Q on its eastern bank. The actual velocity of the boat is 5  i + 12  j 1sm  . Cyclist C travels due north at a constant speed of 3 1sm  along the eastern bank of the river. Find (i) the velocity of C in terms of  i and  j (ii) the velocity of B relative to C in terms of  i and  j (iii) the magnitude and direction of the velocity of B relative to C (iv) the time it takes B to cross the river (v) |PQ|, the distance from P to Q.       m 260 1302 sm 13 125 along speed (v) s 20 5 100 time (iv) N. 60.9 E dirn .310106 95(iii) 95 30125 (ii) 3 0V (i) 1 22 22 C              PQ PQ or V j i j i j i VVV ji BC CBBC      10 5 5 5 5 10 5 5 50 P Q B C 100 m Page 4 3. A particle is projected with initial velocity 72  i + 30  j 1sm  from the top of a straight vertical cliff of height 35 m. It strikes the horizontal ground at P. Find (i) the time taken to reach the maximum height (ii) the maximum height of the particle above ground level (iii) the time of flight (iv) |OP|, the distance from O to P (v) the speed of the particle as it strikes the ground.             sm 4.82 4072 40 70130 (v) m 504 0772 (iv) s 7 076 53035 (iii) m 803545 distance m 45 9 5330 (ii) s 3 10300 (i) 1 22 2 2 1 2 2 2 2 1 2 2 1                    v atu v atutOP t tt tt atut s tfut s t t ftuv y y 10 10 10 10 5 5 50 35 m 30 1sm  72 1sm  O P Page 5 4. (a) Two particles of masses 5 kg and 7 kg are connected by a taut, light, inextensible string which passes over a smooth light pulley. The system is released from rest. Find (i) the common acceleration of the particles (ii) the tension in the string. (i) 2sm 3 5 or 12 20 77 55    a aTg agT (ii) N 58.3 50 3 25 55    T gaT 20 5 5 5 5 5 kg 7 kg Page 8 6. (a) Particles of weight 3 N, 7 N, 1 N and 5 N are placed at the points        8,4 and ,2 ,2, ,,1 qpp  , respectively. The co-ordinates of the centre of gravity of the system are  54,53  . Find (i) the value of p (ii) the value of q. (b) A triangular lamina with vertices A, B and C has the square portion with diagonal [AD] removed. The co-ordinates of the points are      0,12,9,0 ,0,0 CBA and  4,4D . Find the co-ordinates of the centre of gravity of the remaining lamina.                                            .43 , .84 c.g. of ords-co 4.3 216354 38 8.4 216454 38 38 lamina 3 4, 54912 2 2, 1644 square c.g. :rea (b) 3 16 851273 5.4 5 16 4521713 5.3 (a) 2 1              y y x x x, y ABC a q qp p p 10 5 10 5 5 5 5 5 50 A B C D Page 9 7. A uniform rod, [AB], of length 2 m and weight 40 N is smoothly hinged at end A to a vertical wall. One end of a light inelastic string is attached to B and the other end of the string is attached to a horizontal ceiling. The string makes an angle of 30 with the ceiling and the rod makes an angle of 30 with the wall, as shown in the diagram. The rod is in equilibrium. (i) Show on a diagram all the forces acting on the rod [AB]. (ii) Write down the two equations that arise from resolving the forces horizontally and vertically. (iii) Write down the equation that arises from taking moments about point A. (iv) Find the tension in the string. (v) Find the magnitude of the reaction at the hinge, A.       N 1.36 3535 354030sin 3530cos N 10 30sin1402 : about moments Take 4030sin vert 30cos horiz 22         R yTY TX T T A TY TX 50 10 5 5 10 5 5 5 5 A B 30° 30° 40 X Y A B 30° 30° T Page 10 8. (a) A particle describes a horizontal circle of radius r metres with uniform angular velocity ω radians per second. Its speed and acceleration are 6 1sm  and 12 2sm  respectively. Find (i) the value of r (ii) the value of ω . (b) A conical pendulum consists of a particle of mass 3 kg attached by a light inelastic string of length 1 metre to a fixed point P. The particle describes a horizontal circle of radius r. The centre of the circle is vertically below P. The string makes an angle of  with the vertical where 3 4 tan  . Find (i) the value of r (ii) the tension in the string (iii) the angular velocity of the particle. (a)   m3 6 s rad 2 12 onaccelerati 6 speed 1 2 2         r r rr r r r       (b)       1 2 2 srad 08.4 8.030.850 sin )( N 50 306.0 3cos )( m 8.0 5 4 1 5 4 in 3 4 tan )(             mr Tiii TT gTii r r s i 5 5 5 5 5 5 5 5 5 5 50 P 1 m 3 kg  r Page 13 BOOKS BACKs SETUPS 27/7/10 12:03 Page 2