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UNIVERSITY OF KWAZULU-NATAL HOWARD COLLEGE CAMPUS EXAMINATIONS : JUNE 2006 SUBJECT, COURSE AND CODE: ENGINEERING PHYSICS 1A (PHYS151) DURATION: 3 HOURS TOTAL MARKS: 180 Examiners : Internal: Prof. J.P.S. Rash Mr T. Mathe Prof. J.D. Hey Dr K. Naidoo External: Prof. N. Chetty INSTRUCTIONS: 1. Answer ALL questions ON the question paper. 2. Rough work may be done on the back of each page. If you use the back of a page for work which you wish to be marked, please indicate this clearly. 3. Marks have been allocated in such a way that 1 mark corresponds roughly to one minute of time. Candidates are advised not to spend a disproportionate amount of time on any one question. 4. Check that this paper contains 19 numbered pages (excluding the cover page) plus a sheet of graph paper (after page 17) and a formula sheet. 5. The use of calculators (including programmables) is permitted. 6. If there is any word in this examination paper which you do not understand, please ask an invigilator (raise your hand). 7. You may use g = 10 m.s-2 unless otherwise indicated. 8. Where angles of 37° or 53° are given, you may use the approximations sin 37° = cos 53° = 0.6 and cos 37° = sin 53° = 0.8 Page 1 UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 2 SECTION A MECHANICS 75 MARKS QUESTION A 1 : Given two vectors A = 4.00 i + 3.00 j and B = 5.00 i – 2.00 j find the scalar product of these two vectors. (3) [3] QUESTION A 2 : (a) Figure (a) alongside shows a particle moving in a circle, radius R, with constant speed. Its velocity vector is v1 at point P1 and v2 at point P2 a time interval ∆t later. Figure (b) is a vector diagram showing the vector difference ∆v = v2 – v1 . Use these diagrams to derive the expression for the centripetal acceleration in uniform circular motion. State your reasoning clearly. (5) UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 3 (b) The Earth has a radius of 6380 km. If the radial acceleration at the equator was greater than g, objects would fly off the Earth's surface into space. What would the period of the Earth's rotation have to be for this just to occur? (3) [8] QUESTION A 3 : (a) An airplane pilot is practicing to fight forest fires by dropping a canister of water onto a target on the ground in front of and below his plane. If the plane is flying in a horizontal path 90 m above the ground with a speed of 64 m/s, at what horizontal distance from the target should the pilot release the canister in order to hit the target? (Ignore air resistance.) (4) UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 4 (b) Another pilot sets off in her airplane from A in the direction of B which is due East of A. The plane’s airspeed (i.e. magnitude of the velocity of the plane relative to the air) is 420 km/hr. However there is a wind of 200 km/hr blowing due South. (i) What is the velocity of the plane relative to the Earth (magnitude and direction)? (3) (ii) If B is 800 km East of A, how much time will it take the plane to reach a point with the same longitude as B (i.e. somewhere due North or South of B)? (2) (iii) How far North or South of B will the plane be at this time? (1) [10] QUESTION A 4 : (a) Which of Newton’s Laws is also known as the “Law of Inertia”? (1) (b) State, in words, Newton's Third Law, without using the terms "action" and "reaction". (2) [3] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 5 QUESTION A 5 : Block A, of mass 5.0 kg, slides on a plane inclined at 37° to the horizontal. It is connected to block B, of mass 7.0 kg and hanging vertically, by a light inextensible string passing over a frictionless pulley (see figure). The coefficient of kinetic friction between block A and the surface of the inclined plane is 0.30. (a) Draw a free body diagram for block A using clearly labelled arrows: (2) A (b) Show that the magnitude of the frictional force on A is approximately 12 N. (3) (c) Calculate the acceleration of the blocks. (5) (d) Find the tension in the string. (2) [12] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 6 QUESTION A 6 : (a) Using the kinematic equations and Newton's Laws, 'derive' the basic form (i.e. excluding potential energy) of the Work-Energy Theorem for the case of a constant force F acting over a displacement s. Define clearly any quantities which are introduced in the course of the derivation. (5) (b) A car is travelling on a level road with speed v0 when the brakes lock, so that the tyres slide (“skid”) rather than roll over the road surface. (i) Use the Work-Energy Theorem to derive an expression for the stopping distance of the car in terms of v0 , g and the coefficient of kinetic friction µk between the tyres and the road. (4) (ii) The car stops in a distance of 51 m if v0 = 60 km/h. What will be the stopping distance if v0 = 80 km/h? (2) [11] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 7 QUESTION A 7 : (a) State two properties of the work done by a conservative force. (2) (b) A 2.00 kg block is pushed against a spring with negligible mass and spring constant k = 400 N/m, compressing it 0.220 m. When the block is released it moves along a frictionless horizontal surface, and then up a frictionless incline with slope 37°, as shown in the figure alongside. (i) What is the speed of the block as it slides along the horizontal surface after having left the spring? (3) (ii) How far does the block travel up the inclined surface before starting to slide back down? (3) [8] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 8 QUESTION A 8 : (a) State the Principle of Conservation of Momentum (including the conditions under which it applies). (2) (b) A car of mass 1400 kg travelling east on East Street at 35 km/h collides with a truck of mass 2800 kg that is travelling north on North Road at 50 km/h. The two vehicles stick together after the collision. Calculate the magnitude and direction of the velocity of the vehicles after the collision. [Assume that friction forces between the vehicles and the road during the collision can be neglected as the road was wet and slippery.] (6) [8] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 9 QUESTION A 9 : When drilling a 12 mm diameter hole in wood, plastic or aluminium, a workshop manual recommends a drill rotation speed of 1250 rev/min. For a 12 mm diameter drill bit turning at a constant 1250 rev/min, find (i) the maximum linear speed of any part of the bit (3) (ii) the maximum radial acceleration of any part of the bit. (2) [5] QUESTION A 10 : Calculate the net torque about point O for the two forces applied as in the figure below. The rod and both forces are in the plane of the page. (6) [6] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 10 SECTION B WAVES & HEAT 65 MARKS ANSWER ALL QUESTIONS QUESTION B 1 : (a) A body acted upon by a force undergoes simple harmonic motion (SHM). (i) What physical requirements must be met to ensure that the motion is in fact SHM? (Use words and symbols.) (3) (ii) Define 3 of the physical parameters associated with SHM. (3) UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 11 (b) A 100 g object, attached to the end of a horizontal ideal spring with force constant 40 N/m, undergoes SHM. Its speed when it is displaced from the equilibrium position by half its amplitude is 3 m/s. Determine: (i) the amplitude of the SHM (4) (ii) the maximum speed of the object (1) (iii) the acceleration when its displacement x = –3.0 cm (1) (iv) the potential energy when the object’s speed is 1.0 m/s. (4) [16] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 12 QUESTION B 2 : (a) A uniform wire of mass m and length L is fully extended and under tension F. The one end of the wire is plucked, and a pulse travels along the wire. mL Show that time taken for the pulse to travel from one end of the wire to the other end is ∆t = . F Is the pulse longitudinal or transverse? (4) (b) The wire is used in the space programme for an experiment on the Moon. It is mounted vertically on the Moon's surface by an astronaut, and put under tension by attaching a mass M, which is much greater than the mass of the wire. The transit time for a pulse to travel from the upper to the lower end of the wire is measured by the astronaut, and found to be ∆t = 36.1 ms . Other numerical data: M = 3.00 kg, m = 4.00 g, L = 1.60 m. From the data, calculate the gravitational acceleration on the Moon, g Moon . What would one expect for the corresponding value of ∆t on Earth? (6) [10] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 13 QUESTION B 3 : (a) Assuming a value B = γ p0 for the bulk modulus of a gas in terms of the equilibrium pressure, show how the following expression is obtained for the speed of sound in the gas: γ RT v= , where M is the molar mass and the gas is assumed to be ideal. M (3) (b) Sketch the first three standing-wave modes in a hollow tube closed at one end and open to the atmosphere at the other. With each sketch, relate the wavelength of the normal mode to the length of the tube. (3) (c) A long tube contains air at a temperature of 77 °C and a pressure of 1 atmosphere. The tube is open at one end, and closed at the other end by a movable piston. A tuning fork near the open end is vibrating at a frequency of 500 Hz. Resonance is produced when the piston is measured to be at distances of 18.0 cm, 55.5 cm and 93.0 cm from the end of the tube. From the average value of the wavelength of the sound from these data, determine the speed of sound in the air at the given temperature and pressure. (3) UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 14 (d) Using the information above, obtain a value of γ for the air, assuming M = 28.8 kg/mol. From this result, which types of energy do most molecules in air have? (4) [13] QUESTION B 4 : (a) One end of an insulated metal rod is maintained at 100 °C, and the other end is maintained at 0 °C by an ice-water mixture. The rod is 60.0 cm long and has a cross- sectional area of 1.25 cm2. The heat conducted by the rod melts 8.50 g of the ice in 10.0 min. Find the thermal conductivity k of the metal. [ Lf (water) = 334 kJ/kg] (5) UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 15 (b) The original insulated rod in (a) is now replaced by a copper rod of length 30.0 cm joined onto a brass rod of 30.0 cm, to make a combined rod of 60.0 cm length and the same cross-sectional area as previously. The copper end of the new rod is maintained at 100 °C, while the brass end is kept in the ice-water mixture. If the rate at which the ice melts remains as before, what is the temperature at the copper-brass junction in the middle of the new rod? [ k (copper) = 400 W/m °C, k (brass) = 100 W/m °C ] (5) [10] QUESTION B 5 : (a) List six assumptions made in the kinetic-molecular model of the ideal gas. (6) UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 16 (b) From the result ( ) p V = N m v 2 Av x show that the mean translational kinetic energy per molecule is 3 k T . 2 Clarify any assumptions made in obtaining this result. (6) (c) The speed required for any particle to escape from the Earth's surface is given in terms of the gravitational constant G, Earth's radius RE and Earth's mass ME by 2G M E v esc = . RE Use this result to formulate an 'escape temperature', at which a gas (molecular mass m) would be hot enough to escape from the Earth's atmosphere. By comparing the result for a light gas (e.g. hydrogen) with that for a heavier gas (e.g. oxygen), can you explain the practical absence of light gases from the earth's atmosphere? (4) [16] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 17 SECTION C EXPERIMENTAL 40 MARKS QUESTION C 1 : A Mechanical Engineering student set out to measure the elastic properties of the steel used in hacksaw blades. She clamped one end of a hacksaw blade horizontally, attached a series of masses m to the other end, and set the blade oscillating, measuring the time t for 20 oscillations in each case, so that she could determine the period of oscillation T as recorded in the table below. Mass m Time t for 20 Period T (g) oscillations (s) (s) 50 17.6 0.88 100 21.4 1.07 150 25.0 1.25 200 28.0 1.40 250 30.6 1.53 The student suspected she could not use the relationship between the period of oscillation and the mass which holds for the spiral spring, but she was sure that the relationship between T and m should be a power law of the form T=amn Plot a suitable straight line graph to show that the student's supposition was correct, and from the graph determine the values of a and n in the above equation. [You should use the empty columns of the above table for any values deduced from those in the table which you wish to plot. Note that marks are allocated for: sensible choice of scales, labelling of axes, clear and accurate plotting of points, showing how values are found from the graph, as well as consistency with significant figures and inclusion of units where appropriate.] (20) [20] UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 18 QUESTION C 2 : (a) What instrument would you use to measure the mass of the following? (i) a suitcase full of clothes (which should be around 30 kg to be allowed on your plane flight) to the nearest 1 kg (ii) an empty polystyrene cup (into which you plan to pour a small amount of liquid), to the nearest 0.01 g (iii) a physics text book (which you have to carry around the campus), to the nearest 1 g (3) (b) (i) In measuring the period of the simple pendulum, why does it improve the accuracy of the result if you measure the total time for a number of oscillations (10, say) and divide by that number, rather than timing one oscillation? (3) (ii) Why is the length of the simple pendulum measured to the centre of the bob, rather than the point of attachment of the string to the bob? How could you measure the position of the centre more accurately than by estimating with a metre stick? (3) UNIVERSITY OF KWAZULU-NATAL EXAMINATIONS: JUNE 2006 CAMPUS: HOWARD COLLEGE SUBJECT, COURSE AND CODE: ENG. PHYSICS 1A (PHYS151) Page 19 (c) In the Spiral Spring experiment, where a mass is hung from the end of the spring and the extension measured ('static experiment') and then the time of oscillations measured ('dynamic experiment'), does the mass of the 'holder' (onto which 50 g brass mass pieces are added) need to be included in the measurements in both cases? Explain what errors would arise if it was not. (5) [14] QUESTION C 3 : The "Method of Mixtures" is used to determine the specific heat capacity of copper in the experiment SHC, after the specific heat capacity of water has been determined. What is meant by the "Method of Mixtures"? Explain briefly the principles of the method. (You do not need to give a detailed description of the experimental procedure, but you may wish to mention some precautions taken to ensure that the assumptions made in the procedure are valid.) (6) [6] END

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