This mini-lesson covers CCEA Unit A2 1: deformation of solids (Hooke's law, the Young modulus, strain energy), thermal physics (gas laws, pV = nRT, kinetic theory, Q = mcΔθ), uniform circular motion, simple harmonic motion (resonance and damping), and atomic and nuclear physics (decay, half-life, E = Δmc², binding energy, fission and fusion).
Big picture: this unit takes the mechanics of AS 1 and pushes it two ways — outwards into rotation and oscillation, and inwards into the microscopic world of molecules and nuclei. The same conservation laws hold throughout.
Press Start when you are ready.
4.1 Deformation of solids
Hooke's law, the Young modulus and strain energy
Hooke's law: up to the limit of proportionality, the extension of a spring or wire is directly proportional to the load:
F = kxk is the stiffness (force per unit extension), in N m⁻¹.
To describe the material rather than the sample, use stress and strain:
stress σ = F / A · strain ε = x / L · E = σ / εThe Young modulus E is measured in pascals (Pa). Strain has no units.
Limit of proportionality — the point where the F–x graph stops being a straight line.
Elastic limit — just beyond it; the point past which the material will not return to its original length.
Elastic deformation returns; plastic deformation is permanent.
strain energy E = ½Fx = ½kx²= the area under the force–extension graph.
Measuring E (4.1.5): a long, thin wire is used so the extension is measurable. Find the diameter with a micrometer at several points and average (A = πd²/4 — remember the % uncertainty in A is twice that in d). Plot F against x; gradient × L / A gives E.
Calculate
Your turn — Young modulus
1A wire of length 2.00 m and diameter 0.50 mm stretches by 2.0 mm when a force of 40 N is applied. Calculate the Young modulus, in GPa.
GPa
Hint: A = π(0.25 × 10⁻³)² = 1.96 × 10⁻⁷ m². stress = 40/1.96 × 10⁻⁷ = 2.04 × 10⁸ Pa. strain = 2.0 × 10⁻³ / 2.00 = 1.0 × 10⁻³. E = stress ÷ strain, then ÷ 10⁹ for GPa.
Quick check
How far can you pull it?
?A wire is loaded past its elastic limit and then the load is removed. What happens?
4.2 Thermal physics
Gas laws, the ideal gas equation and kinetic theory
Experiments on a fixed mass of gas give: pV = constant at constant T (Boyle), p/T = constant at constant V, and V/T = constant at constant p — which combine to pV/T = constant, and hence
pV = nRT · pV = NkTn = number of moles, R = 8.31 J K⁻¹ mol⁻¹; N = number of molecules, k = 1.38 × 10⁻²³ J K⁻¹. T must be in kelvin: T(K) = θ(°C) + 273.
Kinetic theory models a gas as many tiny molecules in random motion, colliding elastically with the walls. Their momentum changes give rise to the pressure:
pV = ⅓ N m ⟨c²⟩Combined with pV = NkT this gives ½m⟨c²⟩ = (3/2)kT: the mean kinetic energy of a molecule is proportional to the absolute temperature.
Internal energy is the sum of the randomly distributed kinetic and potential energies of the molecules. At absolute zero (0 K) the internal energy is a minimum. Heating a solid or liquid without a change of state:
Q = mcΔθc = specific heat capacity (J kg⁻¹ K⁻¹). Electrical method: Q = VIt, so c = VIt / (mΔθ).
Calculate
Your turn — ideal gas
20.50 mol of an ideal gas occupies 0.010 m³ at a temperature of 300 K. Calculate the pressure, in kPa. (R = 8.31 J K⁻¹ mol⁻¹)
kPa
Hint: p = nRT/V = (0.50 × 8.31 × 300) ÷ 0.010 = 1246.5 ÷ 0.010 Pa. Then divide by 1000 for kPa.
Calculate
Your turn — specific heat capacity
3How much energy is needed to raise the temperature of 0.50 kg of water by 20 K? Give your answer in kJ. (c = 4200 J kg⁻¹ K⁻¹)
kJ
Hint: Q = mcΔθ = 0.50 × 4200 × 20 = 42 000 J.
Quick check
Hotter gas
?The absolute temperature of an ideal gas is doubled at constant volume. What happens to the mean kinetic energy of its molecules?
Sort it
Alpha, beta or gamma?
Tap a property, then tap the radiation it belongs to.
☢️ Alpha
⚡ Beta-minus
〰️ Gamma
4.3 Circular motion
Uniform circular motion
An object moving in a circle at constant speed is still accelerating — its velocity (a vector) is constantly changing direction. The acceleration points towards the centre of the circle.
ω = 2π / T = 2πf · v = rωω is the angular velocity in rad s⁻¹.
a = v² / r = ω²r · F = mv² / r = mω²rThe centripetal force is not a new force — it is the resultant of the real forces (tension, gravity, friction, normal contact) acting towards the centre.
Common error: there is no outward "centrifugal force" on the object. The object is pushed/pulled inwards; it is your frame of reference that is accelerating. Remove the inward force (cut the string) and the object flies off along a tangent, not radially outwards.
Calculate
Your turn — centripetal force
4A ball of mass 0.20 kg is whirled on a string of radius 0.80 m at a constant speed of 4.0 m s⁻¹. Calculate the centripetal force.
Simple harmonic motion is motion in which the acceleration is proportional to the displacement from a fixed point and always directed towards that point:
a = −ω²xThe minus sign is the whole point: acceleration is always opposite to displacement (a restoring force).
x = A cos ωt · vmax = ωA · amax = ω²ADisplacement–time is a cosine (starting from maximum displacement); velocity is the gradient of that graph, and is maximum as the object passes through the centre, where a = 0.
Two standard systems: the simple pendulum, T = 2π√(l/g), and the loaded spiral spring, T = 2π√(m/k). Note that the period does not depend on the amplitude.
Free vibration — the system oscillates at its natural frequency.
Forced vibration — a periodic driver imposes its own frequency.
Resonance — the driving frequency equals the natural frequency; the amplitude becomes very large as energy is transferred most efficiently.
Damping — dissipative forces remove energy. Light damping: amplitude decays slowly. Critical damping: returns to equilibrium in the shortest time without oscillating (car suspension, door closers). Over-damping: returns slowly, without oscillating.
Damping and resonance together: increasing the damping makes the resonance peak lower and broader, and shifts it very slightly to a lower frequency. That is exactly why bridges and buildings are damped.
Calculate
Your turn — SHM
5An object oscillates with simple harmonic motion of amplitude 0.050 m and frequency 2.0 Hz. Calculate its maximum speed.
?A vibrating system is given heavier damping. How does its resonance curve change?
4.5–4.6 The nucleus & radioactive decay
Nuclear decay, activity and half-life
Rutherford's alpha scattering: most alpha particles passed straight through the gold foil (the atom is mostly empty space), but a tiny fraction were deflected through very large angles — evidence for a small, dense, positively charged nucleus.
Radioactive decay is random (you cannot predict which nucleus decays next) but exponential on a large scale, because the rate depends on how many undecayed nuclei remain:
A = λN · A = A₀e−λtλ = the decay constant: the probability per second that a given nucleus decays. Activity A is in becquerels (Bq).
λ T½ = ln 2Half-life T½ = the time for the number of undecayed nuclei (or the activity) to fall to half its initial value.
Radiation types:alpha (helium nucleus) — most ionising, stopped by paper, range a few cm of air. Beta (fast electron) — stopped by a few mm of aluminium. Gamma (high-energy photon) — least ionising, reduced (never fully stopped) by several cm of lead. Alpha: A drops by 4, Z by 2. Beta-minus: A unchanged, Z increases by 1.
Calculate
Your turn — radioactive decay
6A source has an activity of 480 Bq and a half-life of 8.0 days. Calculate its activity after 20 days.
Bq
Hint: 20 days = 2.5 half-lives. A = A₀ × (½)^2.5 = 480 ÷ 2^2.5 = 480 ÷ 5.657.
4.7–4.8 Nuclear energy
Binding energy, fission and fusion
Mass and energy are equivalent:
E = Δmc²The mass defect Δm is the difference between the mass of a nucleus and the total mass of its separate nucleons. The energy equivalent of that defect is the binding energy — the energy needed to pull the nucleus apart.
Plot binding energy per nucleon against nucleon number and you get a curve that rises steeply to a maximum around iron-56 (about 8.8 MeV per nucleon) and then falls slowly. The higher the binding energy per nucleon, the more stable the nucleus.
Fusion — join light nuclei (left of the peak). Products are higher up the curve, so energy is released.
Fission — split a heavy nucleus (right of the peak). The fragments are higher up the curve, so energy is released.
Fission reactor: a neutron splits U-235, releasing more neutrons → a chain reaction. Moderators (e.g. graphite, water) slow the neutrons so they can be captured; control rods (boron) absorb neutrons to keep the reaction just critical; a coolant removes the heat; shielding (thick concrete) protects the workers. The fuel must exceed a critical size or too many neutrons escape.
ITER / tokamak fusion: deuterium–tritium fuel, heated to over 10⁸ K to form a plasma, confined magnetically inside a vacuum vessel (gravitational and inertial confinement are the other two methods). Fusion offers abundant fuel and no long-lived high-level waste, but sustaining the plasma long enough remains extremely difficult.
Quick check
Reading the binding energy curve
?Why does fusing two very light nuclei release energy?
Quick check
Inside the reactor
?What is the job of the moderator in a thermal fission reactor?
Match it
Match each equation to its topic
Tap an item on the left, then its partner on the right.
Equation
Topic
Recap
Unit A2 1 — the big ideas
Deformation: F = kx; σ = F/A, ε = x/L, E = σ/ε; strain energy = ½Fx = area under F–x