Forces, energy, fields, waves and modern physical models
mechanics
What this domain tests
This mechanics review develops kinematics, Newton's laws, momentum, rigid-body equilibrium, circular motion, oscillations and gravitational ideas. State sign conventions, resolve vectors and justify the choice of force, energy or momentum methods. These are shared foundations rather than a claim about any current regional paper.
Kinematics and projectiles
Use the constant-acceleration equations only when acceleration is uniform. For a projectile, resolve the launch velocity into u cosθ (horizontal, constant) and u sinθ (vertical, changing at g). Time of flight comes from the vertical motion; the range comes from horizontal displacement over that time.
Forces, momentum and equilibrium
- Newton's second law uses the resultant force: F = ma. Weight W = mg acts at the centre of gravity.
- Total momentum p = mv is conserved separately in each direction when the net external impulse on the chosen system is zero; elastic collisions also conserve kinetic energy, inelastic ones do not.
- A body in equilibrium has ΣF = 0 and Σmoments about any point = 0. Take moments about the point with the most unknown forces so those unknowns vanish.
- For circular motion the radial resultant force is mv²/r = mω²r toward the centre. Speed is constant only in uniform circular motion; nonuniform motion also has tangential acceleration.
- SHM is defined by a = −ω²x, with T = 2π/ω; energy shuttles between kinetic and potential.
Worked example
In a uniform-gravity model with no air resistance, a ball leaves level ground at 25 m/s at 40° above the horizontal and lands at the same height. Find its range (g = 9.81 m/s²).
Vertical component: 25 sin40° = 16.07 m/s. Time to the top: 16.07/9.81 = 1.638 s, so total flight = 3.276 s. Horizontal component: 25 cos40° = 19.15 m/s. Range = 19.15 × 3.276 = 62.7 m. Cross-check with R = u² sin2θ/g = (625 × sin80°)/9.81 = 62.7 m. ✓
Traps that cost marks
- Swapping sin and cos: the component next to the angle uses cosine.
- Forgetting that at maximum height the vertical velocity is zero while the horizontal velocity is unchanged.
- Mixing g = 10 with data intended for 9.81, or dropping units mid-calculation.
- Sign errors - fix a positive direction and apply it to displacement, velocity and acceleration consistently.
- Calling circular motion "constant velocity", or omitting the direction of the centripetal force.
- Ignoring the mass of a rod or pulley, or treating a rough surface as smooth when friction is asked for.
energy
Energy methods begin with a system boundary and a list of interactions crossing that boundary. They can avoid tracking every intermediate acceleration, but they do not remove the need to identify forces, displacements and the assumptions of the model. Mechanical energy is not automatically conserved just because the word energy appears in the question.
For a constant force acting through a displacement, work is the scalar product of force and displacement: W = F s cos(theta). The angle belongs to the force and displacement vectors, not an arbitrary incline drawing. A perpendicular force does no work through that displacement. When the relevant force component varies with position, work is the signed area under its force-displacement graph, or the corresponding integral. Negative work means that interaction removes mechanical energy from the chosen moving system.
For a constant-mass particle in the ordinary nonrelativistic model, kinetic energy is one half mass times speed squared. The net work by all forces equals the change in this kinetic energy. If using only the work of an applied force, separately account for gravity, friction or other forces; applied work need not equal the entire kinetic-energy change. Near a planet's surface, a uniform-gravity approximation gives a gravitational potential-energy change m g delta-h. This approximation must not replace the inverse-square potential in a large orbital-distance problem.
For example, a hypothetical 2.0 kg object falls from rest through 3.0 m with g = 9.8 m/s squared, while air resistance is neglected. Its gravitational potential energy decreases by 58.8 J. Its kinetic energy increases by that amount, giving speed sqrt(2 times 58.8 divided by 2.0), approximately 7.67 m/s. The mass cancels from the corresponding ideal speed formula. If resistance dissipates energy, the same height drop produces a smaller speed and a thermal-energy transfer, rather than a failure of overall energy conservation.
An ideal spring stores one half k x squared relative to its relaxed state in the specified linear range. Friction, deformation and electrical resistance can transfer organized mechanical energy into internal energy. For an isolated total system, all relevant forms remain in the accounting even when mechanical energy alone decreases.
Average power is transferred energy or work divided by elapsed time. Instantaneous mechanical power is the scalar product of force and velocity. Efficiency in an ordinary energy-conversion calculation is useful output energy divided by supplied input energy over the same process. State which output is useful and which time interval is used. A heat pump's coefficient of performance uses a different definition and must not be confused with this efficiency ratio.
waves-optics
What this domain tests
This domain covers wave properties, superposition, stationary waves, interference, diffraction, refraction, lenses and optical instruments. Practice relates path difference to phase, uses grating and lens equations consistently, and explains fringe patterns through superposition; the chapter does not reproduce a particular examination format.
Core wave relations
- v = fλ; intensity is proportional to amplitude²; polarisation is possible only for transverse waves.
- Phase difference = 2π × (path difference)/λ.
- Superposition: displacements add vectorially. Two identical waves travelling in opposite directions give a stationary wave with nodes every λ/2 and adjacent node–antinode spacing λ/4. No net energy is transported.
- Young's double slit: fringe spacing x = λD/a, where a is the slit separation and D the slit-to-screen distance.
- Diffraction grating: d sinθ = nλ, with d = 1/N and N the number of lines per metre.
Refraction and lenses
Snell's law n₁ sinθ₁ = n₂ sinθ₂; refractive index n = c/v; critical angle sinθc = n₂/n₁ for n₁ > n₂. The special form 1/n applies when the second medium has index approximately one. For lenses use 1/f = 1/u + 1/v with a consistent real-is-positive convention, and signed transverse magnification m = -v/u = image height/object height. For ordinary thin glass lenses in air, a convex lens converges; a concave lens gives a virtual diminished image for a real object.
Worked example
A grating has 500 lines per mm. Find the angle of the second-order maximum for light of wavelength 589 nm.
d = 1/(500 × 1000) = 2.0 × 10⁻⁶ m. sinθ = nλ/d = (2 × 589 × 10⁻⁹)/(2.0 × 10⁻⁶) = 0.589. θ = 36.1°.
Traps that cost marks
- Using degrees inside small-angle formulae that require radians.
- Confusing the order n in d sinθ = nλ with a refractive index.
- Mixing sign conventions in 1/f = 1/u + 1/v, then reporting a virtual image as real.
- Claiming a stationary wave transfers energy along the string.
- Confusing path difference with phase difference (a factor of 2π/λ).
- Forgetting that sinθ ≤ 1 sets a maximum observable order.
modern-physics
What this domain tests
The photoelectric effect, photon energy, atomic energy levels and spectra, wave–particle duality, nuclear binding energy, radioactive decay and half-life. Questions test whether you keep energies in the correct unit (J or eV) and whether you can write balanced nuclear equations.
Photons and the photoelectric effect
- Photon energy: E = hf = hc/λ.
- Einstein's equation: hf = φ + KEmax, where φ is the work function and KEmax = eVs (Vs is the stopping potential).
- Threshold frequency f₀ = φ/h; below it, ordinary single-photon emission does not occur in the specified weak-illumination model.
- Increasing intensity increases the number of photoelectrons per second, not their maximum kinetic energy.
- Ordinary photoemission does not exhibit the long energy-accumulation delay predicted by the elementary classical picture; this supports the photon description.
Energy levels and nuclear physics
- Spectral lines arise from hf = E₂ − E₁; line spectra are evidence for discrete levels.
- de Broglie wavelength: λ = h/p; for a nonrelativistic massive particle, p = mv.
- Mass defect Δm gives binding energy E = Δmc²; 1 u c² is approximately 931.5 MeV.
- α decay: A falls by 4, Z falls by 2. β⁻ decay: Z rises by 1, A unchanged. γ emission changes neither A nor Z.
- Decay law: N = N₀e^(−λt); half-life t½ = ln2/λ; activity A = λN.
Worked example
Light of wavelength 400 nm falls on sodium (work function 2.28 eV). Find the maximum kinetic energy of the emitted electrons.
Photon energy: E = hc/λ = (6.63 × 10⁻³⁴ × 3.00 × 10⁸)/(4.00 × 10⁻⁷) = 4.97 × 10⁻¹⁹ J = 3.11 eV. KEmax = 3.11 − 2.28 = 0.83 eV (= 1.3 × 10⁻¹⁹ J). Since 0.83 eV > 0, emission occurs; the stopping potential is 0.83 V.
Traps that cost marks
- Claiming brighter light gives faster photoelectrons.
- Mixing eV and J in the same calculation - convert once, then stay consistent.
- Using the count of half-lives incorrectly (20 days with a 5-day half-life is four half-lives, not twenty).
- Changing the wrong number in a nuclear equation: mass number and proton number must both balance.
- Using p = mv outside its nonrelativistic regime, or confusing de Broglie momentum with the separate kinetic-energy expression mv²/2.
- Forgetting that γ emission accompanies α or β decay and does not change A or Z.