Senior Physics Challenge Past Paper 2023 PDF

Sponsored by SENIOR PHYSICS CHALLENGE (Year 12) 10th MARCH 2023 This question paper must not be taken out of the exam room Name: School: Total Mark /50...

Sponsored by SENIOR PHYSICS CHALLENGE (Year 12) 10th MARCH 2023 This question paper must not be taken out of the exam room Name: School: Total Mark /50 Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Draw diagrams. Marks allocated for each question are shown in brackets on the right. Calculators: Any standard calculator may be used, but calculators must not have symbolic algebra capability. If they are programmable, then they must be cleared or used in “exam mode”. You may use any public examination formula booklet. Scribbled or unclear working will not gain marks. This paper is about problem solving and the skills needed. It is designed to be a challenge even for the top Y12 physicists in the country. If you find the questions hard, they are. Do not be put off. The only way to overcome them is to struggle through and learn from them. Good Luck.  Important Constants Constant Symbol Value Speed of light in free space c 3.00 × 108 m s−1 Elementary charge e 1.60 × 10−19 C Planck constant h 6.63 × 10−34 J s Mass of electron me 9.11 × 10−31 kg Mass of proton mp 1.67 × 10−27 kg Acceleration of free fall at Earth’s surface g 9.81 m s−2 Avogadro constant NA 6.02 × 1023 mol−1 Radius of Earth RE 6.37 × 106 m Radius of Earth’s orbit R0 1.496 × 1011 m T(K) = T(◦ C) + 273 4 Volume of a sphere = πr3 3 Surface area of a sphere = 4πr2 v 2 = u2 + 2as v = u + at s = ut + 21 at2 s = 12 (u + v)t ρℓ E = hf R= A P = Fv P = E/t P =VI V = IR v = fλ P = ρgh 1 1 1 R = R1 + R2 = + R R1 R2 PV P V = const. = const. T Qus. 1-4 Circle the best answer. 1. A car tyre lasts typically 40 000 km. Estimate the number of rotations it makes during its lifetime. A. 105 B. 106 C. 107 D. 108 E. 109 2. Wine can be produced in large vats, shaped as in Fig. 1. The graphs below are suggested indications of the pressure as a function of depth. Which is the most suitable graph? A. (a) B. (b) C. (c) D. (d) E. (e) Figure 1 3. A steady sound of 165 Hz is produced by a loudspeaker at one end of a field and it is received 157 m away. By what fraction of a cycle (measured in degrees from 0 to 360◦ ) is the received signal out of phase? The speed of sound in air is 330 m s−1 A. 0◦ B. 45◦ C. 90◦ D. 135◦ E. 180◦ 3 4. A girl standing on a cliff throws two balls, one up and one down, at the same speed. How do the final velocities of each compare as they hit the sea? A. The ball thrown down has a greater velocity than the ball thrown up B. If the height going up is greater than the drop down to the sea, then the ball thrown up will have a greater velocity C. If the height up is less than the drop down, then the upwards ball will have a lower velocity D. The same E. The result depends upon the magnitude of the speed of the throw of the two balls 5. The idea of centre of mass is an important concept and is more appreciated with examples of its use. (a) A 34 cm long uniform straight rod lies on a smooth horizontal surface and it is seen to be spinning round whilst also moving across the surface (translating). At one particular moment in time it is observed that the velocities of the ends of the rod are normal to the rod and have values, 2.6 m s−1 and 4.2 m s−1 as illustrated in Fig. 2. Figure 2: A uniform rod which is rotating and translating across a smooth horizontal surface. i. Sketch several diagrams of the rod as it would be seen sliding across the surface (i.e. across the page here). 4  ii. At what speed would you need to fly over the rod as an observer to see only its rotational motion? iii. At what frequency does it rotate? iv. If we observe the rod a quarter of a rotation later, what is the magnitude of the velocity of one of its ends? (b) A cable of mass m hangs from two fixed points A and B and forms a smooth curve, as in Fig. 3(a). In (b), force F is applied to the centre of the cable in order to straighten it. Figure 3: A massive cable suspended from two fixed points. In (a) it hangs under its own weight, in (b) a force F is added to straighten the cable. i. On the sketch of Fig. 3(a), mark on with an (X) the approximate location of the centre of mass. 5  ii. When a force F is applied to straighten the cable, explain any change this makes to the centre of mass, and why. 6. A rod of mass m1 is constrained to move vertically by a pair of guides, as shown in Fig. 4. The rod is in contact with a smooth wedge of mass m2 and angle θ, which itself sits on a smooth horizontal surface. At time t = 0 the rod is released and moves downwards, whilst the wedge accelerates to the right. (a) The weight of the rod is constant, and the force acting on the smooth slope of the wedge is constant. What significant conclusion can be made about the type of motion of the rod and the motion of the wedge as a result? Figure 4: Rod guided vertically to slide down the Figure 5: The motion of the wedge and rod. smooth slope of a massive wedge. (b) As the wedge slides to the right at speed v, the rod slides down at speed u. Copy Fig. 5 and mark on it the motion of the contact point P on the slope, as it moves to the right in time ∆t. Similarly, add the new contact point of the end of the rod which moves downwards at speed u in time ∆t. Use this, or an alternative idea, to relate u, v and the angle of the slope θ. 6 (c) If the rod falls through height h and the rod and slope reach speeds u and v respectively, write down an energy equation for the system in terms of m1 , m2 , u, v, g and h. (d) Now obtain an expression for the speed of the wedge v in terms of m1 , m2 , g, h and θ. (e) If m1 = m2 and θ = 30◦ , what fraction of the GPE lost by the rod in falling is gained by the wedge? (f) From this, write down an expression for the speed of the rod, u, Using this and the ideas introduced earlier, write down an expression for the acceleration of the rod. 7 7. A bulldozer runs on a continuous track, sometimes called a caterpillar track, as shown in the image of Fig. 6. The driving wheel at the front has a diameter of 1.0 m and rotates once in 0.84 s. A person standing at the side of the bulldozer as it drives past sees a large piece of mud stuck to the top side of the moving track (at about 1 m above the ground). At what speed relative to the person is the mud moving past them? Figure 6: The moving caterpillar track on a bulldozer. 8. For many questions, drawing a diagram is the key to unlocking the ideas and unwrapping the question. A diagram should be large, should represent the scales described in the question and should be correct. It may require improving several times to get it right. In the following, you are asked to draw the diagram for this situation and calculate an angle only. Three uniform beams AB, BC and CD, of the same thickness and of lengths ℓ, 2ℓ and ℓ respectively, are connected by smooth hinges at B and C , and rest on a perfectly smooth sphere of radius 2ℓ so that the middle point of BC and the extremities, A and D are in contact with the sphere. Sketch a diagram of the beams and sphere in the space below, and calculate the obtuse angle between beams AB and BC. 8 9. (a) The terminal voltage of a dc power supply is measured as 5.00 V when it is on open circuit. A 2.00 Ω resistor is connected across the terminals and the voltage drops by 0.100 V. i. If the supply is treated as a simple emf and internal resistance, what would be the value of the internal resistance? ii. If the load resistor is reduced to 0.400 Ω, what would be the terminal voltage now? (b) Some electronic devices are designed to take a constant current, irrespective of the voltage applied. Such a circuit with a constant current device attached is illustrated in Fig. 8, in which the constant current device takes 0.40 mA. A moving coil voltmeter with a needle moving over a scale, such as that illustrated on the right in Fig. 7, in fact works by taking a small sample of current and is factory calibrated to show a voltage on the scale. At full scale deflection, the meter draws a current of Figure 7: Moving coil voltmeter 1.0 mA. On the 300 V range, when connected in the circuit of Fig. 8, it reads 90 V. Figure 8: Constant current device connected in a circuit with and then without the moving coil voltmeter. 9  What is the voltage applied to the device with the voltmeter removed from the circuit? 10. A spherical shaped party balloon can be filled by blowing air into it. We observe that it is difficult to start the balloon expanding, but it becomes easier once the rubber has stretched a little. It is easier to inflate the balloon a second time. This behaviour is illustrated by the graph of Fig. 9 and is described by the equation,   r 6  C 0 Pin − Pout = 2 1 − (1) r0 r r where Pin is the pressure inside the balloon, Pout is the external atmospheric pressure, r0 is the uninflated radius of the balloon, r is the radius of the balloon, and C is a constant. (a) What are the dimensions of C in terms of [m], [kg],and [s]? Figure 9: Pressure curves for a rubber balloon. Circles are experimental points of several inflations. The solid curve is equation 1 adjusted to pass through the pressure maximum for the initial inflation. ref. Merritt DR, F Weinhaus Am. J. Phys. 46(10), pp 976-7 Oct1978 10 (b) The pressure curve for a rubber balloon is shown in Fig. 9. This 1978 paper by Merritt and Weinhaus uses old cgs (cm, g, s) units of pressure for Pin − Pout on the vertical axis. Give an estimate from the graph of the maximum value of the pressure shown, giving your answer in pascals. (c) By taking two regions on the graph, estimate the work done in blowing up the balloon to a radius of 6 cm. From the equation WD = F ∆x we obtain WD = P ∆V. (d) From equation (1) above, what is the relation between the uninflated radius r0 and the radius at maximum pressure rp ? (e) In the case of two similar balloons filled so that they are of unequal radii, and joined by an open tube, they will reach the same pressure. i. If the lower pressure balloon is of initially greater radius, in what configuration of radii will the two balloons finish up? 11  ii. If the lower pressure balloon is now of initially lesser radius, under what condition would the balloons end up with equal radii? END OF PAPER The BPhO is sponsored by Page 12  SENIOR PHYSICS CHALLENGE (Year 12) 4th MARCH 2022 This question paper must not be taken out of the exam room Name: School: Total Mark /50 Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Draw diagrams. Marks allocated for each question are shown in brackets on the right. Calculators: Any standard calculator may be used, but calculators cannot be programmable and must not have symbolic algebra capability. You may use any public examination formula booklet. Scribbled or unclear working will not gain marks. This paper is about problem solving. It is designed to be a challenge for the top Y12 physicists in the country. If you find the questions hard, they are. Do not be put off. The only way to overcome them is to struggle through and learn from them. Good Luck.  Important Constants Constant Symbol Value Speed of light in free space c 3.00 × 108 m s−1 Elementary charge e 1.60 × 10−19 C Planck constant h 6.63 × 10−34 J s Mass of electron me 9.11 × 10−31 kg Mass of proton mp 1.67 × 10−27 kg Acceleration of free fall at Earth’s surface g 9.81 m s−2 Avogadro constant NA 6.02 × 1023 mol−1 Radius of Earth RE 6.37 × 106 m Radius of Earth’s orbit R0 1.496 × 1011 m T(K) = T(◦ C) + 273 4 Volume of a sphere = πr3 3 Surface area of a sphere = 4πr2 v 2 = u2 + 2as v = u + at s = ut + 21 at2 s = 12 (u + v)t ρℓ E = hf R= A P = Fv P = E/t P =VI V = IR v = fλ 1 1 1 R = R1 + R2 = + R R1 R2 PV P V = const. = const. T Qus. 1-5 Circle the best answer. 1. Estimate the mass of the Earth. A. 1020 kg B. 1022 kg C. 1024 kg D. 1026 kg E. 1028 kg 2. A block of mass m remains stationary on a rough slope as shown in Fig. 1. Which of the following could be equal to the magnitude of the frictional force Ff on the block? A. N cos θ2 B. N sin θ2 C. mg cos θ2 D. mg sin θ2 E. mg sin θ3 Figure 1 3. Energies in particle accelerators are measured in eV. What is the kinetic energy to an order of magnitude, in eV, of a snail of mass 1 g which crawls along at a rate of 1 cm in 10 s? A. 1 eV B. 1 keV C. 1 MeV D. 1 GeV E. 1 TeV 4. In a modern electric car, the most important reason the batteries are lithium ion rather than lead acid is A. lithium cells are easier to recycle B. lithium is much cheaper than lead C. lithium is less dense so the batteries are much lighter D. the energy density of the lithium battery is much greater E. lead batteries are full of dangerous acid 1 5. A source of high frequency sound from a sonar under the ocean surface sends a 60 kHz sound towards the surface. What is the wavelength of sound in the air above? The speed of sound in air is 330 m s−1 A. 0.18 m B. 0.18 mm C. 5.5 m D. 5.5 mm E. 19.8 m 6. (a) A small ball of mass m is attached to a point by a light string of length ℓ and hangs down under gravity, shown in Fig. 2. The point of attachment is accelerated to the right with a constant acceleration a, so that the string hangs at an angle θ to the vertical, with a tension T in the string. Figure 2: Ball on a light string which is Figure 3: U-tube containing water. accelerated to the right. i. Write expressions for the horizontal and vertical components of the force on the ball, in terms of m, g, T, a and θ. ii. Obtain an expression for the angle of the string to the vertical, θ, in terms of a and g 2  (b) In the U-tube half filled with water of Fig. 3, the tube has a cross-sectional area A. The U-tube has a horizontal acceleration, a, to the right and in the plane of the U-tube. This will cause a height difference h in the levels of the water. i. Sketch the U-tube with the water levels in the tube, showing the water surface on each side. ii. The arms of the tube are a distance L apart, By considering the forces on a thin disc of water in the tube or otherwise, deduce an equation relating h to a, g and L. 7. In an experiment to measure the length of an oil molecule that has a hydrophilic end (one end of the long molecule sticks in a water surface), a drop of the oil of volume 0.1 mm3 is touched on to the surface of some water in a tank. The oil spreads out to give a circular patch of area 1000 cm2. What is the length of the oil molecule? 3 8. One end of a uniform beam, of weight W is placed on a smooth horizontal plane. The other end, to which a light string is attached, rests against another smooth plane inclined at an angle α to the horizontal. The string, passing over a frictionless pulley at the top of the inclined plane, hangs vertically, and supports a weight, P. Sketch a diagram of the beam and the planes, marking on it the forces acting on the beam and on P. (There is no calculation required.) 9. Gamma radiation such as that from a Co-60 source is a penetrating radiation which requires shielding for safety purposes. The radiation is reduced in intensity when it passes through a material by a factor x S = 2− a where x is the distance travelled through the material and a is a constant which depends on the material and gamma ray energy. th What thickness x of lead will reduce the intensity of the same gamma rays to 18 that of concrete of thickness y = 1.0 m? For gamma rays produced by cobalt-60: alead for lead is 12 mm aconcrete for concrete is 60 mm 4 10. (a) A battery is connected to a lamp, a moving coil ammeter (illustrated in Fig. 4) and a switch, all in series. The needle of the ammeter hits the end stop when the switch is first closed, but then returns to read the normal value for the particular lamp. A thermistor is then included in series in the circuit. The lamp again runs at almost normal brightness when the switch is closed, but the needle of the ammeter no longer hits the end stop. Explain why. Figure 4 (b) A filament lamp has a resistance which we can assume is proportional to its temperature in kelvin. A 50 W bulb operates on 230 V at a temperature of 2250 K. What is the resistance of the bulb at room temperature of 27 ◦ C? A circuit of two resistors R and RC in series is connected to a supply as shown in Fig. 5. The potentials at three points are marked as 0 V, VA , VB. The current I in the circuit depends upon the value of RC. Figure 5 (c) i. Obtain a relation between VA , VB , I and R. 5  ii. Sketch a graph of the current I (y−axis) against the potential VB (x−axis). Mark on values where the line crosses each axis, and the gradient. iii. RC is now replaced by a filament light bulb. On your sketch graph above, add another line with an arrow that shows how I and VB vary from the moment the bulb is switched on until its steady illumination. (d) A relay is an electro-mechanical device in which a small current flowing through a coil magnetically operates a switch. A relay is illustrated in Fig. 6. With no current in the coil, the switch is open. In the circuit shown in Fig. 7, a small current flowing through the coil will close the switch and light the bulb. Figure 6 Figure 7 A potential divider circuit is made using a resistor, a cell and light dependent resistor (LDR). The resistor is in parallel with the coil. i. In version A of the circuit, shown in Fig. 8, explain what would happen if light was shone on the LDR. 6  Figure 8: Potential divider arrangement connected to a relay. Version A. ii. In version B of the circuit, the LDR and resistor are interchanged. The bulb is now placed physically over the LDR, and the light bulb starts flashing on and off. The light of the bulb affects the LDR which switches the relay. A very simple estimate can be made of the flashing rate by calculating the time taken for the filament to heat up, and assuming it cools down instantaneously. The bulb is 24 W, 12 V with a filament length of 12 cm. The temperature rise is from 300 K to 2300 K and the specific heat capacity of tungsten is 134 J kg−1 K−1. The density of tungsten is 19 300 kg m−3 and the resistivity can be taken as 66 × 10−8 Ω m. Calculate the time taken for the filament to heat up to estimate the flashing frequency. Ignore any heat loss. Show intermediate steps in your calculation. 7 11. A linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths ℓ1 , ℓ2 , ℓ3 ,... along the beam and with a fixed small separation d between each tube. The tubes are connected to a high voltage, constant radio frequency AC supply where the peak voltage of the AC is V0. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge e and mass me is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it “drifts” through the tube. It passes through a large potential difference between the tubes and, if the charged particle’s motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10 Figure 9 Figure 10 (a) i. Preliminary: a resistor has a potential difference of 5 V across it and a single electron flows through it. What is the thermal energy generated? ii. To generate 2 W, how many electrons flow through in a second? (b) i. Sketch a graph of the AC voltage against time for two cycles of the AC. ii. What is the maximum potential difference between adjacent tubes connected to the AC supply as shown? 8  iii. The frequency of the AC is f. State in terms of f , the shortest time T that the electron should take drifting through a tube in order that it experiences this maximum potential between tubes. (c) i. The electrons leave the electron source in bunches starting with zero velocity, and are accelerated towards the first drift tube by the AC potential difference. In terms of e, V0 and me , what is the maximum speed v1 that the electron bunch can enter the first tube? ii. Using your drift time from (b) iii and the speed from (c) i, obtain an expression for the length of the first drift tube, ℓ1 in terms of e, V0 , me and f. (d) i. The electron emerges from the end of the first tube with the same speed as it entered. What would be the speed of the electron as it leaves the second drift tube? 9 ii. What are the lengths of the drift tubes, ℓ2 , ℓ3 in terms of the length of ℓ1 ? END OF PAPER BPhO Sponsors Page 10  SENIOR PHYSICS CHALLENGE (AS CHALLENGE PAPER) 6th MARCH 2020 This question paper must not be taken out of the exam room Name: School: Total Mark /50 Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Draw diagrams. Marks allocated for each question are shown in brackets on the right. Calculators: Any standard calculator may be used, but calculators cannot be programmable and must not have symbolic algebra capability. You may use any public examination formula booklet. Scribbled or unclear working will not gain marks. This paper is about problem solving. It is designed to be a challenge for the top Y12 physicists in the country. If you find the questions hard, they are. Do not be put off. The only way to overcome them is to struggle through and learn from them. Good Luck.  Important Constants Constant Symbol Value Speed of light in free space c 3.00 × 108 m s−1 Elementary charge e 1.60 × 10−19 C Planck constant h 6.63 × 10−34 J s Mass of electron me 9.11 × 10−31 kg Mass of proton mp 1.67 × 10−27 kg Acceleration of free fall at Earth’s surface g 9.81 m s−2 Avogadro constant NA 6.02 × 1023 mol−1 Radius of Earth RE 6.37 × 106 m Radius of Earth’s orbit R0 1.496 × 1011 m T(K) = T(◦ C) + 273 4 Volume of a sphere = πr3 3 Surface area of a sphere = 4πr2 v 2 = u2 + 2as v = u + at s = ut + 21 at2 s = 12 (u + v)t E = hf P = Fv P = E/t v = fλ V = IR 1 1 1 R = R1 + R2 = + R R1 R2 PV P V = const. = const. T Qus. 1-5 Circle the best answer. 1. Estimate the mass of a soap bubble with a film thickness of 300 nm and a radius of 10 cm. The density of water is 1000 kg m-3. A. 4 × 10−2 kg B. 4 × 10−4 kg C. 4 × 10−5 kg D. 4 × 10−6 kg E. 4 × 10−7 kg 2. Which combination of vectors in Fig. 1 could represent the resolved components of the vector V? A. A B. B C. C D. D E. E V A B C D E Figure 1 3. Protons in the CERN LHC orbit the 27 km circumference ring at almost the speed of light. If they were not held up by a magnetic field, through what height would they fall during one orbit? A. 4 × 10−14 m B. 4 × 10−8 m C. 8 × 10−8 m D. 9 × 10−4 m E. 4 × 10−2 m 4. A long rope is held by two students, one at each end, and they begin shaking the rope to send waves along it. As they change the frequency, they sometimes see the waves cancelling out and sometimes adding together, to produce a wave that appears to remain almost stationary. The physics principle used to explain these observed effects is destructive constructive A. refraction B. interference C. interference D. diffraction E. superposition 1 5. Fig. 2 shows a uniform shelf of mass 4 kg supported at one end by a string, and by a hinge with no friction, fixed to a wall at P. If the string is held at an angle of 40◦ to the vertical and the shelf is horizontal, what is the tension in the string? A. 62 N B. 40 N C. 31 N D. 26 N E. 20 N 40o P Figure 2 6. A jet aircraft travelling at Mach 3 (three times the speed of sound) in a horizontal path at a height of 15 km passes directly over an observer on the ground. Calculate the distance between the plane and the observer when they first hear the sound. 2 7. Fig. 3 shows a point P moving in a circle of radius 4 m at a constant speed. It completes one rotation in 2.0 s. What is its average velocity, giving the magnitude and compass direction (or draw an arrow), A (a) from A through one rotation back to A, P D B N W E (b) between A and C, and C S Figure 3 (c) between A and B? 8. Two massive cannon balls are fired from the same point at the same instant, and at the same angle θ to the horizontal, but with different velocities, v1 and v2 with v1 > v2. The horizontal motion of each ball is constant, whilst the vertical motions are subject to gravity, g. Ignore air resistance. (a) Sketch the paths of the two cannonballs to just beyond their maximum heights. 3  (b) Write down the equations of motion for the horizontal (x) and vertical (y) positions of each ball, in terms of time of flight, t, v1 or v2 , θ and g. Use the notation x1 , y1 and x2 , y2 for the faster and slower balls respectively. (N.B. The balls go up and g is down.) (c) Find the direction of the straight line joining the cannonballs at any time t after firing. Hint: this is similar to finding the gradient of a graph from points (x1 , y1 ) and (x2 , y2 ). 9. A vertical length ` of rope of mass m and cross-sectional area A is gradually lowered into water whilst holding the top end of the rope. When 41 of the rope is submerged in water the tension at the top end of the rope is reduced to 56 of the initial tension. What is the ratio of the density of the rope, ρr , to the density of water, ρw ? Hints: find the tension at the top of the rope when not in the water, in terms of g, A, `, ρr. Then find the tension when a length d is in the water, now involving ρr and ρw. 4  10. To study the structure of crystals, materials scientists and biologists need to use X-rays with a wavelength approximately equal to the size of an atom. X-ray photons are produced when an electron, accelerated by a DC voltage, strikes a metal surface. In some instances, all of the kinetic energy of the electron can be used to create a single photon. Use the information given below to calculate the following quantities. For sodium: density ρm = 0.971 × 103 kg m−3 molar mass M = 23.0 g mol−1 (a) Assuming a simple cubic arrangements of the sodium atoms in the crystalline structure, estimate the diameter of a sodium atom. (b) What would be the energy of a photon with this wavelength? (c) Hence determine the accelerating voltage required to produce such photons. 5 11. A spring with spring constant k and of negligible mass has a linear force-extension graph as shown in Fig. 4a. The natural length of the spring is `0 and the spring can be compressed to a minimum length `min when the coils are squeezed together. Thecompression  factor f ` by which the spring is compressed from `0 to `min is given by f = 1 −. The spring `0 can also be stretched beyond `0 , for which the same formula for f is applicable. The spring is fixed to a rigid surface and points vertically upwards, as in Fig. 4b. tension T 0 ℓ ℓ0 extension ℓ (a) (b) Figure 4 (a) Sketch a graph of f against the length of the spring, `, over the range ` = 0 to ` = 2`0. (b) Explain, using the tension-extension graph of Fig. 4a, why the energy stored in a spring stretched or compressed to length ` is given by 12 k(`0 − `)2. (c) A ball of mass m is lowered slowly on to the spring, which decreases to a length `0 with a compression factor f 0 when at equilibrium. Obtain an expression for the spring constant k in terms of `0 , m, f 0 and g. 6 (d) A ball of mass m is released from a height h above the top of the spring and is caught in the spring, as in Fig. 4b. Sketch a graph of the force acting on the ball against height above the ground, as it falls, compresses the spring to a length ` > `min , and then rebounds back to height h. (e) The ball is held at the top of the uncompressed spring, at a height `0 above the table. It is released and at its lowest point it compresses the spring to length ` > `min. Derive an expression, explaining your physics ideas, to relate the compression factor f to f 0. Hint: eliminate k from your equations. (f) The spring is now hung from the ceiling instead, and the same ball is attached to the spring, and held with the spring at its natural length, `0. When the ball is released, what will be the value of f in terms of f 0 when the ball reaches the lowest point of its motion (whilst remaining attached to the spring)? 7 12. An ideal LED is connected in series with a resistor R and a supply of e.m.f. ε and zero internal resistance, shown in Fig. 5a. The characteristics of an ideal LED are shown in the graph of Fig. 5b, with the conduction of the LED being infinite when the potential across it is equal to Vc. Vc is known as the forward conduction voltage. 0 0 volts (a) (b) Figure 5 (a) If Vc = 1.8 V, the e.m.f. is 5.0 V and the resistor limits the current in the LED to 24 mA, what is the value of the series resistor R that is required? (b) Calculate the fraction of the power from the supply that is dissipated in the LED. (c) The supply is replaced with one that varies as a sawtooth, with a linear rise in voltage and a vertical fall, with a period of 40 ms, as shown in Fig. 6. If the peak voltage applied is 5.0 V, calculate (i) the time at which the LED first switches on, and (ii) the fraction of time for which the LED is lit. 5.0 olts 0 0 40 80 120 ms Figure 6 8 (d) Sketch a graph of the current through the LED against time, for two cycles. Mark numerical values on both of your axes. (e) Calculate the average power of the LED in this circuit with the sawtooth e.m.f. A real LED has an approximate linear I − V characteristic above Vc , but the line is not vertical. This is shown in Fig. 7 with Vc = 1.8 V. The LED is now connected without the series resistor R in the circuit, directly to the power supply. The gradient of the slope has a value of 0.040 A V−1. (f) If the maximum current now permitted to flow through the LED is 10 mA, what would be the e.m.f. of the power supply? Hint: you might redraw the graph in Fig. 7. 0.040 A V-1 0 0 = 1.8 volts Figure 7 END OF PAPER Page 9  BPhO Sponsors Worshipful Company of Scientific Instrument Makers Page 10  AS CHALLENGE PAPER 2019 Name School Total Mark / 50 Friday 8th March Time allowed: One hour Attempt as many questions as you can. If you are stuck, move on to the next question. Write your answers on this question paper. Draw diagrams. Marks allocated for each question are shown in brackets on the right. You may use any calculator. You may use any public examination formula booklet. Allow no more than 5 or 6 minutes for section A. Scribbled or unclear working will not gain marks. This paper is about problem solving. It is designed to be a challenge for the top AS physicists in the country. If you find the questions hard, they are. Do not be put off. The only way to overcome them is to struggle through and learn from them. Good Luck.  Useful constants and equations 𝑐 = 3.00 × 10 m 𝑠 ℎ = 6.63 × 10 Js 𝑒 = 1.60 × 10 C 𝑔 = 9.81 m s Avogadro constant 𝑁 = 6.0 × 10 surface area of a sphere = 4𝜋𝑟 volume of a sphere = 𝜋𝑟 𝒗𝟐 = 𝒖𝟐 + 𝟐𝒂𝒔 𝒗 = 𝒖 + 𝒂𝒕 𝟏 𝟏 𝒔 = 𝒖𝒕 + 𝟐 𝒂𝒕𝟐 𝒔 = 𝟐 (𝒖 + 𝒗)𝒕 power = force × velocity 𝑃 = 𝐸/𝑡 𝑣 = 𝑓𝜆 𝑉 = 𝐼𝑅 𝑅 =𝑅 +𝑅 = + Answers Qu 1 Qu 2 Qu 3 Qu 4 Qu 5 2 Section A: Multiple Choice Circle the correct answer to each question and write your answers in the table on page 2. Each question is worth 1 mark. There is only one correct answer to each question. 1. The mass of a car is approximately A. 10 kg B. 10 kg C. 10 kg D. 10 kg 2. The Planck constant, ℎ, is measured in units of joule.second in the SI system. If it was to be measured in units of (centimeters, grams, second) instead of (metres, kilograms, seconds) by how much would its numerical value increase? A. 10 B. 10 C. 10 D. 10 3. A large boulder of mass 𝑚 lies in a riverbed. It can be rolled over by the water in the river flowing over it at speed 𝑣. Which of the following equations could relate the mass of the boulder to the speed of the river, 𝑣, its density 𝜌 and the gravitational field strength, 𝑔? 𝑘 is a constant with no units. A. 𝑚 = B. 𝑚 = C. 𝑚 = D. 𝑚 = 4. The number of molecules in a teaspoonful of sugar is approximately A. 10 B. 10 C. 10 D. 10 5. You knock a plate of food off the table and observe that it lands upside down. This might be due to: A. The weight of the B. The air drag on C. When you D. As it slides off the food on top of the the plate makes push the plate, edge of the table, plate makes it turn it turn over. you provide a there is a turning over. spinning force applied by motion to it. gravity. /5 3 Section B: Written Answers Question 6. a) A cyclist wants to carry a heavy bag of books on her bicycle, using a shopping bag hanging from the handlebars. When she attaches the bag, it swings from side to side with a period of 1.2 s. When she rides the bicycle, her body swings from side to side each time she turns the pedals. If the bag swings with the same period, it makes the bike wobble dangerously from side to side. The diameter of the back wheel is 650 mm. There are 15 teeth on the rear cog and 48 teeth on the chain ring. What speed on the road should the cyclist try to avoid? Rear cog (15 teeth) Chain ring (48 teeth) Figure 1 /4 4  Question 7. A light, rigid rod with two equal masses, 𝑚, at the ends, is held with one end on a horizontal surface. The other end rests on a circular curve of radius 3.4 m, at a distance from the horizontal corresponding to of the circumference of a circle, as shown in Fig 2. If the surface is frictionless, what is the speed of the rod and masses when released and both masses slide on the flat surface? The masses remain in the same vertical plane. Figure 2 /4 Question 8. A ball is dropped from rest at height ℎ, and accelerates towards the ground, reaching it with final speed 𝑣 after time 𝑡. Ignore air resistance. a) Sketch a 𝑣 − 𝑡 graph and on the graph mark on the average speed, 𝑣 (𝑡). State its value in terms of 𝑣. 𝑣 𝑡 5 b) Describe, referring to the area under the curve in your graph, what is meant by this time averaged speed, 𝑣 (𝑡). c) It can be of interest to determine the average speed over a distance instead. Sketch a suitable graph to help explain the idea of distance averaged speed, 𝑣 (𝑠). d) (i) Referring to your graph, use it to explain what is meant by distance averaged speed, 𝑣 (𝑠). (ii) Calculate the distance averaged speed, 𝑣 (𝑠) in terms of 𝑣. /9 6  Question 9. A circuit with two resistors, 𝑅 and 𝑅 connected in parallel, is shown in Fig 3 below. Current 𝐼 flows through 𝑅. a) Obtain an expression for the total power, 𝑃 , dissipated in 𝑅 and 𝑅 , in terms of 𝑅 , 𝑅 and 𝐼. 𝐼1 𝑅 𝑅2 Figure 3 b) Sketch a graph of 𝑃 against 𝑅 , as 𝑅 is varied from 0 to ∞. Mark on any values. 𝑃 W /6 7  Question 10. The Sun can be treated as a large ball of gas, in which 99.9% of the nuclear energy generation is within the core. The process is well understood, but experiments to confirm the theory are important. Photons from the core are absorbed by atoms and then reradiated at lower energy, reaching the surface after about 100 000 years of travelling through the Sun’s layers. The (simplified) structure of the Sun is shown in Fig 4. Densities, 𝜌, are all in kg m-3. To obtain the wavelength of a photon radiated, we can use the equation for thermal radiation, 𝜆𝑇 = 2.9 × 10 m K in which 𝑇 is the temperature measured in kelvin, and 𝜆 is the wavelength of the radiation. CONVECTION ZONE Radiation 1𝑅⊙ 𝑇 = 1.5 × 10 K 𝜌 = 1.6 × 10 RADIATIVE 0.7𝑅⊙ ZONE 0.25𝑅⊙ Radiative Energy diffusion generated CORE 𝜌 = 8 × 10 𝑇 = 5700 K 𝜌 = 5 × 10 Figure 4. Simplified structure of the Sun. Densities, 𝝆, are in kg m-3. Temperatures are in kelvin. The solar radius is the symbol 𝑹⊙ a) (i) Photons are radiated from the surface of the Sun, at a temperature of 5700 K. Calculate their wavelength, 𝜆, using the formula for thermal radiation, In which region of the electromagnetic spectrum are these photons? 8 (ii) What would be the wavelength of a photon radiated in the core of the Sun? In which region of the electromagnetic spectrum is this? Calculate the energy of this photon (in joules) b) How is the motion of the gas different in the radiative zone from the convective zone? c) The density and temperature of the core can be found in the diagram in Fig 4. By what factor is the density of the (hydrogen) gas in the core of the Sun greater than the density of (the metal) lead, 𝜌 ? 𝜌 = 11300 kg m d) Despite the high density of the hydrogen gas in the core, due to the high temperature, we can use the equation for an ideal gas (a combination of Boyle’s Law 𝑃𝑉 = const. and = constant). This is 𝑘 𝑃 = 𝜌𝑇 𝜇 where 𝜇 is the mass of a hydrogen atom, and 𝑘 is a constant. Calculate a value for the pressure, 𝑃, of the gas at the centre of the Sun and compare it with atmospheric pressure on the Earth (𝑃 = 1.01 × 10 Pa). 𝜇 = 1.67 × 10 kg 𝑘 = 1.38 × 10 JK 9 e) From the values on the diagram, estimate the percentage of the volume of the Sun that lies within the core f) Energy is generated in the core by a series of nuclear reactions. We can summarise the beginning and end points by the following nuclear reaction. 4 1H → 4He + 2e + 2𝜈 + 4.27 × 10 joules 1 H is a hydrogen nucleus 4 He is a helium nucleus 𝑒 is an electron 𝜈 is a neutrino i.e. four hydrogen nuclei fuse together to form a helium nucleus and release a pair of electrons, a pair of neutrino particles and some energy. (i) The Sun radiates in all directions, illuminating an imaginary spherical surface. If the Earth is at a distance of 1.50 × 10 m from the Sun, and receives about 1.3 kW m-2, show that the Sun generates a power output of about 4 × 10 W. (ii) From the energy released in the reaction, calculate how many neutrinos are generated in the core each second. 10  (iii) If these neutrinos do not interact with anything, they will pass out of the core, straight through the Sun, and reach the Earth (after about 8 minutes of travel time). Calculate how many neutrinos pass through 1 cm each second at the Earth. Neutrinos are very difficult to detect. Neutrino physicists construct detectors the size of a small factory and are delighted to detect a few neutrinos each day. /15 Question 11. The Specific Heat Capacity (SHC), 𝑐, of a material is defined as the energy needed to raise the temperature of 1 kg of material by 1∘ C: Δ𝐻 = 𝑚𝑐ΔT where Δ𝐻 is the amount of thermal energy supplied to the material, 𝑚 is its mass, and Δ𝑇 is the temperature rise. An experiment to measure the SHC of a sample of a solid is shown in Fig 5. Figure 5. Heating a sample of material surrounded by insulating lagging. 11 The insulation (lagging) surrounding the sample reduces thermal energy loss. The rate at which thermal energy is lost to the surroundings can be described by Newton’s Law of Cooling: Δ𝑄 = 𝑘(𝑇 − 𝑇 ) Δ𝑡 where 𝑇 is the temperature of the material, 𝑇 is the temperature of the surroundings and is the rate at which thermal energy is lost. is a constant. a) What would affect the value of ? An accurate method of correction for loss of thermal energy is attributed to the Scottish scientist, Thomas Charles Hope. The material is heated from room temperature, 𝑇 , using apparatus as above, and the temperature recorded against time. In Fig 6, the ideal curve shows what would happen in the absence of any thermal energy being lost to the surroundings. Figure 6 b) The times 𝑡 , 𝑡 and 𝑡 are equally spaced, with 𝑡 being the time at which the temperature reaches its maximum. Explain why 𝑡 is some time after the change in gradient of the ideal curve. 12 c) Use Newton’s Law of Cooling to explain why the area 𝐴 is proportional to the thermal energy lost between times 𝑡 and 𝑡. d) Hence explain why = e) Hence find an expression for 𝑐 in terms of the quantities you could measure in the experiment and off the graph. /7 END OF PAPER 13 BLANK PAGE 14  BPhO sponsors Worshipful Company of Scientific Instrument Makers 15  AS CHALLENGE PAPER 2018 Name School Total Friday 2nd March Mark/50 Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Draw diagrams. Marks allocated for each question are shown in brackets on the right. You may use any calculator. You may use any public examination formula booklet. Allow no more than 5 or 6 minutes for section A. Scribbled or unclear working will not gain marks. This paper is about problem solving. It is designed to be a challenge for the top AS physicists in the country. If you find the questions hard, they are. Do not be put off. The only way to overcome them is to struggle through and learn from them. Good Luck. Students: Again, this year, this paper is being used to select students to invite to the BAAO Astronomy & Astrophysics Training Camp at Oxford from Monday 9th to Thursday 12th April 2018. Previous experience in these subjects is not required.  Useful constants and equations c = 3.00 × 10 m 𝑠 h = 6.63 × 10 Js e = 1.60 × 10 C g = 9.81 m s Avogadro constant 𝑁 = 6.0 × 10 surface area of a sphere = 4𝜋𝑟 volume of a sphere = 𝜋𝑟 𝑣 = 𝑢 + 2𝑎𝑠 𝑣 = 𝑢 + 𝑎𝑡 𝑠 = 𝑢𝑡 + 𝑎𝑡 𝑠 = (𝑢 + 𝑣)𝑡 power = force × velocity 𝑃 = 𝐸/𝑡 𝑣 = 𝑓𝜆 𝑉 = 𝐼𝑅 𝑅 =𝑅 +𝑅 = + Answers Qu 1 Qu 2 Qu 3 Qu 4 Qu 5 2 Section A: Multiple Choice Circle the correct answer to each question. Write your answers in the table on page 2. Each question is worth 1 mark. There is only one correct answer to each question. 1. The thickness of a page of this exam paper is approximately A. 0.01 mm B. 0.1 mm C. 1 mm D. 10 mm 2. A volt is a joule per coulomb. The unit of electrical resistance, 𝑅, in terms of base units (m, kg, s, A) is A. B. C. D. 3. A student has to solve a difficult problem on calculating a speed of an astronomical object 𝑣, which involves masses 𝑚 and 𝑚 , an acceleration 𝑎, the age of the object 𝑡, and the speed of light 𝑐. He tries several times, each time getting a different answer. Finally, he runs out of time and has to pick one of his answers. Which one is the best option? ( ) A. 𝑣 = 𝑎𝑡 B. 𝑣 = 𝑚 𝑡 + C. 𝑣=𝑚. D. 𝑣 =. 4. The number of atoms in a typical course grain of sand found on a beach is approximately A. 10 B. 10 C. 10 D. 10 5. A beaker of water sits on a top pan balance. When a student sticks his finger in the water, the reading on the balance A. Decreases B. Increases C. Remains the D. Increases or decreases same depending on the relative depth of the finger and the water 3 Section B: Written Answers Question 6. A car travels along a straight road at constant speed 𝑢. It passes a stationary motorbike, which immediately begins to accelerate from rest with a constant acceleration, 𝑓. Thus they move off from the same starting point at the same time. a) Sketch two speed-time graphs on the same axes below, of the speeds of the car and motorbike before the time 𝑡 , when the motorbike overtakes the car. 𝑣 𝑡 𝑡 b) Sketch, on the same axes below, distance-time graphs of the distances travelled by the car and motorbike, from the start until they pass each other. 𝑠 𝑡 4 c) On your graph in (b), mark with a dotted line the time 𝑡 when the vehicles have the greatest separation. State in words how you have chosen this time. d) Without using calculus, show that the vehicles have maximum separation (∆𝑠 ) at time 𝑡 =. e) Without using calculus, obtain an expression for the maximum separation of the vehicles, ∆𝑠 , in terms of 𝑢 and 𝑓. /12 5  Question 7. A glass prism has a cross section that is an isosceles triangle. One of the equal faces is silver coated to reflect a ray internally. A ray of light is incident on the prism, normal to the unsilvered face, with the incident ray being reflected twice within the prism, and emerging from the base of the prism at normal incidence. a) Sketch a large (realistic) diagram of the path of the light ray, marking on angles 𝛼 and 𝛽 for the apex and base angle of the prism respectively. Hint: the angle 𝛼 is less than 60∘. (You may wish to practise your diagram on the back page of the exam paper) b) Calculate the value of angle 𝛼, the apex of the prism. /5 6  Question 8. The European Space Agency runs experiments on Earth which require a weightless environment (freefall) for a few seconds. It uses compressed air to fire a container with the apparatus inside, upwards from a long, vertical tube. The lower end of the tube rests on the ground, as shown in Fig. 1. The container falls back onto thin cushions on the ground. The tube is 8.0 m long and the container is fired upwards with a vertical acceleration of 25𝑔. You should ignore air resistance in this question. 𝑔 = 9.81 m s Thin cushions Figure 1. Sketch of the up and down trajectory. a) Calculate the exit velocity of the container from the tube. b) Calculate the maximum height reached above the ground. 7  c) Calculate the time for which the apparatus experiences the effect of weightlessness; that is, the time for which it is in free fall. /6 Question 9. A circuit consists of a battery of emf 𝜀 = 4.5 V and negligible internal resistance, connected in series with three resistors, 𝑅 , 𝑅 , 𝑅 of values 200 Ω, 300 Ω and 400 Ω respectively. A digital voltmeter is connected between points A and B, shown in Fig. 2. 𝜀 = 4.5 V B 𝑅 = 200 Ω 𝑅 = 400 Ω V A 𝑅 = 300 Ω Figure 2. 8  a) What is the potential difference between A and B measured by the voltmeter in Fig. 2? The voltmeter is replaced with an older moving coil meter, which itself has a resistance 𝑅 of only 500 Ω, shown in Fig. 3. This lower resistance affects the circuit, but the voltmeter is calibrated to give the correct reading of the voltage across its terminals, which are connected to A and B. 𝜀 = 4.5 V B 𝑅 = 200 Ω 𝑅 = 500 Ω 𝑅 = 400 Ω V A 𝑅 = 300 Ω Figure 3. b) What would be the potential difference between A and B measured by the voltmeter in Fig. 3? /4 9  Question 10. When viewed from a point above the North Pole of the Earth, all the motions of the Earth and the Moon appear anticlockwise. a) Sketch a diagram of the Moon and Earth, relative to the Sun’s position (not to scale), with arrows to show the directions of the orbital and rotational motions of the Moon and Earth. b) As the Moon orbits the Earth, the same face of the Moon always points towards the Earth. Because of this, the Moon is said to be “tidally locked” to the Earth. The far side of the Moon is often called the dark side of the Moon. Explain why the phrase “dark side of the Moon” is misleading. You may use a diagram. 10 c) A solar day is 24 hours, from noon when the Sun is overhead in the sky, until the Sun is again overhead at noon the following day. However, if any other star in the sky is used (working from midnight to midnight when the star could be seen overhead), the measured time from the star being overhead from one day to the next day is a few minutes shorter. This is called a sidereal day. Explain why the sidereal day is shorter (you may use a diagram), and calculate by how many minutes the length of a sidereal day is less than 24 hours. Hint: the Earth takes 365.25 days to orbit the Sun. d) The Moon orbits the Earth once every lunar month. A new moon, which cannot be seen, occurs when the Moon lies between the Earth and the Sun. A few days after the new moon, a faint crescent Moon can be seen in the sky in the West. Explain, with the aid of a diagram, why this new crescent moon always appears to the west of an observer on Earth. /8 11  Question 11. New ultra low power devices (watches and calculators for example) need miniature long- lasting electrical sources. The system described here, uses a thin Pu-238 alpha source, in which the kinetic energy of the alpha particles produces photons of light in the thin layer of phosphor on a glass plate. Then photons emitted from the phosphor then produce a voltage in the photovoltaic layer, illustrated in Fig. 4 below. Figure 4. Alpha indirect conversion setup. Ref. Alpha Indirect Conversion Radioisotope Power Source, Maxim Sychov et al. Applied Radiation and Isotopes Vol 66, Issue 2, February 2008, Pages 173-177 A thin, uniform layer of a zinc sulfide (ZnS) phosphor layer is deposited on the glass slide. A thin layer of an alpha emitting radioactive source placed close to the phosphor layer will cause light to be emitted from the phosphor. The light goes in both directions out of the phosphor layer (up and down in Fig. 4), but a very thin reflecting sheet of aluminium (thin enough to allow most alphas to penetrate) reflects light emitted in the phosphor back towards the photovoltaic array. Figure 5. Radioluminescent intensity vs ZnS phosphor layer thickness, H, for an alpha source (Pu-238) and a beta source (Sr-90). Ref. ibid. 12 a) From the graph of Fig. 5, read off a value for the optimum thickness to give maximum light intensity from the layer of ZnS phosphor with an alpha source. b) The thickness of the layer is given in mg/cm2 rather than a length measurement. If the density of ZnS is 4.1 g cm-3, calculate the thickness of the ZnS layer in μm (1 μm = 1 × 10 m.) You may find a simple diagram of the ZnS layer helpful. c) The alpha particle range in a material can be calculated using the formula 10 𝐴 𝐸 𝑅 = 𝜌 in which 𝑅 is the range of the alphas in cm, 𝜌 is the density of the material in g cm-3, and 𝐸 is the initial energy of the alphas in MeV (1 MeV is 1.6 × 10 J.) -3 If 𝐸 = 5.5 MeV, 𝜌 = 4.1 g cm and 𝐴 is 48.7 (the average atomic weight of ZnS), calculate a value of 𝑅. How does this value compare with the optimal thickness of ZnS from part (b)? 13 d) In Fig. 5, the Sr-90 is a source of beta radiation. How does this explain the difference in the shape of the two curves? Hint: the penetration of beta radiation is different from alpha radiation, with alpha being stopped by a thin sheet of paper. e) A 2.15 mW Pu-238 source of power was used in each radioisotope power source (each cell). Five of these cells were connected in series to form a battery with the following three characteristics; 1. The maximum power output of the battery was 21 μW. 2. A short circuit current of 14 μA could be obtained 3. The open circuit voltage was 2.3 V i. What would be the efficiency of the power conversion? ii. What would be the internal resistance of a single cell? /10 END OF PAPER 14 BPhO sponsors 15  AS CHALLENGE PAPER 2017 Name School Total Friday 10th March Mark/50 Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Draw diagrams. Marks allocated for each question are shown in brackets on the right. You may use any calculator. You may use any public examination formula booklet. Allow no more than 6 or 7 minutes for section A. Scribbled or unclear working will not gain marks. This paper is about problem solving. It is designed to be a challenge for the top AS physicists in the country. If you find the questions hard, they are. Do not be put off. The only way to overcome them is to struggle through and learn from them. Good Luck. Students: this year this paper is being used to select students to invite to the Astronomy & Astrophysics Training Camp at Oxford Tuesday 3rd to Friday 7th April 2017. Previous experience in these subjects is not required.  Useful constants and equations c = 3.00 × 10 m h = 6.63 × 10 Js e = 1.60 × 10 C g = 9.81 m s surface area of a sphere = 4"# volume of a sphere = "# & = ' + 2* & = ' + *+ power = force × velocity 0 = 1/+ & = 34 5 = 67 7 =7 +7 = + 8 89 8: Answers Qu 1 Qu 2 Qu 3 Qu 4 Qu 5 Qu 6 2 Blank page 3 Section A: Multiple Choice Circle the correct answer to each question. Write your answers in the table on page 2. Each question is worth 1 mark. There is only one correct answer to each question. 1. Which are the correct dimensions of force in terms of mass [M], length [L] and time [T]? ;< ;=: ;= C B. > = ? &@ C. > = ?& @ D. > = ?&@ The LHC accelerator at CERN has two beams of protons circulating in opposite directions, with each proton in a beam having an energy of 7.0 TeV (T is terra which is 1012). The circumference of the circular accelerator is 27 km and the particles are travelling at (almost) the speed of light. The protons circulate as bunches, with 2808 bunches per beam. There are 1.15 × 1011 protons per bunch. 1 eV= 1.6 × 10-19 J 3. What is the energy of each beam in joules? A. 180 MJ B. 360 MJ F D. 360 J C. 5.2 × 10 J 4. What is the current in each beam? A. 0.6 μA B. 6 μA C. 6 mA D. 0.6 A 4  5. The energy stored in the LHC at any one moment is about 10 GJ, mainly in the magnetic fields of the 1200 magnets. If the mass of a magnet is about 35 × 10 kg and the energy stored was used to move the magnets in the form of kinetic energy, what would be the speed of a single magnet? A. 6.9 m s B. 11 m s C. 15 m s D. 22 m s 6. The energy content of an explosive called TNT is 4.7 MJ kg-1. What mass of TNT is equivalent to the 10 GJ of energy stored in the LHC? A. 2.1 kg B. 21 kg C. 2100 kg D. 2.1 × 105 kg Section B: Written Answers Question 7. When cutting a hard piece of cheese with a knife, a rocking motion of the knife is often used. Give an explanation why, even with a sharp knife, it is easier to cut cheese in this way. /3 5 Question 8. A spider of mass J hangs from the end of a single, elastic, thread obeying Hooke’s Law, of a web attached to the ceiling of a room. The extension of the thread is equal to the natural length of the thread ℓL , when the spider is at the bottom end of the thread. a) Explain why the work done by spider in climbing the thread all the way to the ceiling is less than the work done to climb a vertical distance 2ℓL without the thread. b) What fraction of the work that would be required to climb to the ceiling does the spider save by using the elastic thread to climb up? /4 6 Question 9. a) A mass can be measured on Earth using both a spring balance and a lever balance. If these two instruments are used on the Moon, what difference would it make to the value of the weight that you measure? Explain your answer. spring balance lever balance Figure 1. Examples of two types of balance used to weigh objects. b) A lever balance of the same type as shown above in Figure 1, is used for weighing objects. It consists of two small, unequal pans at the ends of a beam balanced on a fulcrum. The arms of the balance are of unequal length, but the beam remains horizontal when the pans are not loaded. An object of true weight M is to be weighed. When placed in one pan, the balance is levelled with a weight M in the other pan, and in the other pan, the weight M is balanced by a weight M. Find a symbolic expression that relates W to W1 and W2 before inserting numbers. If M = 1.22 kg and M = 1.90 kg, what is the true weight M? /5 7  Question 10. In 1639 J. Marc Marci (de Crownland) laid down the rules for the collision of elastic and inelastic bodies, which were again investigated by Wallis, Wren and Huygens in the late 1660s. Their work was made more accessible by Wallis in Mechanica, Pars Tertia, 1671, and through written works of other authors of this period. Here is a problem on the topic. A particle is incident on a smooth, rigid, plane surface at angle of incidence N and reflects inelastically with some loss of energy at angle of reflection O. The initial speed is '; the final speed is &. Half of the kinetic energy is lost in the collision with the wall. In addition, the normal component of velocity is reduced by a factor √3 on collision with the wall. a) Show the path of the particle on an annotated diagram, giving the angles and speeds. WALL Figure 2. Diagram to be annotated for the path of the particle. b) Using the information given, write down three equations connecting i. the initial and final speeds, ii. the components of velocities parallel to the wall, and iii. the components of velocities perpendicular to the wall. 8 c) Solve these three equations to determine the values of the angles N and O which are required to satisfy these conditions. /8 9  Question 11. An early measurement of the speed of light was made by a Danish scientist Ole Rømer in the 1670s. He observed the period of orbit of Io, the closest known moon of Jupiter at that time. The mean time interval between successive eclipses of Io by Jupiter is equal to its period of orbit, 42 h 28 min 42 s. However, for some months during the Earth year (region A in the Earth’s orbit), the period of Io’s orbit increased by a few seconds, whilst during other times (region B in the Earth’s orbit) the period decreased. It can be assumed that radius of Jupiter’s orbit about the Sun is much larger than the Earth’s. a) Sketch a diagram of the orbits of the Earth (showing its direction of motion) and Jupiter to indicate where (A and B) these variations from the mean period would be greatest. Diagram: b) The radius of the Earth’s circular orbit is 1.5 × 1011 m and its period can be taken as 365 days. If the variation of the period of the orbit of Io is up to 15 s longer or shorter than the mean, what value does this give for the speed of light? c) Sketch a graph on the axes showing the variation in the period of Io’s orbit for one Earth year. Give values on the axes and mark on A and B. Variation from mean of orbital period of Io Time /6 10  Question 12. Resistors have a very important role to play in circuits and here are four examples in which the characteristics can be put to use. a) In the circuit shown below with the arrangement of resistors 7 and 7 , what is 8 the ratio of 9 so that the power dissipated in 7 is equal to the power dissipated 8: in one of the resistors 7. 7 7 7 Figure 3. Simple arrangement of resistors with a cell. 11  b) An analogue ammeter, with a moving coil and a needle on a scale, has a full scale deflection (f.s.d,. i.e. the needle has swung fully across the scale) when a current of 2.0 mA flows through the coil which has a resistance 5.0 Ω. To measure a current of 0.20 A in a circuit, the ammeter would require a low value resistor to be connected in parallel so that not all the current flows through the coil. What is the value of the resistor Figure 4. Typical moving coil needed? meter. Sketch a diagram to illustrate your approach. c) If the 2.0 mA, 5.0 Ω moving coil meter is to be used as a voltmeter instead to measure 4.0 V f.s.d., what value series resistor would be used? A high gain amplifier is one in which the difference of the small input voltage between the two input leads is multiplied by a large factor and the polarity inverted (typically × T10U ) i.e. 5LVW T10U 5XY. This high gain is not very useful, and it is only approximate. The amplifier A shown in Figure 6 has two input and two output terminals, connected as shown. The polarity of E is opposite to that of C. C E 5XY A 5LVW D F 0V ] ^_` Figure 5. High gain amplifier for which Z[\ T abc d) In Figure 5, if 5LVW T2.0 V, mark clearly on Figure 5 the potentials of terminals C, D, E and F. 12  7 7 5i C 5XY A 5LVW 0V Figure 6. High gain amplifier with two resistors ea f\g eh attached. Two resistors 7 and 7 are connected to the amplifier, as in Figure 6. No current can flow in to terminal C on the amplifier. The potential 5i at terminal C is very small i.e. −5LVW /10U = 5i. e) Write down the currents flowing through resistors 7 and 7 in terms of the polarities across them. What approximation can now be made in order to find a simple expression for the ratio of 5LVW /5XY , which is the gain of the amplifier when controlled by the resistors. /10 13  Question 13. The announcement in 2016 about the discovery of gravitational waves has demonstrated the effects that gravity has in more subtle ways than just dropping a weight on our toe. The Einstein Tower gedanken (thought) experiment illustrates that electromagnetic waves are affected by gravity. It uses the equation for the equivalence of mass and energy, 1 = Jj in which j is the speed of light, J is the mass of an object, and 1 is the amount of energy into which it can in principle be converted. a) If a small mass J is dropped from a height ℎ, it gains kinetic energy as it falls. On reaching the ground, it could all (in principle) be converted into electromagnetic waves in the form of photons (packets of energy), each of energy 1 = ℎ3, and reflected back up. The photons could then be converted back into a mass. Explain using the concept of energy conservation, why this process indicates that photons must be affected by gravity. b) A photon of frequency of 4.2 × 10 Hz is emitted towards the ground from a satellite orbiting the Earth at a height of 450 km. Assume that the acceleration due to gravity, l is constant and the motion of the satellite may be ignored in this calculation. By identifying the energy of the photon as it leaves the satellite with a fictitious mass Jm , calculate the change in frequency, Δ3, of the photon when it reaches the ground. 14 c) When the photon travels the 450 km to Earth, how many wavelengths of light would this be if the photon was not affected by gravity? d) Since the photon is affected (slightly) by gravity, comment on how the wavelength changes as the photon falls. e) What is the change in the number of complete wavelengths along the 450 km path due to the effect of gravity on the photon? /8 END OF PAPER 15  BPhO sponsors Worshipful Company of Scientific Instrument Makers 16  AS CHALLENGE PAPER 2016 Name School Total Friday 11th March Mark/50 Time Allowed: One hour Attempt as many questions as you can. Write your answers on this question paper. Marks allocated for each question are shown in brackets on the right. You may use any calculator. You may use any public examination formula booklet. Allow no more than 6 or 7 minutes for section A. Scribbled or unclear working will not gain marks. This paper is about problem solving. It is designed to be a challenge for the top AS physicists in the country. If you find the questions hard, they are. Do not be put off. The only way to overcome them is to struggle through and learn from them. Good Luck.  BPhO sponsors - Trinity College Cambridge University - Cavendish Laboratory 2  Useful constants and equations c = 3.00 × 10 m s h = 6.63 × 10 Js e = 1.60 × 10 C g = 9.81 m s surface area of a sphere = 4!" volume of a sphere = !" % = & + 2)* % = & + )+ power = force × velocity 0 = 1/+ % = 34 5 = 67 7 =7 +7 = + 8 89 8: Answers Qu 1 Qu 2 Qu 3 Qu 4 Qu 5 Qu 6 3 Section A: Multiple Choice Circle the correct answer to each question. Write your answers in the table on page 3. Each question is worth 1 mark. There is only one correct answer to each question. 1. Which are the correct dimensions of energy in terms of mass [M], length [L] and time [T]?