Physics unit 3 and 4 Theory

Projectile motion

Motion under gravity only. Horizontal and vertical are independent. Horizontal: constant velocity (a = 0). Vertical: constant acceleration g = 9.8 m s⁻² downward.

Projectile at maximum height

Vertical velocity = 0, horizontal velocity unchanged. Use v = u + at, v² = u² + 2as, s = ut + ½at² for the vertical part.

Range of a projectile

Horizontal distance = horizontal velocity × total time of flight (x = vₓt).

Newton's first law

An object stays at rest or at constant velocity unless a net external force acts on it.

Newton's second law

Fnet = ma. Acceleration is proportional to net force and inversely proportional to mass.

Newton's third law

Forces come in equal and opposite pairs that act on different objects.

Forces on an inclined plane

Weight components: parallel to slope = mg sinθ, perpendicular to slope = mg cosθ. Normal force N = mg cosθ (if no other vertical forces).

Friction on a slope

f = μN opposes motion. Net force along slope = mg sinθ − f.

Centripetal force

The net force toward the centre of a circular path. It is not a separate force, it is provided by tension, gravity, friction, normal force, etc.

Centripetal acceleration

a = v²/r = 4π²r/T². Always directed toward the centre. Changes direction of velocity, not speed.

Centripetal force formula

Fc = mv²/r = 4π²mr/T²

Banked curve (no friction)

Horizontal component of normal force provides centripetal force. tanθ = v²/(rg).

Vertical circle motion

Net force toward centre = difference between weight and normal/tension. At the top: Fnet = mg + N (or T). At the bottom: Fnet = N − mg.

Newton's law of universal gravitation

F = Gm₁m₂/r². Every mass attracts every other mass. Force is proportional to the product of the masses and inversely proportional to the distance squared. G = 6.67 × 10⁻¹¹ N m² kg⁻².

Gravitational field strength

g = F/m = GM/r². Force per unit mass, in N kg⁻¹. Direction is toward the mass creating the field.

Weight

Gravitational force on a mass, W = mg. Varies with location.

Orbital speed (circular orbit)

v = √(GM/r). Independent of the satellite's mass.

Orbital period

T = 2πr/v. Gravity provides the centripetal force, so GM/r² = 4π²r/T².

Kepler's third law

r³/T² = GM/4π² = constant for all objects orbiting the same central mass.

Geostationary satellite

Orbits above the equator, T = 24 h, same direction as Earth's rotation, so it stays above one point on Earth.

Gravitational potential energy (general)

Ep = −GMm/r. Zero at infinity, negative closer in. Use ΔEp = area under the F–r graph for changes.

Gravitational potential energy (near surface)

Ep = mgh. Only valid when g is approximately constant.

Total energy of a satellite

Etotal = Ek + Ep = −GMm/2r. Ek = GMm/2r, Ep = −GMm/r. Negative means it is bound.

Escape velocity

Minimum speed to reach infinity with zero speed left over. Ek + Ep = 0, so v = √(2GM/r).

Weightlessness (apparent)

Occurs in free fall, such as astronauts in orbit. Gravity still acts but there is no normal force.

Electric field

Region where a charge experiences a force. Direction is the direction a positive test charge would move. Field lines go from + to −.

Electric field strength

E = F/q (N C⁻¹ or V m⁻¹). Force per unit positive charge.

Electric field of a point charge

E = kQ/r². k = 8.99 × 10⁹ N m² C⁻². Field decreases with the square of distance.

Coulomb's law

F = kq₁q₂/r². Like charges repel, unlike attract.

Uniform electric field between parallel plates

E = V/d. Field lines are parallel and evenly spaced. Direction is from the positive to the negative plate.

Work done on a charge in an electric field

W = qΔV = qEd. Energy gained by a charge moved through a potential difference.

Charged particle in a uniform E field

Constant force F = qE gives constant acceleration a = qE/m. Path is parabolic if it enters perpendicular to the field.

Electron-volt (eV)

Energy gained by one electron crossing 1 V. 1 eV = 1.60 × 10⁻¹⁹ J.

Magnetic field

Region where magnetic materials, moving charges or current-carrying wires experience a force. Field lines go from N to S outside a magnet.

Magnetic field direction around a straight wire

Right-hand grip rule. Thumb is the conventional current, fingers curl in the direction of the field.

Magnetic field in a solenoid

Uniform inside, similar to a bar magnet outside. Direction by right-hand grip rule (fingers = current, thumb = north end). B = μ₀nI.

Force on a moving charge in a magnetic field

F = qvB sinθ. Force is perpendicular to both v and B, so it does no work and the speed stays constant.

Direction of magnetic force

Right-hand palm rule: fingers = B, thumb = v (positive charge), palm = force. Reverse the direction for negative charges.

Circular path in a magnetic field

Magnetic force provides centripetal force: qvB = mv²/r, so r = mv/qB.

Force on a current-carrying conductor

F = BIl sinθ. Direction by the right-hand palm rule with the thumb as conventional current.

Magnetic force between parallel wires

Currents in the same direction attract, opposite directions repel.

DC motor

Current-carrying coil in a magnetic field experiences opposing forces on its sides, producing torque. A split-ring commutator reverses current every half turn to keep rotation in one direction.

Magnetic flux

Φ = BA cosθ (Wb). Amount of magnetic field passing through an area. θ is the angle between B and the normal to the area.

Electromagnetic induction

A changing magnetic flux through a conductor induces an emf (and a current if the circuit is closed).

Faraday's law

emf = −N ΔΦ/Δt. Induced emf is proportional to the rate of change of flux and the number of turns.

Lenz's law

The induced current opposes the change in flux that caused it. Explains the negative sign in Faraday's law and follows conservation of energy.

Ways to change flux

Change B, change area A, or change angle θ. Moving a magnet, rotating a coil or changing coil area all induce an emf.

Emf induced in a moving conductor

emf = Blv for a conductor of length l moving at speed v perpendicular to B.

AC generator

Coil rotating in a magnetic field. Flux changes continuously, producing a sinusoidal emf. Slip rings keep the output alternating (a commutator gives DC).

Maximum emf of a generator

Occurs when the coil is parallel to B (flux is zero but changing fastest). Emf is zero when the coil is perpendicular to B (flux is a maximum).

Transformer

Uses changing flux in a soft iron core to change AC voltage. Vp/Vs = Np/Ns. Ideal transformer: VpIp = VsIs.

Step-up and step-down transformer

Step-up: Ns > Np, voltage increases and current decreases. Step-down: Ns < Np, voltage decreases and current increases.

Why transformers need AC

Only a changing current produces changing flux in the core, which is required to induce an emf in the secondary coil.

Transformer energy losses

Resistive heating in coils (I²R), eddy currents in the core, hysteresis, flux leakage. Reduced by laminated cores, thick wire and soft iron.

Power transmission at high voltage

P = VI, so a higher V means a lower I for the same power. Power loss = I²R falls greatly, so less energy is wasted as heat.

Eddy currents

Circulating currents induced in a conductor by changing flux. They oppose the change (Lenz) and cause heating and magnetic braking.

RMS values (AC)

Vrms = Vpeak/√2, Irms = Ipeak/√2. The equivalent DC value that gives the same heating effect.

Electromagnetic wave

Transverse wave of oscillating electric and magnetic fields at right angles to each other and to the direction of travel. Travels at c = 3.00 × 10⁸ m s⁻¹ in a vacuum.

Origin of EM waves

Produced by accelerating charges. A changing E field produces a B field and vice versa (Maxwell).

Wave equation

v = fλ, so c = fλ for EM waves.

Electromagnetic spectrum order

Radio, microwave, infrared, visible, ultraviolet, X-ray, gamma. Frequency and energy increase, wavelength decreases.

Special relativity: postulate 1

The laws of physics are the same in all inertial reference frames.

Special relativity: postulate 2

The speed of light in a vacuum is the same for all observers, regardless of the motion of the source or observer.

Inertial reference frame

A frame that is at rest or moving at constant velocity (not accelerating).

Proper time

Time interval measured by an observer who sees both events at the same place (in the rest frame of the clock). It is the shortest time.

Time dilation

Moving clocks run slow as seen by an observer in a different frame. t = t₀/√(1 − v²/c²) = γt₀.

Proper length

Length measured in the frame where the object is at rest. It is the longest length.

Length contraction

Length is shortened in the direction of motion as seen by an observer in a different frame. L = L₀√(1 − v²/c²) = L₀/γ.

Lorentz factor

γ = 1/√(1 − v²/c²). γ = 1 at rest, increases without limit as v approaches c.

Relativistic momentum

p = γm₀v. Momentum increases without limit as v approaches c, so massive objects cannot reach c.

Mass–energy equivalence

E = mc². Mass is a form of energy. Rest energy E₀ = m₀c².

Total relativistic energy

E = γm₀c². Kinetic energy = (γ − 1)m₀c².

Evidence for time dilation

Muons made in the upper atmosphere reach the surface even though their half-life is too short without dilation. Also atomic clock experiments on aircraft.

Muon decay explanation

In Earth's frame the muon's clock runs slow (time dilation). In the muon's frame the atmosphere is length-contracted. Both explain the same observation.

Photoelectric effect

Emission of electrons from a metal surface when light of sufficient frequency shines on it.

Photoelectric effect: key observations

Electrons only emitted above a threshold frequency, no time delay, kinetic energy depends on frequency (not intensity), current depends on intensity.

Why the wave model fails

It predicts any frequency would work if intense enough, that there would be a delay, and that Ek would depend on intensity. None of these are observed.

Photon

A quantum (packet) of EM energy. E = hf = hc/λ. h = 6.63 × 10⁻³⁴ J s.

Work function

Minimum energy needed to remove an electron from a metal surface (φ or W). φ = hf₀, where f₀ is the threshold frequency.

Photoelectric equation

Ek max = hf − φ. Maximum kinetic energy of an emitted electron.

Stopping voltage

Voltage needed to stop the fastest photoelectrons. eVs = Ek max.

Photoelectric graph (Ek vs f)

Straight line, gradient = h, x-intercept = threshold frequency, y-intercept = −φ. Same gradient for all metals.

Intensity vs frequency (photoelectric)

Intensity sets the number of photons per second, so it sets the photocurrent. Frequency sets the energy per photon, so it sets Ek max.

de Broglie wavelength

λ = h/p = h/mv. All matter has wave properties. The wavelength is only noticeable for very small masses.

Evidence for matter waves

Electron diffraction and interference patterns (Davisson–Germer, double slit with electrons).

Wave–particle duality

Light and matter show both wave-like (diffraction, interference) and particle-like (photoelectric effect, discrete energy) behaviour.

Emission spectrum

Bright lines on a dark background. Produced when excited electrons drop energy levels and emit photons of specific energy.

Absorption spectrum

Dark lines on a continuous spectrum. Atoms absorb photons that match the gaps between energy levels.

Atomic energy levels

Electrons can only exist at discrete energies. Transition energy ΔE = hf = E_upper − E_lower.

Photon emitted in a transition

hf = Ei − Ef, so λ = hc/ΔE. Larger energy drop means higher frequency and shorter wavelength.

Ionisation energy

Energy needed to remove an electron from an atom (move it to the n = ∞ level).

Bohr model of the atom

Electrons orbit in fixed energy levels without radiating. Photons are emitted or absorbed only during jumps between levels.

Line spectra and elements

Each element has a unique set of spectral lines, so spectra can identify elements (including in stars).

Antiparticle

Has the same mass but opposite charge to its particle. Particle and antiparticle annihilate into energy (photons).

Four fundamental forces

Strong (gluons), electromagnetic (photons), weak (W and Z bosons), gravity (graviton, theorised). Strong is strongest, gravity is weakest.

Mass defect

The mass of a nucleus is less than the sum of the masses of its separate nucleons. Δm = (sum of nucleon masses) − (nucleus mass).

Binding energy

Energy needed to separate a nucleus into its nucleons. E = Δm c². Higher binding energy per nucleon means a more stable nucleus.

Binding energy per nucleon curve

Peaks near iron-56 (most stable). Fusion of light nuclei and fission of heavy nuclei both move toward the peak and release energy.

Nuclear fission

A heavy nucleus splits into smaller nuclei and releases energy and neutrons. Can cause a chain reaction.

Nuclear fusion

Light nuclei join to form a heavier nucleus, releasing energy. Needs very high temperature and pressure to overcome electrostatic repulsion.