24   12 In 2011, a tsunami was caused by a massive earthquake centred some distance off the coast of Japan. The tsunami caused a cooling system failure at the Fukushima Nuclear Power Plant. This resulted in a nuclear meltdown and radioactive materials were released into the surroundings. (a) A reservoir beside one of the reactor buildings contained a large volume of water. In 2013, this water was found to have an extremely high concentration of caesium-137. Caesium-137 is a radioactive isotope of caesium. (i) Complete the nuclear equation for the decay of caesium-137. 137 55Cs → ............. .............Ba + ............. .............β− + 0 0νe̅ (2) (ii) An activity of 2.35 × 1012 Bq per m3 of water in the reservoir was measured. It is suggested that a safe level for the activity of all water in the reservoir would be 100 Bq. Calculate the time in years for the caesium-137 to decay to a safe level. volume of water in reservoir = 5000 m3 half-life of caesium-137 = 30 years (4) .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. Time = ....................................................... years Turn over 25   (b) The most common radionuclide amongst the fission products in the fuel was iodine-131, which decays with a half-life of 8.0 days to form a stable isotope of the gas xenon. Deduce whether enough xenon would have collected in 32 days to exert a pressure of 1.0 × 105 Pa in a volume of 450 m3. Assume that no gas escapes. temperature = 20 °C initial number of iodine nuclei = 1.25 × 1028 (6) .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. 26   (c) Buildings in nearby Tohoku University suffered structural damage during the 2011 earthquake. The graph shows how the acceleration of one of the buildings, measured on the 9th floor, varied with time during the earthquake. 90 88 86 84 82 80 t / s 10 5 0 −5 −10 a / m s−2 (Source: https://www.sciencedirect.com/science/article/pii/S0038080612001035) At the time it was reported that during the earthquake the 9th floor of the building displaced by more than 30 cm from its normal position. Assess the accuracy of this report. 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(Total for Question 12 = 17 marks) Turn over 27   13 A student used a sonometer to investigate the properties of a stretched wire. The sonometer is a long hollow wooden box. A steel wire is attached to one end of the box and rests on two wooden bridges. The wire is placed under tension T by hanging a mass from the end of the wire, as shown. wire pulley mass wooden bridges bench sonometer L The student placed the base of a vibrating tuning fork in contact with the wire, at one of the bridges. This set the wire into oscillation. He adjusted the position of the other bridge until a single-loop standing wave was produced on the wire between the bridges. (a) Explain how an antinode is produced at the mid-point of the wire between the bridges. (3) .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. 28   (b) The student repeated this for a series of tuning forks with different frequencies f. For each fork he measured the distance L between the bridges. The steel wire, of mass per unit length μ, was placed under tension T by hanging a mass of 2.10 kg from the end of the wire. (i) State one safety precaution that should be taken when carrying out the investigation. (1) .................................................................................................................................................................................................................................................. (ii) The student plotted a graph of L2 against 1/f 2. Show that the gradient of this graph is equal to 4 T μ (3) .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. Turn over 29   (iii) The student’s graph is shown below. 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 1 / f 2 / 10−6 s2 0.05 0.04 0.03 0.02 0.01 0.00 L2 / m2 The value of μ for different standard wire gauge (SWG) steel wire is shown in the table. SWG μ / g m−1 22 3.15 24 1.95 26 1.31 Deduce which wire the student used in the investigation. (4) .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. 30   (c) The student then found a value of μ for a brass wire, using a different method. (i) He measured the diameter d of the wire using a micrometer. Explain one technique the student should use when measuring d. (2) .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. 31   (ii) The student obtained the following data. d / mm 0.55 0.59 0.57 0.58 The stated value of μ for the brass wire used by the student was 2.14 × 10−3 kg m−1. Deduce whether the student’s data supports this value for μ. density of brass = 8700 kg m−3 ± 200 kg m−3 (6) .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................. (Total for Question 13 = 19 marks) TOTAL FOR PAPER = 120 MARKS 32   List of data, formulae and relationships Acceleration of free fall g = 9.81 m s−2 (close to Earth’s surface) Boltzmann constant k = 1.38 × 10−23 J K−1 Coulomb law constant k = 1 4πε0 = 8.99 × 109 N m2 C−2 Electron charge e = −1.60 × 10−19 C Electron mass me = 9.11 × 10−31 kg Electronvolt 1 eV = 1.60 × 10−19 J Gravitational constant G = 6.67 × 10−11 N m2 kg−2 Gravitational field strength g = 9.81 N kg−1 (close to Earth’s surface) Permittivity of free space ε0 = 8.85 × 10−12 F m−1 Planck constant h = 6.63 × 10−34 J s Proton mass mp = 1.67 × 10−27 kg Speed of light in a vacuum c = 3.00 × 108 m s−1 Stefan-Boltzmann constant σ = 5.67 × 10−8 W m−2 K−4 Unified atomic mass unit u = 1.66 × 10−27 kg Mechanics Kinematic equations of motion s = (u + v)t 2 v = u + at s = ut + 1 2 at2 v2 = u2 + 2as Forces ∑F = ma g = F m W = mg moment of force = Fx Momentum p = mv Work, energy and power ΔW = FΔs Ek = 1 2 mv2 ΔEgrav = mgΔh P = E t P = W t efficiency = useful energy output total energy input efficiency = useful power output total power input Turn over 33   Electric circuits Potential difference V = W Q Resistance R = V I Electrical power and energy P = VI P = I 2R P = V 2 R W = VIt Resistivity R = ρl A Current I = ΔQ Δt I = nqvA Materials Density ρ = m V Stokes’ law F = 6πηrv Hooke’s law ΔF = kΔ x Young modulus Stress σ = F A Strain ε = Δ x x E = σ ε Elastic strain energy ΔEel = 1 2 FΔ x Waves and particle nature of light Wave speed v = f λ Speed of a transverse wave on a string v = T μ Intensity of radiation I = P A Power of a lens P = 1 f P = P1 + P2 + P3 + … Thin lens equation 1 u + 1 v = 1 f Magnification for a lens m = image height object height = v u Diffraction grating nλ = d sin θ Refractive index n1 sin θ1 = n2 sin θ2 n = c v Critical angle sin C = 1 n Photon model E = h f Einstein’s photoelectric equation hf = ϕ + 1 2 mv2 max de Broglie wavelength λ = h p 34   Further mechanics Impulse FΔt = Δp Kinetic energy of a non-relativistic particle Ek = p2 2m Motion in a circle v = ωr T = 2π ω F = ma = mv2 r a = v2 r a = rω2 F = mrω2 Fields Coulomb’s law F = Q1Q2 4πε0r2 Electric field strength E = F Q E = Q 4πε0r2 E = V d Electric potential V = Q 4πε0r Capacitance C = Q V Energy stored in a capacitor W = 1 2 QV W = 1 2 CV 2 W = 1 2 Q 2 C Capacitor discharge Q = Q0e−t/RC I = I0e−t/RC V = V0e−t/RC ln Q = ln Q0 −  t RC ln I = ln I0 −  t RC ln V = ln V0 −  t RC In a magnetic field F = BIl sin θ F = Bqv sin θ Faraday’s and Lenz’s laws E = −d(Nϕ) dt Root-mean-square values Vr ms = V0 √2 Ir ms = I0 √2 35   Nuclear and particle physics In a magnetic field r = p BQ Thermodynamics Heating ΔE = mcΔθ ΔE = LΔm Molecular kinetic theory 1 2 mác2ñ = 3 2 kT pV = 1 3 Nmác2ñ Ideal gas equation pV = NkT Stefan-Boltzmann law L = σAT 4 L = 4πr2σT 4 Wien’s law λmaxT = 2.898 × 10−3 m K Space Intensity I = L 4πd 2 Redshift of electromagnetic radiation z = Δλ λ  ≈ Δf f  ≈ v c Cosmological expansion v = H0d Nuclear radiation Mass-energy ΔE = c2Δm Radioactive decay A = λN dN dt = −λN λ = ln 2 t½ N = N0 e−λt A = A0 e−λt Gravitational fields Gravitational force F = Gm1m2 r 2 Gravitational field strength g = Gm r 2 Gravitational potential Vgrav = −Gm r Oscillations Simple harmonic motion F = −k x a = −ω2x x = A cos ωt v = −Aω sin ωt a = ‒Aω2 cos ωt T = 1 f = 2π ω ω = 2π f Simple harmonic oscillator T = 2π m k T = 2π l g 36   BLANK PAGE