Dec2005 - FACULTY OF SCIENCE FINAL EXAMINATION PHYS 131...

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Unformatted text preview: FACULTY OF SCIENCE FINAL EXAMINATION PHYS 131 MECHANICS AND WAVES Examiner: Prof. R. Harris Thursday December 8th, 2005 Associate Examiner: Prof. M. Kilfoil 9:00 am. — 12 noon Answer all questions: Part I: 15 questions at 1 point each for a total of 15 points Part II: 5 questions at 5 points each for a total of 25 points VERSION 1 INSTRUCTIONS: Answer Part I by filling out the pink computer sheet. Be sure to write your name, student ID and version number on the sheet, and also enter them on the sheet at the left, so that they can be machine-read. The Examination Security Monitor Program detects pairs of students With unusually similar answer patterns on multiple-choice exams such as this one. Data generated by this program can be used as admissible evidence, either to initiate or corroborate an investigation or a charge of cheating under Section 16 of the Code of Student Conduct and Disciplinary Procedures. Answer Part II in the booklet provided. Calculators are permitted. Formulae are provided on the last page. You may keep the question paper. This exam comprises 10 pages, with questions on pages numbered 1 through 8. £7047” PHYS 131 1 Part I: Questions Each question is worth 1 point. Mark your answers on the pink computer cards. 1. A toy train travels at constant speed around a circular track. The magnitude of its velocity is 120. Points A and B are exactly one quarter of the circle apart. What is the magnitude of the train’s average velocity as it travels between them? ”HRH / \‘x 4.," \'\:“ (1) 27 2 v0 3 i _ A o (2) 00 > v > 0 1 (3) '17 > ’00 (4:) If} : 0 \“'~.\\ B 1/11, "mm-a... ..—--""'i 2. The position and velocity of a car on a straight road are measured at 3 times, as shown in the table. 3. A flock of Canada geese are flying South for the winter. Their objective is 5,000 km due South of their starting point. However, strong winds blow them off course, and they end up 7,500 km South—East of their starting point. Which option corresponds to the distance and to the direction of their objective from this point? (a) 11,588 km, 62.760 N of West (b) 11,588 km, 6276" S of East (c) 5,310 km, 324° N of West (d) 5,310 km, 324° S of West PHYS 131 2 4. A small ball, mass m, traveling with velocity 22m, collides head on with a heavier ball, mass 2M, which is at rest. The collision is so violent that the second mass breaks into two equal pieces. One piece becomes attached to the small ball, and remains at rest. The second piece moves off with velocity VM. Which one of the following statements correctly describes what happens? A B ( ) Only momentum is conserved. ( ) Only energy is conserved. (C) Neither energy nor momentum are conserved. (D) Both energy and momentum are conserved. 5. Superman has a challenge: he must divert an incoming piece of space-junk which is headed directly for New York City. He knows that it began at rest at an infinite distance from the Earth, and that its velocity is due only to its interaction with Earth’s gravitational field. He decides to change its direction, so that it goes into an (almost circular) orbit around planet Earth. In order to do so, without changing its distance from Earth, he must (A increase its kinetic energy (B ( ( decrease its kinetic energy C D increase its gravitational potential energy vvvv decrease its gravitational potential energy 6. A mountain—bike rider is traveling in a horizontal circular path on the side of a conical hill. (Of course, it is a perfect cone!) The diagram shows the geometry of the situation. Which one of the following statements is correct? A The value of the normal force is not important. If there is no friction, the motion is impossible. ) B) The friction force must exceed the normal force. a ) D) ( ( (C ( The friction force must point down the slope. 7. A mass is connected to two springs as shown in the diagram. Spring 1 is stronger than spring 2: so that K1 > K2. The mass is at equilibrium when |K1x1i = |K2x2|. It is then set in motion with kinetic energy E, moving in the direction towards the stronger spring, as shown. When it is a distance a: away from its equilibrium position, some of the energy E is divided between the elastic potential energy of the two springs. K2 —>- K1 I—IIIIIIIIIII—‘HllllllllII—l What is the expression for this elastic potential energy? PHYS 131 3 (A) %(K1 +K2):132 (B) %K2x2 (C) $(K1 — [(2)962 (D) %le2 8. A snow—vehicle works on the same principle as a jet engine: it has no wheels, and propels itself (and passengers) by sliding on large skis. In order to accelerate, the motor must provide a force larger than the force of sliding friction Fk. The mass of the machine and passengers is M. The power of the motor has a constant value P. A student (in an advanced mechanics course) is asked to to calculate how long the vehicle will take to reach velocity 1) if it starts from rest. The student writes down the following four equations, as his solution. Where is his first mistake? A Work required is 1M ’U + Figs, where s is the distance traveled C ( ) (B) F16: M a where a is the instantaneous acceleration. ( )a m d’u (D) Pt— dle2 + MEES 9. Two identical twins are swinging on identical swings. The diagram shows the two swings at a particular time when both are instantaneously at rest. Which one of the following statements correctly describes what happens next? (Note that angles are positive measured counterclockwise from the vertical.) (A Not enough information is given to predict the motion. ( ) B) When OB becomes zero, 6A is already positive. C) When 63 becomes zero, 6A is still negative. ) ( (D When 63 becomes zero, 0A is also zero. 9A 93 10. A geosynchronous satellite remains exactly over the same spot on the Earth’s surface. The acceleration of such a satellite is PHYS 131 4 11. 12. 13. A performer at the “Circus on Ice” does her act standing on a three—legged table. Either one, two or three of the table—legs are on the ice at different times. Her act depends on the legs sliding across the (frictionless) ice at constant velocity. However, during a particular show, the ice is melting, so that there is friction between the table—legs and the ice. The performer tries to reduce the friction by doing her entire act with only one table leg in contact with the ice. ( ) Her strategy increases the total friction force. ( ) Her strategy makes no change to the total friction force. (O) Her strategy eliminates the friction force completely. ( ) Her strategy reduces the total friction force. A long stick is lying flat on a frictionless surface. It is struck by a small flat disc which is sliding across the surface, and starts to rotate. The collision is elastic. Which of the following statements are true? A B ( ) Only energy and momentum are conserved. ( ) (C) Energy, momentum and angular momentum are all conserved. ( ) ( ) Only momentum is conserved. D Only momentum and angular momentum are conserved. E Only energy is conserved. A low—flying aircraft is flying faster than the speed of sound. It is flying parallel to the (level) ground, and passes directly overhead of an observer. See the diagram. Observer I Which one of the following statements describes what happens? (A) The shock wave arrives before the aircraft is directly overhead. (B) The shock wave arrives after the aircraft is directly overhead. (C) Depending on the Mach angle (0), the shock wave arrives either before or after the aircraft is directly overhead. (D) The shock wave arrives at the same time that the aircraft is directly overhead. PHYS 131 5 14. A rope is tied between two posts, and oscillates as a standing wave. Which one of the following statements describes what happens? (Ignore the effect of gravity.) If none of the statements are correct, then answer (E) (A) On only one occasion during each period, the Potential Energy of the rope is zero. (B) On only one occasion during each period, the Kinetic Energy of the rope is zero. (C) On only one occasion during each period, the Kinetic Energy of the rope equals the total energy. (D) On only one occasion during each period, the Kinetic Energy and the Potential of the rope are equal. 15. A wave on the surface of a pond has the equation a: t : 2 — — -— y ACOSl 7r(A T)l At which of the following values oft and a: is the displacement not above the average water level? (A) t : T/2; a: = 5A/16 (B) tzT; x : A/16 (C) t : 2T; m = 7A/16 (D) t = 3T/2; x : 9A/16 PHYS 131 6 Part II: Problems Each question is worth 5 points. Write your answers in the booklets provided. You must give reasons for your answers. Credit will not be given for answers with no supporting reasoning, even if they are numerically correct. In all questions, you may take the acceleration due to gravity to be 10 m/s2. 1. The football team’s kicker is practicing: he has to kick a ball placed on the ground so that it clears the crossbar between the goal—posts. The cross—bar is 2 metres above the ground. He places the ball 40 metres from the foot of the goal—posts. He kicks the ball at an angle 0 : 23° to the ground. (a) Show that the initial speed of the ball so that it just clears the cross—bar must be approximately 25 m/s. (b) How long will the ball take to reach the goal? With this approximate speed, a kick with an angle of (9 = 69° degrees will also just clear the cross—bar. (c) With this angle of 69", what is the maximum height of the football above the ground? (d) Where (at what horizontal distance from the goal) will this occur? 2. A 2000 kg Cadillac automobile and a 1000 kg Volkswagen collide at a city intersection. The Cadillac was traveling due North, and the VW due East. The VW struck the left front fender of the Cadillac, and then the two cars stuck together and slid in a direction 35° North of East. The Cadillac was traveling at 3.0 m/s before the collision. (a) Write down the equation(s) for conservation of momentum. Clearly identify the unknown velocities. (b) Solve the equations. (c) To check your answer, compute the magnitude and direction of the total momentum of the two cars before the collision. ((1) Why is your answer correct? (If it seems not to be, say why you think so!) PHYS 131 7 3. In a manufacturing plant, a cylindrical container is made by pouring concrete vertically downwards into a ring—shaped mold placed on a circular rotating platform. The platform and mold have a moment of inertia I : 104 kg—m2, and a motor keeps them rotating at 1 rpm. The radius of the mold is 1 m: assume that all the concrete arrives at exactly that distance from the centre of the ring. concrete _ 1 Top View Side view (a) How much work must be done by the motor while the first 1 kg of concrete is being poured? (Ignore any friction in the bearings.) (b) If the concrete is poured at 5 kg/ second, what is the power required of the motor? On one occasion, the motor breaks down, just as the concrete begins to pour. (c) What is the change in the angular momentum of the platform and mold after 1 kg of concrete has fallen? (Note: be careful * the change is very small!) 4. The brakes of a farm tractor fail, so that it rolls down a steep slope. Because of its construction, the tractor can be described as two very large drive wheels, connected by a single massless axle. Each wheel has moment of inertia I : 100 kg—m2, with outside radius R = 0.5 m, and the mass of the complete tractor, wheels included, is M : 1000 kg. The slope is 10 metres high and 25 metres long, so that 6, see the diagram, is 23.6 degrees. 10m 25m 0 (a) What is the speed of the tractor at the bottom of the slope? (b) Therefore, what is its acceleration down the slope? After the brakes are repaired, the tractor’s motor is started, and it is driven back up the slope with constant acceleration a : 2 m/s2. Draw a free body diagram for the tractor, then answer the following question: (c) What is the magnitude of the friction force required to produce this acceleration? PHYS 131 8 5. A traveling wave on a river has the form y : 0.1003(393 — 1575 + 7r/4) where the displacement y is measured in metres. (a) Draw a 3/ vs :3 plot of this wave at t : 0. (b) Draw a y vs t plot of this wave at as : 7r/ 3. At x = 7r/ 3, there is a gate beneath a low bridge. At time t = 0, the gate drops, so that there is a vertical gap of 0.05 metres between it and the wave. Vertical gap II M (c) At what time does the wave hit the gate? (d) At what time does the crest of the wave arrive at the gate? To avoid the wave hitting the gate, the gate operator raises it at constant velocity. He chooses the velocity so that the gate will be just high enough at the instant that the crest arrives. (e) Does the wave hit the gate? Explain your answer with a y vs t plot, showing both the motion of the water and of the gate. A numerical answer is not required. < >_A$ <a>*& 2’ _At’ _At v=v¢+at,122=vi2+2ax 1 2 II} 2.437; +0215 + E at R = ’0? sin20/g 1 , w2=£, wzzfl m l x=Acoswt, vz—wAsinwt 2 ac—U—2w2r F 2 ma, F12 2 —F21 pzmv, IzFAtzAp F9 2 mg Ffr S MsFN, Ff'r = MkFN FR 2 —bv; ’UT 2 mg/b 1 2mg F =——DA2~ 21/— R 2 p UQUT DpA W=F-s=/F-ds 1 KEzimzfl AW P—Tt“F'” 1 AUzmgh s=r0,v=rw,aT=ra F : —GMm/r2, U = —GMm/r For angular motion, replace x,vandaby0,wandoz respectively. T=Fri=I04 W2702/7d6 I=cMR2=Mk2 hoopzc:1; disk:c=1/2 I:ICM+MD2 L=Iw €=mUT:pr KE—EI 2 _2 w wzzDMg/I $0M = mei/ Z7774 v=f/\, vsz/p, k227r/A yzAcos(kx:l:wt) y = A sin km sin wt 3 = 1010g I/I0 fb= |f1 —f2| ’U—vo f0 :fs’U—Us sin0 = v/vs Useful constants: g = 10 m/s2 2 10 N/kg G = 6.67 x 10—11Nm2/kg2 ...
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