A2 Physics Definitions PDF

A2: MOMENTUM The (linear) momentum p (kg ms-1) of a body is its mass × its velocity: p = mv, where m is the body’s mass (kg) and v is its velocity (ms-1). It is a vector quantity (same direction as velocity). The total momentum of a system is the sum of the momentums of all bodies in the system. M...

A2: MOMENTUM The (linear) momentum p (kg ms-1) of a body is its mass × its velocity: p = mv, where m is the body’s mass (kg) and v is its velocity (ms-1). It is a vector quantity (same direction as velocity). The total momentum of a system is the sum of the momentums of all bodies in the system. Momentum can be used to work out kinetic energy: Ek = p2/2m, where Ek is the kinetic energy of the body(J), p is its momentum (kgms-1) and m is its mass (kg). You need to be able to derive this equation from Ek = ½ mv2 and p = mv. Newton’s second law of motion: The resultant force acting on a body is proportional to the rate of change of momentum and acts in the direction of the momentum change: F = p/t, where F is the resultant force on the body (N),  is a mathematical operator modelling ‘change of’, p is the momentum (kg ms-1) and t is the time (s). The Newton (N) is defined so that the rate of change of momentum is equal to the resultant force. For a body with constant mass, F = (mv)/t = m v/t = ma, as used in AS Mechanics. For a constant rate of change of mass (e.g. a fluid accelerated to a constant speed): F = (mv)/t = v m/t, where v is the speed reached by the fluid (ms-1) and m/t is the mass of fluid accelerated per second (kg s-1). The impulse of a force in Ns or kg ms-1 is equal to the change of momentum: Ft = p Law of conservation of momentum: The total momentum of a system is constant, provided no resultant external force acts on the system. A collision is elastic if the total kinetic energy of the system is conserved during the collision. The collision is inelastic if kinetic energy is transferred to another type of energy like sound for example. A2: CIRCULAR MOTION A body undergoes uniform circular motion when it travels on a circular path with constant speed. The velocity of a body in uniform circular motion constantly changes direction, therefore the body is accelerated. The angular velocity in radian per second (rad s-1) is calculated with:  = 2/T, where T is the time for one revolution or the period (s). The relation between linear speed v (ms-1) and angular velocity  (rad s-1) is: v = r, where r is the radius of the circular path (m). The acceleration for uniform circular motion is directed towards the centre of the circular path and it is equal to: a = v2/r = r, where a is the acceleration (ms-2), v is the linear speed (ms-1), is the angular speed (rad s-1) and r is the radius of the circular path (m). You need to be able to use vector diagram to derive a = v2/r or a = r Please refer to the lesson notes. The acceleration is perpendicular to the velocity, so the speed stays constant (no work is done). The resultant force acting on a body in uniform circular motion is called the centripetal force. It is directed towards the centre of the circular path and its magnitude is: F = mv2/r = mr, where F is the centripetal force (N), m is the body’s mass (kg), v is its linear speed (ms-1), is its angular velocity (rad s-1) and r is the radius of the circular path (m). The centripetal force is perpendicular to the body’s velocity and motion. A2: ELECTRIC FIELDS Coulomb’s law The force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of their separation: F = k q1 q2 /r2, where F is the electrostatic force (N), q1, q2 are the two charges (C), and r is the distance between the two charges (m). k = 1/4πε0 where ε0 = 8.85 10-12 F m–1 is the electric field constant (also called permittivity of free space). An electric field is an area in which a charge experiences a force. The electric field strength is defined as the electrostatic force per unit charge: E = F/q, where E is the electric field strength (NC-1), F is the force acting on the charge (N) and q is the charge (C). The field is represented by field lines (or lines of force): the lines which positive test charges would follow. The arrows indicate the direction of the force on positive charges. A point like charge q produces a radial electric field. The electric field strength E (NC-1) is: E = kq/r2, where q is the point like charge (C), and r is the distance from the point like charge (m). The diagram on the left shows the field for a negative charge. The electric field between 2 parallel plates to which a potential difference is applied is uniform (identical everywhere). Its strength E (NC-1) is: E = V/d, where V is the potential difference across the plates (V) and d is the distance between the plates (m). In the diagram, the top plate has a positive potential. A charge accelerated through a potential difference gains energy W (J): W = qV, where q is the accelerated charge (C) and V is the potential difference (V). One electronvolt (eV) is the energy gained by one electron accelerated through a potential difference of 1V: 1eV = 1.60 × 10–19 J. Equipotentials are lines of constant potential energy. A charge moving along an equipotential does not gain or lose electric potential energy. Equipotentials are perpendicular to the electrostatic forces, hence to the field lines. When a charge move from equipotential V1 to equipotential V2, it would gain or lose electric potential energy equal to W = q(V2-V1), where q is the charge (C) and W is the energy transferred (J). For a radial field caused by a point charge, the equipotentials are circles centred about the point charge and their potentials are given by: V = k Q / r, where V is the potential (V), Q is the charge of the particle placed in the field (C) and r is the distance between the centres of the point charge and the particle (m). For a uniform field, the equipotentials are parallel to the 2 electrodes and equidistant (e.g. the equipotential exactly in the middle of the 2 plates has a potential equal to half the potential difference between the 2 plates). Equipotentials and electric field strength are linked by: E = - dV/dr, where dV is the potential change between 2 nearby points, dr is the distance between the same 2 points and E is the electric field strength between these 2 nearby points. Where the field is strong, equipotentials are close together. The minus sign gives the direction of the field (e.g. away from a positive point charge OR towards a negative point charge). A2: CAPACITORS The capacitance C of capacitor (e.g. a pair of parallel plates) is: C = Q/V, where Q is the charge stored in the capacitor (C) and V is the potential difference across the capacitor (V). Capacitance is measured in Farad (F). The energy W stored in a capacitor (J) is: W = 1/2 CV2 = 1/2QV = 1/2 Q2/C, where C is the capacitance (F), Q is the stored charge (C) and V is the potential difference across the capacitor (V). You need to be able to derive the equation for the area under the potential difference against charge stored and also W = 1/2 CV2 and W = 1/2 Q2/C from W= 1/2QV and Q=CV. When charging a capacitor from a battery through a fixed resistance, the current follows a exponential decay curve: I = Io e–t/RC where Io is the initial current (A) (Io = Vo/R), t is the time (s), R is the fixed resistance (Ω) and C is the capacitance (F). RC is also called the time constant of the circuit (s). The charge and potential difference will build up towards the voltage of the power supply Vmax or towards final charge Qmax = CVmax. (Note: both V-Vmax and Q-Qmax follow an exponential decay, where V is the potential difference across the capacitor and Q is the charge stored in the capacitor. When discharging a capacitor through a fixed resistance, the charge Q stored in the capacitor (C) follows an exponential discharge curve: Q = Qo e–t/RC where Qo is the initial charge of the capacitor (C), t is the time, R is the fixed resistance (Ω) and C is the capacitance (F). RC is the time constant of the circuit (s). You need to be able to derive: as V =Q/ C, the potential difference across the capacitor follows: V = Qo /C e–t/RC = Vo e–t/RC as I = - dQ/dt, the current out of the capacitor follows: I = Qo /RC e–t/RC = Io e–t/RC and also the corresponding log equations: ln Q = ln Q0 - t/RC, ln V = ln V0 – t/RC and ln I = ln I0 – t/RC. A2: MAGNETIC FIELDS A magnetic field is an area in which a magnetic material experiences a force. It is represented by field lines: the lines along which a tiny bar magnet would lie. The direction of the field lines is the direction towards which the north seeking pole would point, away from North and towards South. The direction of the magnetic field around a current carrying wire and around a solenoid is given by the corkscrew rule (using your right hand). A uniform magnetic field is a field in which the magnetic field strength is constant. All magnetic field lines are parallel to each other and equidistant. Fleming’s left hand rule: The force F on a current carrying wire at right angle to a uniform magnetic field is: F = BIL, where F is the magnetic force on the wire (N), B is the magnetic field strength (T), I is the current (A) and L is the length of wire (m). Fleming’s left hand rule gives the direction of the force: Thumb = force F, First finger = magnetic field B, Second finger = current I. If there is an angle θ between the magnetic field lines and the wire, the force is F = Bsin θ IL, where Bsin θ is the field component perpendicular to the wire. The magnetic field strength B (also called magnetic flux density) is the force per unit length per unit current for a wire at right angle to the magnetic field: B = F/IL. It is measured in Tesla (T). The force F on a charged particle (N) moving at right angle to a uniform magnetic field is directly proportional to the magnetic field strength B (T), to the particle charge q (C) and to its speed v (ms-1): F = Bqv. Fleming’s Left Hand Rule gives the direction of the force with the current in the direction of the velocity for positive charges and in the opposite direction of the velocity for negative charges. If the velocity of the particle is not at right angle to the magnetic field lines: F = Bsin θ qv, where Bsin θ is the field component perpendicular to the particle velocity. When a charged particle moves at right angle to a uniform magnetic field, the force F acts as a centripetal force and makes the particle move in a circle of radius r = p/Bq, where r is the radius of the circle (m), q is the particle’s charge (C), p is the particle’s moment (kgms-1) and B is the magnetic field strength (T). You need to be able to derive r = p / Bq from F = Bqv, p = mv and F = mv2/r. A2: ELECTROMAGNETIC INDUCTION The magnetic flux  is the product of the magnetic flux density B (T) and area A (m2) at right angle to the field lines:  = BA. It is measured in Weber (Wb). The flux linkage  (Wb) through a coil of N turns is: N = NBA, where A (m2) is the area of the coil at right angle to the magnetic field of strength B (T). Faraday’s law of electromagnetic induction: The induced e.m.f. in a conductor is equal to the rate of change of magnetic flux linkage through the conductor: ε = – N∆φ/∆t, where ε is the induced electromotive force (emf) (V), N is the number of turns in the coil (N = 1 for a simple conductor), ∆φ/∆t is the rate of change of magnetic flux (Wb s-1). When a conductor cuts magnetic field lines, there is flux cutting (a change in flux), which induces an emf in the conductor. If this conductor is in a closed circuit, a current is also induced. If a bar magnet moves in/out of a coil, the motion of the bar magnet changes the magnetic field around the coil so there is a change in the flux linkage of the coil. This induces an emf in the coil. If the coil is in a closed circuit, a current is also induced. When a coil is placed close to an electromagnet connected to an AC supply (transformer), the magnetic field of the electromagnet keeps changing and cuts through the coil, this changes the flux linkage of the coil and induces an emf in the coil. If this coil is in a closed circuit, a current is also induced. The explanations MUST be applied/adapted to the situation described in the question. Increased rate of flux cutting and increased rate of change of flux linkage will increase the induced emf. Lenz’s law The direction of an induced current/emf has an effect tending to oppose the change in flux that produces it. Lenz’s law is responsible for the negative sign in the equation for electromagnetic induction: ε = – N∆φ/∆t. You MUST apply Lenz’s law in the situation described in the question. Describing AC supply: The root mean square value of an AC power supply is the voltage/current of a DC supply that would supply the same power to the component: VRMS = Vmax/√2 and IRMS = Imax/√2 where Vmax and Imax are the peak values for voltage and current. A2: PARTICLE PHYSICS The atomic number Z of an atom is the number of protons. The mass number N of an atom is the number of nucleon. A nucleon is a particle forming the nucleus: a proton or a neutron. The atomic mass unit u is defined by: 1u = 1.661 × 10-27 kg (1/12th the mass of a carbon-12 atom). Protons and neutrons have a mass of approximately 1u. The Rutherford experiment:  particles emitted by a radioactive source were fired at a very thin gold foil positioned in a vacuum. The radioactive source was chosen so that all emitted α particles had identical kinetic energy. Most  particles went straight through or suffered small deflections, suggesting that the atom is mainly empty space. Few were deflected at small angles and very few (1 in 8000) were deflected at large angles (more than 90°), suggesting that there is a tiny nucleus in the atom which contains most of the atom mass and which is positively charged. When high energy particles collide or interact with gases or liquids, they ionise the particles, losing some of their kinetic energy. This property is used in particle detectors like cloud chamber and bubble chamber. As a result, neutral particles do not leave a track in particle detectors. When particles are accelerated close to the speed of light, they gain kinetic energy through a change in mass rather than a change in speed, due to relativistic effects. Only massless particles reach the speed of light. In a linear accelerator/linac, an electric field accelerates charged particles in the gap between the electrodes. Particles move at steady speed in the electrodes (drift tubes) and the applied potential difference is reversed whilst the particle is in the tube. The electrodes become longer to be able to compensate for the faster speed of the particles and keep the frequency of the alternating potential difference constant. In a cyclotron, an electric field accelerates the charged particles in the gap between the dees (circular electrodes). The potential difference switches polarity every half-cycle and has a fixed frequency. A uniform magnetic field is applied at right angle to the dees, which applies a centripetal force on the charged particles. The particles complete a half-circle in a dee and the radius increases as the particle’s speed increases. This allows the particles to spend the same time in a dee no matter what their speed. Energy/mass equivalence: In a nuclear reaction, a loss in mass m between the products and the reactants implies that energy has been released, following: E= c2m, where Eis the energy released (J), c is the speed of light (c = 3 × 108 ms-1) and m is the change in mass (kg). A particle with a mass of 1 MeV/c2 would release 1 MeV = 106 eV = 1.6 × 10-13 J with E = mc2. A particle with a mass of 1 GeV/c2 would release 1 GeV = 109 eV = 1.6 × 10-10 J with E = mc2. Conservation laws: In a reaction, momentum, charge and mass/energy are always conserved. In a reaction, baryon and lepton numbers are conserved. Strangeness is conserved when the reaction is caused by a strong interaction (through the strong nuclear force). Antimatter Particles and antiparticles have the same mass, but opposite charge, lepton number, strangeness etc. When a particle and its antiparticle meet, they annihilate. This releases energy in the form of a pair of photons: each photon carries: E= mc2, where m is the mass (kg) of the annihilated particle. Similarly, a pair of particle and antiparticle can be created from a single high energy photon carrying: E= 2mc2. High energy beams are required to investigate the fine structure of atoms and particles because high speed may be needed to overcome electrostatic repulsion and more energy is available for creating new particles. Also, higher speed means shorter wavelength (λ = h/mv) and diffraction/scattering requires the wavelength λ to be approximately equal to the particle spacing/internal structure. THE STANDARD MODEL A fundamental particle is a particle which cannot be further divided. Fundamental particles are the building blocks of matter. There are two types of fundamental particles: quarks and leptons. Leptons are light fundamental particles which are not affected by the strong nuclear force. There are 12 leptons: electron (e−), muon (µ−), tau (τ−), their corresponding antiparticles, their corresponding neutrino and antineutrino. All leptons have a lepton number of 1 and all antileptons have a lepton number of −1. All neutrino and antineutrino have a charge of 0. Quarks are fundamental particles that are affected by the strong nuclear force. There are 6 quarks and their corresponding antiquarks: down, up, strange, charm, bottom and top. Quarks can change flavour (= type) through the weak interaction by emitting a lepton: this causes beta decay (n → p + e + νe ). Down, strange and bottom quarks have a charge of −1/3 e. Up, charm and top quarks have a charge of +2/3 e. All quarks have a baryon number of 1/3. The strange quark has a strangeness of -1, antistrange has a strangeness of 1 and all other quarks have a strangeness of 0. Hadrons are subatomic particles composed of quarks and antiquarks. Hadrons are either baryons or mesons. A baryon is a heavy subatomic particle made of 3 quarks (3 antiquarks for antibaryons). Their baryon number is 1 (−1 for antibaryons). Protons and neutrons are examples of baryons. A proton is made of one down quark and two up quarks (duu). A neutron is made of two down quarks and one up quark (ddu). A meson is a subatomic particle with a medium mass made of one quark and one antiquark. Its baryon number is 0. Pions are examples of mesons. If a particle moves close to the speed of light, time dilation takes place due to relativistic effects. This increases the expected lifetime of particles (their half-life) and was important to discover the third generation of fundamental particles such as the top quark as heavy particles quickly decay into lighter particles. A2: THERMAL PHYSICS Absolute zero of temperature: - Temperature at which the pressure / volume of a gas is zero. - Temperature at which the kinetic energy of the molecules is zero (gas molecules are stationary) The Kelvin scale is the absolute thermodynamic scale. 0K is the absolute zero (-273°C). The specific heat capacity is the energy required when a kg of the substance undergoes a temperature change of 1°C or 1K. It is used in: E = mcΔT, where E is the energy transferred (J), m is the mass of substance (kg), c is the subtance’s specific heat capacity (Jkg-1K-1) and ΔT is the change in temperature (K or °C). The specific latent heat of fusion is the energy transferred to change the state of 1 kg of a substance between solid and liquid without change in temperature. The specific latent heat of vaporisation is the energy transferred to change the state of 1 kg of a substance between liquid and gas without change in temperature. It is used in: E=ΔmL where E is the energy transferred (J), Δm is the mass of substance that changes state (kg) and L is the specific latent heat of fusion/vaporisation (Jkg-1). The internal energy is the sum of the potential and kinetic energies of all molecules / atoms (there is no gpe!) Ideal gases do not have forces between molecules so there is no atomic/molecular potential energy Charles’ law: The volume V of a fixed mass of ideal gas, kept at constant pressure, is proportional to its absolute temperature. V/T = constant for a constant mass of ideal gas at constant pressure, where V = volume of gas (m³) and T = temperature of gas in Kelvin (K) Boyle’s law: The pressure of a fixed mass of gas is inversely proportional to its volume if its temperature is kept constant. PV = constant for a constant mass of ideal gas at constant temperature, where P is the pressure of the gas (Pa or N m-2) and V is the volume of gas (m³). Pressure law: The pressure of a fixed mass of ideal gas kept at constant volume is proportional to its absolute temperature. P/T = constant for a constant mass of ideal gas at constant volume, where P is the gas pressure (Pa or N m-2) and T is the absolute temperature of the gas in Kelvin (K). Ideal gas law: PV = NkT for a constant mass of ideal gas, where P = pressure (Pa or N m-2), V = volume (m3), N = number of molecules, k = Boltzmann constant (1.38 10-23 J K-1), T = absolute temperature (K) Or PV = nRT for a constant mass of ideal gas where P = pressure (Pa or N m-2) , V = volume (m3), n = number of moles (mol), R = molar gas constant (8.31 J.mol-1K-1), T = absolute temperature (K) N = n A, with N = number of molecules, n = number of moles and A = Avogadro number = 6.02 1023 mol-1 You need to be able to derive PV = 1/3 Nm using the kinetic theory model (P = pressure (Pa or N m-2), V = volume (m3), N = number of molecules, is the mean square speed of a molecule). The important assumptions are: elastic collisions between molecules and container, no intermolecular forces, random motion with no preferred directions and the many random collisions give a smooth pressure. You need to be able to use PV = 1/3 Nm and PV = NkT to derive ½ m< v² > = 3/2 k T : where ½ m< v² > is the mean kinetic energy of a gas molecule, is the mean square speed of a molecule, k is Boltzmann’s constant and T is the absolute temperature of the ideal gas in Kelvin The r.m.s. speed of a molecule is √ A2: OSCILLATIONS Simple Harmonic Motion (SHM) happens when the resultant force (hence the acceleration) is proportional to the displacement (the distance from the equilibrium position). The resultant force (acceleration) is directed towards the equilibrium position, in opposite direction to the displacement. As an equation, SHM is defined by: F = – k × x, where F is the resultant force (N), k is a constant and x is the displacement (m), resultant force and displacement are in opposite directions (resulting in the minus sign). When a system follows SHM, a = −ω² x, where a is the acceleration (ms-2), x is the displacement (m) measured from the equilibrium position and ω is the angular frequency of the system (rad s-1). Note that the constant in the previous equation k is equal to k = mω2. ω = 2πf = 2π/T where f is the system’s frequency (Hz) and T is the system’s period (s). The equations of motion for SHM are: x = A cos(ωt) v = −Aω sin(ωt) a = −Aω2 cos(ωt) where x is the displacement (m) from the equilibrium position, A is the motion amplitude (m), ω is the angular frequency (rad s-1), v is the velocity (ms-1) and a is the acceleration (ms-2). Aω is the maximum speed and Aω2 is the maximum acceleration. The natural frequency (fundamental frequency) is the frequency of free (unforced) vibrations, i.e. the frequency when the system oscillates by itself / of its own accord. Forced oscillation happens when the system is being forced to oscillate/vibrate at the driver’s frequency. Resonance happens when the forcing frequency is equal to the natural frequency. During resonance, the system energy is increased by the work done by the driving force during each period, so there is maximum energy transfer. As a result, the amplitude of oscillations increases drastically. When the oscillation is damped, the amplitude of the vibrations is reduced but the frequency stays similar. During resonance, oscillation amplitude is large so energy is wasted at a higher rate by damping forces, until the rate at which energy is supplied by the driving force is equal to the rate at which energy is dissipated by damping forces. As a result, amplitude at resonance is greatly reduced by damping. A2: NUCLEAR DECAY Isotopes have the same proton number (= atomic number) but a different nucleon number (= mass number). Consequently, they have a different number of neutrons. Unstable nuclei [not atoms] are liable to break up and decay because the nucleus has either too much energy or too many nucleons. This will release ionising radiation. Alpha decay: Nucleus loses 2 neutrons and 2 protons. An alpha particle (Helium nucleus) is emitted from the nucleus. Beta decay: One neutron changes into a proton in the nucleus. A beta particle (electron) and an electron antineutrino ( ν e ) are emitted from the nucleus. (For beta + decay, one proton changes into one neutron, a beta + particle (positron) and an electron neutrino is emitted). Gamma decay: Nucleus is not altered; it loses energy in the form of gamma radiation. A gamma photon is emitted. Ionisation: an electron is removed from an atom or molecule by an alpha or beta particle. If the radiation is strongly ionising, it will have a short range because the ionising particle loses some kinetic energy for each ionisation. Nuclear decay is random: - It is impossible to identify which atom/nucleus will be the next to decay - It is impossible to predict when a given atom will decay - It is impossible to predict exactly how many atoms will decay in a set time But we can estimate the probability that a nucleus will decay in a given time Nuclear decay is spontaneous: - It happens independently of / cannot be controlled by chemical conditions (e.g. concentration) and physical conditions (e.g. temperature or pressure) - It happens without stimulation / trigger The half life of a substance is the average time taken for half the original number of nuclei to decay. It is also the average time taken for the activity (count rate) to drop by half. The activity of a source is measured in Becquerel or count per second. 1 Bq = 1 count per second. The binding energy is the energy required to separate a nucleus into nucleons. Background radiation: the ionising radiation surrounding us. It is caused by: radioactive rocks in the ground, food, radon gas, cosmic rays, medical uses, nuclear fall out or leaks (from nuclear power/industry or buried waste or nuclear weapons testing). It is measured using a Geiger Mueller tube and counter in the absence of radioactive source over a long period of time (a minimum of 5 minutes). The activity of a source is the number of count per second, it can be measured with a Geiger Mueller tube and counter by removing the background count rate from the measured count rate. It will follow an exponential decay: A = A0 e-λt where A is the source activity (Bq), A0 is the initial activity (Bq), t is time (s, min, h..) and λ is the decay constant (s-1, min-1, h-1, …). The activity of a source A (Bq) is linked to the number of radioactive nuclei N (no unit): A = - dN/dt. As a result, N also follows an exponential decay: N = N0 e-λt and A = λN, this last equation only works if the decay constant λ is in s-1. The half-life can be worked out as the time t1/2 at which half the nuclei have decayed, giving: t1/2 = ln 2/λ. Nuclear fusion happens when two light nuclei combine to form a single heavier nucleus, energy is released due to a mass loss during fusion (increase in binding energy per nucleon). A sustained fusion reaction needs high temperature to overcome electrostatic repulsion and high density for a large collision rate to sustain the reaction and to contain the fuel within the core (gravitational pull in a star). A fission reaction starts with a heavy nucleus (e.g. Uranium 235) which absorbs a neutron and becomes unstable. It splits into 2 smaller nuclei and more neutrons, releasing energy from mass loss (due to an increase in binding energy per nucleon) and creating a chain reaction (provided there is enough fuel). During fission, the binding energy per nucleon is increased so energy is released. A2: ASTROPHYSICS AND COSMOLOGY A light year is the distance travelled by light in one year. The luminosity of a star is its power; the intensity of a star is the observed power per unit area. A standard candle is an object of known luminosity. Examples of standard candles are Type 1a supernovae and Cepheid variable stars. The distance to standard candles can be worked out using the inverse square law: I = L/4r2 where I is the intensity of the star detected on Earth (W m-2), L is the star’s luminosity (W) and r is the distance between the star and Earth (m). Distances can also be found for objects close to our solar system using trigonometric parallax. You must be able to produce a diagram and the tan Θ equation. For very distant galaxies, the Doppler shift can be used in combination with Hubble’s law (see below). A black body absorbs and emits radiation at all wavelengths. Black body radiation is characterised by 2 laws: Wien’s law determines the colour of the emitted radiation: p × T = 2.90 × 10–3 m K, where T is the surface temperature of the black body in Kelvin and p is the peak wavelength (m). Stefan-Boltzmann law determines the luminosity (power) of the emitted radiation: Q/t = AT4 where Q/t is the energy per second radiated / luminosity (W), A is the surface area of the black body radiator (m²), T is the surface temperature in Kelvin and  = 5.7 × 10-8 W m-2 K-4 is Stefan’s constant. Hubble’s law The speed of recession of a distant galaxy is proportional to the distance to the galaxy. v = H × d, where v is the speed of recession (km s-1), d is the distance to the galaxy (Mpc) and H is Hubble’s constant (The accepted value of H is 65 km s-1 Mpc-1). Hubble’s law can be used to estimate the age of the universe: 1/H is the age of universe, provided the universe has been expanding at a constant rate since the Big Bang. The fate of the universe depends on the density of the universe: If the density is equal to the critical density, the gravitational pull is large enough to prevent the universe expanding for ever, but not too big to cause a collapse of the universe. Closed universe: If the density is above the critical density, the gravitational attraction is sufficient to cause the Big Crunch, i.e. to stop the expansion and cause a collapse of the universe. Open universe: If the density is lower than the critical density, the universe will continue expanding for ever. Cosmologists cannot estimate the density of the universe accurately as they can only use the light of stars and galaxies. We know that some matter is unobserved, for example neutrinos. The expansion of our ‘near universe’ is slower than predicted using the gravitational force of observable mass. This means a significant proportion of the universe’s mass is not observable, about 95%: this is dark matter. Doppler effect: The Doppler shift is the change in frequency of a wave when the source or the receiver is moving. When source or receiver move towards each other, the received frequency is higher than the emitted frequency (blue shift). When source or receiver move away from each other, the received frequency is lower than the emitted frequency (red shift). The Doppler shift is proportional to the relative speed between source and receiver. The change in wavelength (Doppler shift) is directly proportional to the relative velocity between source and observer: ∆λ/λ = v/c for an EM wave, where ∆λ is the observed change in wavelength, λ is the wavelength of the emitted EM wave, v is the relative speed between source and observer and c is the speed of light. This is also approximately true for frequencies: ∆f/f = v/c. The core of a main sequence star like our Sun is powered by the nuclear fusion of hydrogen: hydrogen nuclei / protons fuse into helium, thereby releasing energy. Heavier stars fuse at higher rates, resulting in higher luminosity/power and shorter lifetime, also reaching higher temperature and pressure. A brown dwarf is a star-like object that is composed of hydrogen but has insufficient mass to become a main sequence star. After all hydrogen in the core of a main sequence star has been fused, it becomes a giant or supergiant. Its core reaches a higher temperature and density and is powered by the fusion of helium into heavier elements. Its outer layers expand and cool down, resulting in a large size, low surface temperature and low average density. When a giant/supergiant has fused all elements in its core, it collapses due to gravity, sometimes shedding its outer layers in a planetary nebula, and becomes a white dwarf: a dead star, in which no fusion takes place in the core. A white star is small, dense and hot. It consists of products of fusion such as carbon / oxygen / nitrogen. It still emits light but with lower luminosity as it has no energy source. Over time, it cools down, becomes dimmer and changes colour. When no visible light is emitted, it (hypothetically) becomes a black dwarf. A white dwarf with a mass bigger than 1.4 times the solar mass (the mass of the original star would exceed 8 times the solar mass) is unstable and collapses / implodes suddenly in a supernova: electrons and protons combine in the core into neutrons which are absorbed by nuclei in the outer layers, before the imploding core creates a shock wave that throws off the outer layer. A supernova is detected on Earth as a star which suddenly appears very bright or luminous. The supernova remnant is called a neutron star. It is extremely dense; it rotates rapidly emitting beams of radio waves. It is also called a pulsar. If the mass of a neutron star exceeds 2.5 times the mass of the Sun (the mass of the original star would exceed 10 times the mass of the Sun), the supernova core continues to collapse and to become denser until not even light can escape from it: a black hole has been created. The gravitational force between 2 bodies is proportional to the product of both masses and inversely proportional to their separation squared: F = GMm/ r2 where G = 6.67 10-11 N m2 kg-2 is the gravitational constant, M and m are the masses of the 2 bodies and r is the distance between the 2 bodies. The gravitational field strength g (Nkg-1) is the gravitational force attracting a unit mass: g = F/m where F is the gravitational force attracting the object of mass m (kg). You need to be able to derive g = GM/r2 for the gravitational field strength of a point mass. An equipotential surface is a surface containing all points at the same (gravitational) potential energy. A body moving along an equipotential does not gain or lose GPE. For a uniform gravitational field (at the surface of the Earth where g = 9.81Nkg-1), equipotentials are surface of equal height (e.g. contour lines on a map), when an object of mass m (kg) change height, it will gain or lose GPE: ΔE = mgΔh, where ΔE is the gain/loss in GPE (J) and Δh is the change in height (m). For a radial field (e.g. around a planet), equipotentials are circles centred about the centre of mass and the potential is given by: VGRAV = - GM/r, where VGRAV is the gravitational potential (Nmkg-1), G is the gravitational constant (see above), M is the mass of the body creating the field (kg) and r is the distance to the body’s centre of mass. When an object of mass m (kg) moves between 2 equipotentials, it would gain or lose GPE = m(VGRAV_2 – VGRAV_1). PRACTICALS Accuracy is a measure of how close a measured or calculated value is to the true value. Precision is a measure of the consistency of values obtained by repeated measurements. An error is the difference between the measured result and the true value. An (absolute) uncertainty is the interval/range in which the value can be considered to lie. If repeated readings are taken, the absolute uncertainty is half the range of the measured values. If repeated readings are identical OR if no repeated readings are taken, the absolute uncertainty is half the resolution of the equipment. The absolute uncertainty has the same unit and number of decimal places as your measured or calculated value. percentage uncertainty = absolute uncertainty / mean value expressed as a percentage with 0 or 1 decimal places. When adding or subtracting measurements, absolute uncertainties are added to find the total (absolute) uncertainty. When multiplying or dividing measurements, percentage uncertainties are added to find the total (percentage) uncertainty. Same principle when using powers (such as squaring).