YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY AN INVIGILATOR.

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1 BSc/MSci Examination by course unit.???day??th??? :00 12:30 PHY4116 From Newton to Einstein Duration: 2 hours 30 minutes YOU ARE NOT PERMITTED TO READ THE CONTENTS OF THIS QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY AN INVIGILATOR. Instructions: Answer ALL questions in section A. Answer ONLY TWO questions from section B. Section A carries 50 marks, each question in section B carries 25 marks. If you answer more questions than specified, only the first answers (up to the specified number) will be marked. Cross out any answers that you do not wish to be marked. Only non-programmable calculators are permitted in this examination. Please state on your answer book the name and type of machine used. Complete all rough workings in the answer book and cross through any work that is not to be assessed. Important note: The academic regulations state that possession of unauthorised material at any time when a student is under examination conditions is an assessment offence and can lead to expulsion from QMUL. Please check now to ensure you do not have any notes, mobile phones or unauthorised electronic devices on your person. If you have any, then please raise your hand and give them to an invigilator immediately. It is also an offence to have any writing of any kind on your person, including on your body. If you are found to have hidden unauthorised material elsewhere, including toilets and cloakrooms it will be treated as being found in your possession. Unauthorised material found on your mobile phone or other electronic device will be considered the same as being in possession of paper notes. 'A mobile phone that causes a disruption is also an assessment offence. EXAM PAPERS MUST NOT BE REMOVED FROM THE EXAM ROOM Examiners: D.J. Dunstan M. Baxendale Queen Mary, University of London, 2013

2 Page 2 PHY4116 (2013) Question A1 SECTION A. Answers all questions in Section A Given the vectors u = (2, 0, 1) and v = (1, 1, 0), calculate i) u + v ii) u.v iii) u v State giving reasons which of the following quantities are vectors, iv) iˆ 2 ˆj 3kˆ v) Stress Question A2 i) State Newton s Third Law of Motion and explain what it says about momentum. ii) A falling rock in a quarry gains momentum of 100 kg m s 1 during its descent. What other body must be considered in the application of Newton s Third Law to this problem? What momentum does it gain (state magnitude and direction)? Question A3 A pendulum has a bob of mass 10kg; at rest the bob is at x = 0. We will approximate its potential curve when it is displaced as E = 5x 2 J m 2. i) Sketch this curve for E and label the axes. ii) Calculate the force as a function of position iii) Sketch this curve for F and label the axes. iv) What is the acceleration at x = 0 Question A4 A space station consists of a cylindrical shell of inner diameter 20m. It rotates around its axis to give artificial gravity of 1g at the floor (the inner surface of the shell). i) Calculate the speed v of the floor. ii) Calculate the angular velocity. iii) If ½g were wanted, what would be the speed? iv) If the diameter were 40 m, what would be the speed for 1g? v) How does the artificial gravity vary from the axis to the floor? Question A5 A bicycle wheel weighs 1kg, all of which is at the rim (i.e. neglect the hub). Its diameter 2r is 0.7m. If the bicycle is ridden at the speed of 3.5 m s 1, then in the frame of reference of the rider,

3 PHY4116 (2013) Page 3 i) Calculate the angular speed of the wheel. ii) Calculate the momentum of the rim, p. iii) Calculate the angular momentum of the wheel. iv) Calculate the kinetic energy of the wheel. [4 marks] Question A6 A planet orbits its star in a circular orbit at a speed v at a radius R. i) Give an expression for the centripetal force F C on the planet. ii) Give an expression for the gravitational force FG of the star on the planet, in terms of the mass M of the star and the mass m of the planet. iii) Hence find an expression for the speed v of the planet in its orbit. iv) Putting in appropriate numerical values to your answer to (iii) find the speed of the Earth in its orbit. v) What will escape velocity be at the Earth s orbital distance from the Sun? Question A7 A fast sprinter visiting the Antarctic research base crosses the South Pole, at 10 m s 1. Assuming he weighs 80kg, i) Calculate the Coriolis force he experiences and give its direction. ii) He now goes to a tropical country and runs across the Equator. What is the value of the Coriolis force now? iii) Still at the Equator, he now climbs a tree, at 1 m s 1. What is the value of the Coriolis force now? Question A8 [4 marks] A diatomic molecule consists of two atoms separated by a distance x. i) Sketch the Lennard-Jones potential energy E(x) for two atoms separated by a distance x, and mark the equilibrium separation x 0. ii) On your sketch add a dashed curve showing the force F(x) between the atoms. iii) Approximating the minimum of the potential as the parabolic well E = a + b(x x 0 ) 2, give rough estimates in any convenient units of the constants a, b and x 0 for real atoms. Question A9 [7 marks] A body has a rest mass of m 0. It is accelerated to the speed corresponding to = 5. i) What is now its mass? ii) Give expressions for its total energy at rest and at speed. iii) What is its kinetic energy at speed? Question A10 [4 marks] Turn Over

4 Page 4 PHY4116 (2013) i) Special Relativity rests upon the Principles of Relativity, and the constancy of the speed of light. Give the name of the additional Principle required for General Relativity and say what it asserts. ii) Give a qualitative description and explanation of the path required by this Principle for a beam of light emitted by a horizontal torch or laser in a laboratory that is accelerating upwards. iii) Describe some experimental evidence for General Relativity and this Principle. [6 marks]

5 PHY4116 (2013) Page 5 Question B1 SECTION B. Attempt two of the four questions in this section a) An unstable sub-atomic particle is created in a collision in the beamline in the Large Hadron Collider. It has a lifetime of 0.1ns. It travels at almost the speed of light, and is detected decaying 3m away in a detector element in the ATLAS experiment. i) From its point of view, how long does the particle live? ii) From the point of view of the scientists running ATLAS, how long does it live? iii) Explain your answer to (ii), giving the value of. iv) What is the speed of the particle? v) What is the Proper Time between the creation and decay of the particle? vi) What is the distance between the creation and decay of the particle in the particle s frame of reference? vii) What is the Proper Length between the beamline and the detector elemen viii) What is the spacetime Lorentz-invariant interval between the creation and decay of the particle? Is this spacelike or timelike? [8 marks] b) A roller-coaster includes a complete vertical circle as part of the track, of radius 5 m. A car coasting down the track weighs 200 kg. What is the minimum speed of the car, i) At the top of the circle? ii) At the bottom of the circle? iii) What is the force of the car on the track at the top and at the bottom of the circle? c) The figure shows a beam of negligible weight pivoted at O with weights attached as indicated. 1.1 m 1.2 m 2.0 m O A 15 kg m 3 kg i) Calculate the mass m at A, and the normal reaction N of the system if it is to satisfy these equilibrium conditions. ii) The mass m is now removed. Calculate the angular acceleration of the beam. iii) After two seconds, a second pivot O CG is inserted at the position of the centre of gravity of the beam. Where is this, relative to O iv) What will be the angular momentum about O CG? [12 marks] Turn Over

6 Page 6 PHY4116 (2013) Question B2 a) A pair of atoms separated by the distance r have the potential energy given by the Lennard-Jones U(r). Find the radius at which the force is at a maximum. What happens if an external tensile force is applied to pull the atoms to this separation? What does this say about the strength of materials? [3 marks] b) Describe the aim and methodology of the Michelson-Morley experiment. c) A hollow cylindrical roller (all its weight is at the rim) weighing 10kg is initially held stationary on the inclined surface of a ramp, a wedge also weighing 10kg, which itself is initially stationary but free to slide without friction on a horizontal surface (air-bed). The wedge angle is 45. The roller is released at time t = 0, and rolls without slipping down the wedge. i) State what Conservation of Momentum requires of the horizontal speeds u of the wedge and v H of the wedge when the roller has descended a height h. ii) State what Conservation of Energy requires of the speeds u of the wedge and v of the roller at subsequent times. iii) Hence or otherwise, find the speed of the wedge when the roller has descended through the distance of 110 cm. [15 marks]

7 PHY4116 (2013) Page 7 Question B3 a) For a body in the gravitational field of a mass M, calculate the escape velocity at a radius r. Compare it with the orbital velocity for a circular orbit at the same radius. b) A crystalline solid may be represented as in the diagram (but continued indefinitely), with atoms at cube corners connected by bonds along the cube edges. We treat the bonds as Hooke s Law springs with a spring constant k and an unstrained length a 0. Calculate the bulk modulus B = V dp / dv for small volume changes. [8 marks] c) In a frame of reference S, an event O occurs at the origin (x = 0, t = 0). Two other events are A (x = 5, t = 13) and B (x = 13, t = 5). Units of x are lightyears and units of t are years. Using a Minkowski spacetime diagram, or otherwise, i) Calculate the three spacetime intervals OA, OB and AB. ii) Identify a frame S in which events O and A occur at the same place. What is the time between events O and A in this frame? Give the value of for the relative speed between S and S and state how the time interval of 13 in S is related to the time interval in S. iii) Identify the frame S in which events O and B occur at the same time. What is the distance between events O and B in this frame? iv) Give reasons why frames S and S are the same frame. v) Explain why the reasoning in (ii) and (iii) is not symmetrical, i.e. why we do not deduce a dilated length in the same way as we deduce a dilated time. How does it come about that the length between O and B is contracted in S from the Proper Length of 12 determined in S. [12 marks] Turn Over

8 Page 8 PHY4116 (2013) Question B4 a) A ladder stands on a floor with a coefficient of friction of = 0.25 and leans against a frictionless wall. The angle the ladder makes to the floor is. i) Find the minimum value of. ii) Neglecting the weight of the ladder, what does the minimum value of become if a man stands on the top rung (approximate this to the very top) of the ladder? iii) An assistant of equal weight now stands on the bottom rung (approximate this to the very bottom), as recommended by Safety Officers. Using your previous results or otherwise, find the minimum value of in this situation. [7 marks] b) The Coriolis force may be expressed as F = 2mv. i) Derive this expression. ii) A naval gun fires a shell at 900 kph (ignore the vertical component of the motion). The target is 10km away to the north. By how much will it miss the target if no correction is made for the Coriolis effect, if the gun is at the South Pole, at the latitude 45 South, and at the Equator [9 marks] c) A circular pendulum consists of a light string attached to the ceiling, by which is suspended a 1kg steel ball. The ball follows a horizontal circular path, radius 2.5 m with a period of seconds. i) Calculate the speed of the ball and the length of the string. ii) Give the angular momentum of the ball and its kinetic energy. iii) The string is now shortened until the radius of the path of the ball is 1.25 m. What are now the angular momentum, the speed of the ball, the kinetic energy, and the length of the string? iv) How much work has been done shortening the string against the tension in the string? [9 marks] End of Paper An appendix of 1 page with formulae and values of physical constants follows

9 PHY4116 (2013) Page 9 FORMULA AND DATA SHEET You may wish to use some of the following formulae and data. u.v = Vector dot product = u x v x + u y v y + u z v z u v = Vector cross product = (u y v z u z v y, u z v x u x v z, u x v y u y v x ) Lennard-Jones potential is 12 6 r0 r0 U ( r) r r g = Acceleration due to gravity = 10 m s 1 c = Speed of light = m s 1 G = Gravitational constant = m 3 kg 1 s 2 M E = Mass of the Earth = kg M S = Mass of the Sun = kg R E = Radius of the Earth = kg ω E = Angular velocity of Earth = rad s 1 L S = Distance of Earth from Sun = m P atm = Atmospheric pressure = 1atm = Pa Turn Over