7408/2 · 2 hours · 85 marks · 34% of A-Level · Sections 6 – 8
Section 6.1 - Further mechanics: circular motion and SHM
An object moving in a circle at constant speed is accelerating because its direction continuously changes. The acceleration points towards the centre of the circle this is centripetal acceleration. Angles are measured in radians (2π rad = 360°). Note: the direction of angular velocity is not examined.
The centripetal force is not a new type of force it is provided by an existing force: tension (in a string), gravity (for orbiting satellites), friction (for a car cornering), or the normal contact force.
The centripetal force does no work because it is always perpendicular to the velocity. Therefore, an object in circular motion at constant speed has constant kinetic energy only direction changes, not speed.
SHM is defined by the condition: a ∝ −x (acceleration proportional to displacement, always directed towards equilibrium).
Graphical representations: the x–t graph is sinusoidal. The v–t graph is derived from the gradient of the x–t graph (v and x are 90° out of phase: v is maximum when x = 0 and vice versa). The a–t graph is derived from the gradient of the v–t graph (a is 180° out of phase with x). KE and PE both vary sinusoidally with time (at twice the frequency of oscillation); total energy is constant.
SHM systems:
Required Practical 7: investigation into simple harmonic motion using a mass–spring system and a simple pendulum.
At maximum displacement (x = A): velocity = 0, acceleration = maximum (= ω²A), PE = maximum, KE = 0. At equilibrium (x = 0): velocity = maximum, acceleration = 0, PE = 0, KE = maximum. Total energy is constant throughout.
Increased damping: lowers the resonance peak amplitude and broadens it; shifts the peak slightly below f0.
Resonance in stationary waves: resonance also occurs in mechanical systems producing stationary waves e.g. a vibrating string or air column resonates when the driving frequency matches a harmonic of the system's natural frequency. The same principles apply: maximum amplitude at resonance, damping reduces the sharpness of the peak.
Real-world examples of resonance: Tacoma Narrows bridge collapse (wind matching natural frequency); tuned strings and organ pipes (stationary wave resonance); microwave ovens (water molecule resonance). Damping is used to prevent unwanted resonance (car suspension, building earthquake protection).
Section 6.2 - Thermal physics
First law of thermodynamics (qualitative): the internal energy of a system is increased when energy is transferred to it by heating or when work is done on it (and vice versa). Both routes are equivalent in their effect on internal energy.
During a change of state, temperature is constant even though energy is being supplied or removed. The energy changes the potential energy (intermolecular bonds) not the kinetic energy of the particles.
The gas laws (Boyle's, Charles's, Pressure law) are experimental relationships between p, V, T and the mass of gas. The ideal gas equation combines them.
Assumptions of an ideal gas:
Required Practical 8: investigation of Boyle's law (constant temperature) and Charles's law (constant pressure) for a gas.
Always use temperature in Kelvin in gas law calculations. A common error is substituting degrees Celsius. Absolute zero is defined as the temperature at which an ideal gas would have zero volume/pressure the concept of absolute zero comes from extrapolating gas law graphs.
Brownian motion (e.g. smoke particles in air, pollen in water) provides direct evidence for the existence of atoms: the random, jerky motion of visible particles is caused by random collisions with invisible, fast-moving molecules.
The gas laws are empirical (derived from experiment). The kinetic theory model is theoretical, derived from applying Newton's laws and conservation of momentum to molecules bouncing off container walls. Together they explain why pV = ⅓Nmcrms².
For an ideal gas, internal energy consists entirely of the kinetic energy of the molecules (there are no intermolecular potential energies). Hence internal energy ∝ T.
Our understanding of gas behaviour has developed over time: from early observations of gas laws (empirical), to Maxwell and Boltzmann's statistical mechanics, building a deeper theoretical foundation that connected temperature to molecular motion.
The mean KE depends only on temperature, not on the type of gas. At the same temperature, all ideal gas molecules have the same mean translational KE regardless of their mass. Heavier molecules move slower (lower rms speed) to have the same KE.
Section 7 (part) - Gravitational and electric fields
A force field is a region in which a body experiences a non-contact force. Force fields can be represented as vectors (the direction must be determined by inspection of the field).
Force fields arise from:
A gravitational field is a region where a mass experiences a force. Field lines point towards the attracting mass (gravity is always attractive).
Equipotential surfaces are surfaces of constant potential. No work is done moving a mass along an equipotential surface. Equipotentials are perpendicular to field lines.
Field strength and potential: g = −ΔV/Δr (g = negative gradient of V–r graph). The change in potential ΔV between two points equals the area under the g–r graph between those points.
Orbits: for a circular orbit, gravitational force provides centripetal force:
Gravitational potential is always negative (zero at infinity). The closer to the mass, the more negative the potential. Work must be done to move a mass away from an attracting body. Escape velocity follows from setting KE = |GPE| at the surface.
An electric field is a region where a charged particle experiences a force. Field lines run from positive to negative charges.
Equipotential surfaces are surfaces of equal potential. No work is done moving a charge along an equipotential. Field lines are always perpendicular to equipotentials.
Trajectory in a uniform field: a charged particle entering a uniform electric field at right angles (e.g. between parallel plates) follows a parabolic path analogous to projectile motion (constant force perpendicular to initial velocity).
Comparison with gravity: same inverse-square-law form; both use potential concept and equipotentials. Key difference: gravity is always attractive, but electric force can attract or repel. The gravitational force between subatomic particles is negligible compared to the electrostatic force between them.
Electric potential is positive near a positive charge and negative near a negative charge. The closer to the charge, the larger the magnitude of the potential. Equipotentials are closer together where the field is stronger.
Section 7 (part) - Capacitance and magnetic fields
A capacitor stores electric charge and energy. It consists of two conducting plates separated by an insulator (dielectric).
Charge and discharge through a resistor:
Required Practical 9: investigation of the charge and discharge of capacitors. Analysis should include log-linear plotting (ln Q vs t) leading to determination of the time constant RC.
Energy stored = area under Q–V graph = ½QV. On a discharge Q–t graph, the gradient gives current (I = −dQ/dt). Plotting ln Q against t gives a straight line of gradient −1/RC, which is the key experimental method for finding RC.
A magnetic field exerts a force on moving charges and current-carrying conductors. The force is always perpendicular to both the velocity/current and the field direction.
Required Practical 10: investigate how the force on a wire varies with flux density, current, and length of wire using a top-pan balance.
A charged particle moving perpendicular to a uniform magnetic field follows a circular path (magnetic force = centripetal force):
The magnetic force does no work on a moving charge (always perpendicular to velocity), so speed and KE do not change only direction. This is why charged particles move in circles, not spirals, in a uniform magnetic field.
Applications: a straight conductor moving through a magnetic field has an EMF induced across it (Faraday's law flux linkage of the circuit changes). Simple dynamos and generators use a rotating coil.
Required Practical 11: investigate the effect on magnetic flux linkage of varying the angle between a search coil and a magnetic field direction, using a search coil and oscilloscope.
Alternating current (AC): sinusoidal waveforms only.
An oscilloscope can be used as a DC and AC voltmeter, to measure time intervals and frequencies, and to display AC waveforms. Familiarity with the controls (timebase, Y-gain) is expected; knowledge of the internal structure is not required.
Transformers:
Causes of inefficiency in real transformers:
Lenz's law is a consequence of energy conservation. Transmitting power at high voltage reduces current; since Ploss = I²R, even doubling the voltage quarters the resistive losses for the same power delivered.
Section 8 - Nuclear physics
Rutherford's alpha scattering experiment (1909–1911): alpha particles fired at thin gold foil. Results showed most passed straight through (nucleus is mostly empty space), some deflected slightly, and a very small number deflected by more than 90° proving the nucleus is small, dense, and positively charged. This replaced Thomson's “plum pudding” model, in which positive charge was spread throughout the atom.
Understanding of nuclear structure has continued to evolve: from Rutherford's nuclear model, to Chadwick's discovery of the neutron (1932), to the current quark model of nucleons.
| Radiation | Nature | Range in air | Stopped by | Ionisation | Hazard |
|---|---|---|---|---|---|
| Alpha (α) | 42He nucleus | ~5 cm | Paper / skin | Strongly ionising | Very dangerous if ingested/inhaled |
| Beta (β−) | Fast electron | ~1 m | ~3 mm Al | Moderately ionising | Skin burns; less hazardous internally than α |
| Gamma (γ) | EM radiation | Unlimited | Many cm Pb | Weakly ionising | Penetrates body; requires shielding |
Applications of radiation:
Safe handling: keep sources at distance (use tongs), minimise exposure time, use appropriate shielding (α: paper; β: aluminium; γ: lead/concrete), store in lead-lined containers, use dosimeters.
Inverse-square law for γ radiation:
Background radiation is radiation from natural and artificial sources present at all times. Sources include: cosmic rays, radon gas (from rocks/soil), building materials, food, medical X-rays, nuclear industry. It must be subtracted from measured count rates to find the count rate due to the source alone.
Required Practical 12: investigation of the inverse-square law for gamma radiation.
Alpha particles are most ionising because of their large charge (+2) and slow speed. Gamma is least ionising but most penetrating. A magnetic field deflects α and β in opposite directions (by Fleming's left-hand rule) but not γ (uncharged). Always subtract background count rate before calculations.
Radioactive decay is a random and spontaneous process: it cannot be predicted when a particular nucleus will decay, and it is not affected by external conditions (temperature, pressure, chemical state). Each nucleus has a constant probability of decaying per unit time (λ, the decay constant).
The half-life is the average time for the number of undecayed nuclei (or the activity) to halve. It is constant for a given nuclide and independent of the initial amount. Half-life can be determined from a decay curve by reading off when the count rate halves, or from a ln A vs t graph (gradient = −λ, so T½ = ln2/λ). Modelling with constant decay probability (e.g. using dice or a computer simulation) demonstrates the random nature of decay.
Applications: radioactive dating (e.g. carbon-14 for organic materials, uranium-lead for rocks) uses known half-lives to estimate age. Long half-lives make certain nuclides a challenge for radioactive waste storage (must remain isolated for thousands of years).
Nuclear instability decay modes:
N–Z stability graph: stable nuclei lie on a “valley of stability”. For light nuclei (Z ≤ 20), N ≈ Z. For heavier nuclei, N > Z (extra neutrons needed). Nuclei above the line (β− emitters) or below it (β+ emitters / electron capture) decay towards the stable valley.
Nuclear energy levels and γ emission: after α or β decay the daughter nucleus may be in an excited state. It de-excites by emitting a γ photon (no change in A or Z). Example: technetium-99m (Tc-99m) is a metastable excited state that emits only γ rays widely used in medical diagnosis (short half-life ~6 h, no α/β emission minimises patient dose).
On an A–t or N–t graph, the quantity falls by half in each successive half-life. From a ln A vs t graph: gradient = −λ, y-intercept = ln A0. Always subtract background before plotting decay data.
Nuclear radius two experimental methods:
E = mc² applies to all energy changes not just nuclear reactions. Any system that loses energy loses mass (though the mass change is undetectable in everyday reactions).
Binding energy per nucleon graph: peaks around A = 56 (iron-56), the most stable nucleus. Fission (heavy nuclei, A > ~100, split into two medium-mass nuclei) and fusion (light nuclei combine) both move product nuclei towards the peak increasing BE/nucleon and releasing energy. Students should be able to identify fission and fusion regions on the graph.
The physics of nuclear energy allows society to use science to inform decision-making about energy policy, weighing benefits (low CO&sub2;, large energy output) against risks (waste, accident risk, proliferation).
Binding energy is the energy RELEASED when a nucleus forms (or equivalently, the energy NEEDED to separate it). Higher BE/nucleon = more stable. In fission/fusion calculations, find the total mass before and after; Δm × c² = energy released.
Induced fission: a slow (thermal) neutron is absorbed by a fissile nucleus (e.g. U-235), making it unstable. It splits into two daughter nuclei, releasing typically 2–3 fast neutrons and a large amount of energy.
A chain reaction occurs when released neutrons cause further fissions. Critical mass is the minimum mass of fissile material needed to sustain a chain reaction (at least one neutron per fission causes another). In a reactor, control rods absorb excess neutrons to keep the reaction at the critical condition (exactly one neutron per fission continues the reaction).
Moderation (slowing neutrons): fast neutrons produced by fission must be slowed to thermal energies by a moderator. The mechanism is elastic collisions with moderator nuclei the neutron transfers more kinetic energy when it collides with a nucleus of similar mass (most efficient when masses are equal, e.g. hydrogen).
Safety aspects:
Appreciation of the balance between risks and benefits in the development of nuclear power: large energy output with low CO&sub2;; risks include accidents, long-lived waste, and proliferation concerns.
Nuclear fusion: two light nuclei combine to form a heavier nucleus. The product has a higher binding energy per nucleon than the reactants, so energy is released. Both fission (of heavy nuclei) and fusion (of light nuclei) move towards the peak of the BE/nucleon curve at iron-56.
In the reactor: the moderator slows neutrons; the control rods absorb neutrons; the coolant carries heat to the steam generator. Know which component does which function. For fusion: high temperature overcomes Coulomb repulsion; high pressure ensures sufficient collision rate.