PHYSICAL QUANTITIES Quantity Gravitational acceleration on Earth Radius of Earth Mass of Earth Mass of Moon Radius of Moon Mean Radius of Moon Orbit Universal constant of gravitation Speed of light in vacuum Speed of sound in air Symbol g RE ME MM RM Value 9·8 m s–2 6·4 × 106 m 6·0 × 1024 kg 7·3 × 1022 kg 1·7 × 106 m 3·84 × 108 m G c v 6·67 × 10–11 m3 kg–1 s–2 3·0 × 10 m s 3·4 × 10 m s
2 8 –1 –1
Quantity Mass of electron Charge on electron Mass of neutron Mass of proton Mass of alpha particle Charge on alpha particle Planck’s constant Permittivity of free space Permeability of free space
Symbol me e mn mp ma h
Value 9·11 × 10–31 kg –1·60 × 10–19 C 1·675 × 10–27 kg 1·673 × 10–27 kg 6·645 × 10–27 kg 3·20 × 10–19 C 6·63 × 10–34 J s 8·85 × 10–12 F m–1 4p × 10–7 H m–1
REFRACTIVE INDICES The refractive indices refer to sodium light of wavelength 589 nm and to substances at a temperature of 273 K. Substance Diamond Glass Ice Perspex SPECTRAL LINES Element Hydrogen Wavelength/nm 656 486 434 410 397 389 589 Colour Red Blue-green Blue-violet Violet Ultraviolet Ultraviolet Yellow Element Cadmium Wavelength/nm 644 509 480 Lasers Element Carbon dioxide Helium-neon PROPERTIES OF SELECTED MATERIALS Substance Density/ kg m–3 Melting Point/ K Boiling Point/K Specific Heat Specific Latent Specific Latent Capacity/ Heat of Heat of J kg–1 K–1 Fusion/ Vaporisation/ J kg–1 J kg–1 9·02 × 102 3·86 × 102 6·70 × 102 2·10 × 103 2·43 × 103 2·52 × 103 3·93 × 103 4·19 × 103 …. 1·43 × 104 1·04 × 103 9·18 × 102 3·95 × 105 2·05 × 105 …. 3·34 × 105 1·81 × 105 9·9 × 104 …. 3·34 × 105 …. …. …. …. …. …. …. …. 8·30 × 105 1·12 × 106 …. 2·26 × 106 …. 4·50 × 105 2·00 × 105 2·40 × 104 Wavelength/nm 9550 10590 633 Colour Infrared Red Colour Red Green Blue Refractive index 2·42 1·51 1·31 1·49 Substance Glycerol Water Air Magnesium Fluoride Refractive index 1·47 1·33 1·00 1·38
Aluminium Copper Glass Ice Glycerol Methanol Sea Water Water Air Hydrogen Nitrogen Oxygen
2·70 × 103 8·96 × 103 2·60 × 103 9·20 × 102 1·26 × 103 7·91 × 102 1·02 × 103 1·00 × 103 1·29 9·0 × 10–2 1·25 1·43
933 1357 1400 273 291 175 264 273 …. 14 63 55
2623 2853 …. …. 563 338 377 373 …. 20 77 90
The gas densities refer to a temperature of 273 K and a pressure of 1·01 × 105 Pa.
Marks 1. A stunt driver is attempting to “loop the loop” in a car as shown in Figure 1. Before entering the loop the car accelerates along a horizontal track. P loop
X Figure 1 The radius r of the circular loop is 6·2 m. The total mass of the car and driver is 870 kg.
(a) Show that the car must have a minimum speed of 7·8 m s–1 at point P to avoid losing contact with the track. (b) During one attempt the car is moving at a speed of 9·0 m s–1 at point P. (i) Draw a labelled diagram showing the vertical forces acting on the car at point P. (ii) Calculate the size of each force. (c) When the car exits the loop the driver starts braking at point X. For one particular run the displacement of the car from point X until the car comes to rest at point Y is given by the equation s = 9·1t – 3·2 t2 Sketch a graph to show how the displacement of the car varies with time between points X and Y. Numerical values are required on both axes.
2. The entrance to a building is through a revolving system consisting of 4 doors that rotate around a central axis as shown in Figure 2A.
Figure 2A The moment of inertia of the system about the axis of rotation is 54 kg m2. When it rotates, a constant frictional torque of 25 N m acts on the system. (a) The system is initially stationary. On entering the building a person exerts a constant force F perpendicular to a door at a distance of 1·2 m from the axis of rotation as shown in Figure 2B.
Figure 2B The angular acceleration of the system is 2·4 rad s–2. (i) Calculate the magnitude of the applied force F. (ii) The applied force is removed and the system comes to rest in 3·6 s. Calculate the angular displacement of the door during this time. 3 3
Marks 2. (continued) (b) On exiting the building the person exerts the same magnitude of force F on a door at the same distance from the axis of rotation. The force is now applied as shown in Figure 2C.
Figure 2C How does the angular acceleration of the door system compare to that given in part (a)? Justify your answer.
Marks 3. Planets outside our solar system are called exoplanets. One exoplanet moves in a circular orbit around a star as shown in Figure 3.
Figure 3 The period of orbit is 14 days. The mass Ms of the star is 1·7 × 1030 kg.
(a) (i) Show that the radius of the orbit can be given by the relationship
2 r 3 = GMs T 2 4p
where the symbols have their usual meaning. (ii) Calculate the radius of this orbit.
m and its mass is 5·4 × 1026 kg. (b) The radius of the exoplanet is 1·2 × 108 Calculate the value of the gravitational field strength g on the surface of the exoplanet. 2 (c) Astrophysicists have identified many black holes in the universe.
(i) State what is meant by the term black hole. kg and a radius of (ii) A newly discovered object has a mass of 4·2 × 1030 2·6 × 104 m.
Show by calculation whether or not this object is a black hole.
Marks 4. A “saucer” swing consists of a bowl shaped seat of mass 1·2 kg suspended by four ropes of negligible mass as shown in Figure 4A.
Figure 4A When the empty seat is pulled back slightly from its rest position and released, its motion approximates to simple harmonic motion. (a) Define the term simple harmonic motion. (b) The acceleration-time graph for the seat with no energy loss is shown in Figure 4B. a/m s–2 1·28 1
Figure 4B (i) Show that the amplitude of the motion is 0·29 m. (ii) Calculate the velocity of the seat when its displacement is 0·10 m. 3 2
(c) Calculate the displacement of the seat when the kinetic energy and potential energy are equal. 3 (9)
Marks 5. The Bohr model of the atom suggests that the angular momentum of an electron orbiting a nucleus is quantised. A hydrogen atom consists of a single electron orbiting a single proton. Figure 5 shows some of the possible orbits for the electron in a hydrogen atom.
n=3 n=2 n=1 nucleus
Figure 5 The table shows the values of the radii for the first three orbits. Orbital radius/10–10 m 0·53
Orbit number, n 1 2 3 (a)
2·1 4·8 2 2
(i) Calculate the speed of the electron in orbit number 3. (ii) Calculate the de Broglie wavelength associated with this electron.
(iii) What is the name given to the branch of physics that treats electrons as 1 waves and predicts their position in terms of probability? (b) Compare the magnitudes of the electrostatic and gravitational forces between an electron in orbit number 1 and the proton in the nucleus. Justify your answer by calculation. 3 (8)
[Turn over for Question 6 on Page ten
Marks 6. A research physicist is investigating collisions between protons and the nuclei of metallic elements. Protons are accelerated from rest across a potential difference of 4·0 MV. The protons move through a vacuum and collide with a metal target as shown in Figure 6A. vacuum proton source target
4·0 MV Figure 6A (a) (i) Calculate the maximum speed of the protons as they hit the target. (ii) In one test the researcher uses zirconium as the target. A proton of charge q and velocity v travels directly towards a zirconium nucleus as shown in Figure 6B. The zirconium nucleus has charge Q. 2
q v proton Figure 6B Show that the distance of closest approach r to the metal target is given by qQ r= 2peo mv2 where the symbols have their usual meaning. 1 zirconium nucleus
(iii) Calculate the distance of closest approach for a proton travelling towards a zirconium nucleus in the target. 3
Marks 6. (continued) (b) At CERN protons are accelerated to speeds approaching the speed of light. Calculate the relativistic energy of a proton moving at 0·8c. 4 (c) A student visiting CERN asks why the protons in the nucleus of an atom do not just fly apart. Explain fully why protons in a nucleus do not behave in this way. 3 (13)
Marks 7. In a nuclear power station liquid sodium is used to cool parts of the reactor. An electromagnetic pump keeps the coolant circulating. The sodium enters a perpendicular magnetic field and an electric current, I, passes through it. A force is experienced by the sodium causing it to flow in the direction shown in Figure 7A. B
cross-section showing liquid sodium I
direction of flow of liquid sodium I 0·40 m
The magnetic induction B is 0·20 T. The current I in the sodium is 2·5 A and is perpendicular to the magnetic field. (a) Define one tesla.
(b) Calculate the force acting on the 0·40 m length of sodium within the magnetic 2 field. (c) The pump is moved during maintenance and as a result the direction of the magnetic field is changed so that it is no longer perpendicular to the current. What effect does this have on the rate of flow of sodium passing through the pump? You must justify your answer. 2
Marks 7. (continued) (d) An engineer must install a long, straight, current carrying wire AB close to the pump and is concerned that the magnetic induction produced may interfere with the safe working of the pump. The wire is 750 mm from the pump and carries a current of 0·60 A. Show by calculation that the magnetic induction at this distance is negligible. 2
(e) A second long straight wire CD is installed parallel to the first wire AB as shown in Figure 7B. A C
0·010 m 0·60 A 0·60 A
B Figure 7B
(i) It also carries a current of 0·60 A in the same direction as in the first wire AB. Calculate the size of the force per unit length exerted on wire CD by 2 wire AB. (ii) State the direction of the force on the wire CD. Justify your answer. 2 (11)
Marks 8. In 1909 Robert Millikan devised an experiment to investigate the charge on a small oil drop. Using a variable power supply he adjusted the potential difference between two horizontal parallel metal plates until an oil drop was held stationary between them as shown in Figure 8.
+ 16 mm oil drop 2·0 kV –
Figure 8 (a) What was Millikan’s main conclusion from this experiment? (b) Draw a labelled diagram showing the forces acting on the stationary oil drop. (c) The parallel plates are fixed 16 mm apart. In one experiment the charge on the oil drop was found to be 2·4 × 10 –18 C. Calculate the mass of the oil drop. 3 (5) 1 1
9. The charge Q on a hollow metal sphere is (–15·0 ± 0·4) µC. The sphere has a radius r of (0·65 ± 0·02) m.
Figure 9 (a) Calculate the electrostatic potential at the surface of the metal sphere. (b) Calculate the absolute uncertainty in the electrostatic potential. (c) State the electrostatic potential at the centre of the sphere. 2 2 1 (5)
Marks 10. A 0·40 H inductor of negligible resistance is connected in a circuit as shown in Figure 10. Switch S is initially open. + 9·0 V – S
0·40 H Figure 10
(a) (i) The switch S is closed. Sketch a graph of current against time giving numerical values on the current axis. 2 (ii) Explain fully the shape of the graph. (b) Calculate the initial rate of change of current when switch S is closed. 2 2 (6)
Marks 11. High quality optical flats made from glass are often used to test components of optical instruments. A high quality optical flat has a very smooth and flat surface. (a) During the manufacture of an optical flat, the quality of the surface is tested by placing it on top of a high quality flat. This results in a thin air wedge between the flats as shown in Figure 11A. monochromatic light flat being tested thin air wedge d high quality flat 50 mm Figure 11A The thickness d of the air wedge is 6·2 × 10–5 m. Monochromatic light is used to illuminate the flats from above. When viewed from above using a travelling microscope, a series of interference fringes is observed as shown in Figure 11B. not to scale
1·2 × 10–3 m Figure 11B Calculate the wavelength of the monochromatic light. (b) A second flat is tested using the same method as in part (a). This flat is slightly curved as shown in Figure 11C. 3
Figure 11C Draw the fringe pattern observed. 1
Marks 11. (continued) (c) Good quality optical flats often have a non-reflecting coating of magnesium fluoride applied to the surface as shown in Figure 11D. magnesium fluoride
optical flat Figure 11D (i) With the aid of a diagram explain fully how the coating reduces 2 reflections from the flat for monochromatic light. (ii) Calculate the minimum thickness of magnesium fluoride required to 2 make the flat non-reflecting for yellow light from a sodium lamp. (8)
Marks 12. A water wave of frequency 2·5 Hz travels from left to right. Figure 12 represents the displacement y of the water at one instant in time. direction of travel
S 0 T x/m 0·060 m
0·28 m Figure 12 Points S and T are separated by a horizontal distance of 0·28 m. The phase difference between these two points is 3·5 radians. (a) Calculate the wavelength of this wave. (b) A second wave with double the frequency travels in the same direction through the water. This wave has five times the intensity of the wave in part (a). Calculate: (i) the speed of this wave; (ii) the amplitude of this wave. 1 2 (5) 2
Marks 13. A student is investigating polarisation of waves. (a) State what is meant by plane polarised light. (b) The student wishes to investigate polarisation of sound waves and asks a teacher for suitable apparatus. The teacher says that sound waves cannot be polarised. Why can sound waves not be polarised? 1 1
(c) (i) While doing some background reading the student discovers that the
Brewster angle ip for the liquid solvent triethylamine is given as 54·5 °. Explain using a diagram what is meant by the Brewster angle. 2 (ii) Calculate the refractive index of triethylamine. 1 (5)