AS Level Physics formulas

Physical quantities and units

Physical quantities

• physical quantity = numerical magnitude × unit, e.g. 5.0 m s⁻¹
• Masses: apple about 0.1 kg (weight about 1 N), adult about 70 kg, car about 1000 kg
• Speeds: walking about 1.5 m s⁻¹, sprinter about 10 m s⁻¹, car on a motorway about 30 m s⁻¹
• Densities: air about 1.2 kg m⁻³, water 1000 kg m⁻³
• Atmospheric pressure is about 1 × 10⁵ Pa
• One floor of a building is about 3 m high; a sheet of paper is about 0.1 mm thick

SI units

• Base quantities and units: mass (kg), length (m), time (s), current (A), temperature (K)
• newton: N = kg m s⁻² (from F = ma)
• joule: J = N m = kg m² s⁻²; pascal: Pa = N m⁻² = kg m⁻¹ s⁻²
• coulomb: C = A s (from Q = It); volt: V = J C⁻¹ = kg m² s⁻³ A⁻¹
• Small prefixes: pico p 10⁻¹², nano n 10⁻⁹, micro μ 10⁻⁶, milli m 10⁻³, centi c 10⁻², deci d 10⁻¹
• Large prefixes: kilo k 10³, mega M 10⁶, giga G 10⁹, tera T 10¹²
• Homogeneous equation: every term has the same base units

Errors and uncertainties

• percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%
• Adding or subtracting quantities: add the absolute uncertainties
• Multiplying or dividing quantities: add the percentage uncertainties
• Quantity raised to a power n: multiply its percentage uncertainty by n (x² gives 2 × the percentage, √x gives ½ ×)
• Zero error: corrected reading = reading − zero error
• Systematic error affects accuracy; random error affects precision
• Timing n oscillations: the percentage uncertainty in the period is the same as in the total time

Scalars and vectors

• Scalars: distance, speed, mass, time, energy, work, power, temperature
• Vectors: displacement, velocity, acceleration, force, weight, momentum
• Two vectors at right angles: resultant² = a² + b²
• Component next to the angle θ = F cos θ; component opposite the angle = F sin θ
• The resultant of two vectors a and b (a larger) lies between a − b and a + b
• change in a vector = final − initial (reverse the initial vector and add)

Kinematics

Equations of motion

• v = u + at
• s = ½(u + v)t
• s = ut + ½at²
• v² = u² + 2as
• Gradient of displacement–time graph = velocity; gradient of velocity–time graph = acceleration
• Area under velocity–time graph = displacement
• Projectile: horizontal velocity constant; vertical acceleration = g downwards; at the top the vertical velocity is zero

Dynamics

Momentum and Newton's laws of motion

• momentum = mass × velocity (p = mv), unit kg m s⁻¹ or N s
• resultant force = change in momentum ÷ time taken (F = Δp/Δt)
• resultant force = mass × acceleration (F = ma), for constant mass
• weight = mass × g (W = mg), with g = 9.81 m s⁻² on Earth
• First law: an object stays at rest or at constant velocity unless a resultant force acts on it
• Third law: if A exerts a force on B, B exerts an equal and opposite force of the same type on A
• A third law pair acts on two different objects, so the two forces never cancel each other

Non-uniform motion

• Drag force increases as speed increases; it acts opposite to the velocity
• Falling object: resultant force = weight − drag, so acceleration = (weight − drag) ÷ mass
• At terminal velocity: drag = weight, resultant force = 0, acceleration = 0
• Falling from rest: acceleration starts at g and decreases to zero
• Same size and shape, larger weight: higher terminal velocity
• Constant velocity on a rough surface: friction = applied force
• Thrown upwards with air resistance: deceleration going up is greater than g, and the time up is shorter than the time down

Linear momentum and its conservation

• total momentum before = total momentum after: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
• Elastic collision: total kinetic energy is conserved
• Elastic collision: relative speed of approach = relative speed of separation
• Inelastic collision: momentum is conserved, total kinetic energy is not
• kinetic energy = ½ × mass × speed² (E_k = ½mv²)
• Explosion from rest: total momentum stays zero, so the parts move in opposite directions
• Two dimensions: apply conservation of momentum to the x and y directions separately

Forces, density and pressure

Turning effects of forces

• moment = force × perpendicular distance from the pivot to the line of action of the force, unit N m
• Force at angle θ to the lever of length d: moment = F × d × sin θ
• torque of a couple = one of the forces × perpendicular distance between the two forces
• A couple has zero resultant force: it causes rotation only, no linear acceleration
• Centre of gravity: the point where the whole weight of the object may be taken to act
• A uniform object has its centre of gravity at its geometric centre

Equilibrium of forces

• Equilibrium: resultant force = 0 and resultant torque = 0
• Principle of moments: for an object in equilibrium, sum of clockwise moments about any point = sum of anticlockwise moments about the same point
• Vertical forces balance: total upward force = total downward force
• Three forces in equilibrium form a closed triangle when drawn head to tail
• Resolving: components to the left = components to the right; components up = components down
• The weight of a uniform beam acts at its midpoint

Density and pressure

• density = mass ÷ volume (ρ = m/V), unit kg m⁻³
• pressure = force ÷ area (p = F/A), unit Pa, where 1 Pa = 1 N m⁻²
• change in pressure = density of fluid × g × change in depth (Δp = ρgΔh)
• upthrust = density of fluid × g × volume of fluid displaced (F = ρgV)
• Floating object: upthrust = weight of the object
• Total pressure at a depth = atmospheric pressure + ρgh
• 1 cm³ = 1 × 10⁻⁶ m³; 1 g cm⁻³ = 1000 kg m⁻³

Work, energy and power

Energy conservation

• work done = force × displacement in the direction of the force (W = Fs); unit joule, J
• Force at angle θ to the displacement: W = Fs cos θ
• Conservation of energy: energy cannot be created or destroyed, only transferred from one form to another
• efficiency = useful energy output ÷ total energy input (× 100 for a percentage); the same ratio works with powers
• power = work done ÷ time taken (P = W/t); 1 W = 1 J s⁻¹
• power = force × velocity (P = Fv), because P = W/t = Fs/t and s/t = v
• At constant velocity, driving force = total resistive force

Gravitational potential energy and kinetic energy

• change in gravitational potential energy = mass × g × change in height (ΔE_p = mgΔh); g = 9.81 m s⁻²
• kinetic energy = ½ × mass × speed² (E_k = ½mv²)
• Deriving ΔE_p: use W = Fs with F = weight mg and s = vertical height Δh
• Deriving E_k: from rest, v² = 2as and F = ma, so W = Fs = ma × v²/(2a) = ½mv²
• No resistive forces: mgΔh = ½mv², so v = √(2gΔh)
• E_k ∝ v²: doubling the speed gives four times the kinetic energy
• change in kinetic energy = ½m(v² − u²)

Deformation of solids

Stress and strain

• Hooke's law: force is proportional to extension, provided the limit of proportionality is not exceeded (F = kx)
• spring constant = force ÷ extension (k = F/x); unit N m⁻¹
• stress = force ÷ cross-sectional area (σ = F/A); unit Pa (N m⁻²)
• strain = extension ÷ original length (ε = x/L); no unit
• Young modulus = stress ÷ strain (E = FL/(Ax)); unit Pa
• Area of a wire: A = πd²/4, so doubling the diameter gives four times the area and one quarter of the extension
• Two identical springs in series: total extension doubles. In parallel: total extension halves

Elastic and plastic behaviour

• Elastic deformation: returns to original shape when the load is removed. Plastic deformation: permanent change of shape
• Elastic limit: the maximum load beyond which the object does not return to its original length
• work done = area under the force–extension graph
• elastic potential energy = ½ × force × extension (E_p = ½Fx), within the limit of proportionality
• E_p = ½kx², because F = kx
• Work done in stretching from x₁ to x₂ = ½k(x₂² − x₁²)
• For a curved graph, find the area by counting squares

Waves

Progressive waves

• frequency = 1 ÷ period (f = 1/T)
• wave speed = frequency × wavelength (v = fλ): the wave travels one wavelength in one period, so v = λ/T = fλ
• phase difference = (distance apart ÷ wavelength) × 360°, so λ/4 apart is 90° and λ/2 apart is 180°
• Oscilloscope: period = divisions for one cycle × time-base
• Oscilloscope: amplitude = half the crest-to-trough height × y-gain
• intensity = power ÷ area (I = P/A); unit W m⁻². For a point source the area is 4πr²
• intensity ∝ amplitude²: double the amplitude gives four times the intensity

Transverse and longitudinal waves

• Transverse: vibrations perpendicular to the direction of energy transfer
• Longitudinal: vibrations parallel to the direction of energy transfer
• Wavelength of a longitudinal wave = distance between the centres of two neighbouring compressions (or two neighbouring rarefactions)
• Compression centre to the nearest rarefaction centre = half a wavelength
• Displacement–distance graph gives λ; displacement–time graph gives T; then v = λ/T
• To find which way a point on a transverse wave is moving, sketch the wave a moment later, shifted in the direction of travel
• On a displacement–distance graph for sound, the centres of compressions and rarefactions are where the displacement is zero

Doppler effect for sound waves

• observed frequency = source frequency × speed of sound ÷ (speed of sound ± speed of source): f_o = f_sv/(v ± v_s)
• Source moving towards the observer: use v − v_s, so f_o is higher than f_s
• Source moving away from the observer: use v + v_s, so f_o is lower than f_s
• The speed of the sound waves through the air does not change; the wavelength does
• Ratio of approaching to receding frequency = (v + v_s)/(v − v_s)

Electromagnetic spectrum

• speed = frequency × wavelength (c = fλ), with c = 3.00 × 10⁸ m s⁻¹ in free space
• Radio waves: longer than about 10⁻¹ m. Microwaves: about 10⁻³ m to 10⁻¹ m
• Infrared: about 7 × 10⁻⁷ m to 10⁻³ m
• Visible light: 4 × 10⁻⁷ m to 7 × 10⁻⁷ m (400 nm violet to 700 nm red)
• Ultraviolet: about 10⁻⁸ m to 4 × 10⁻⁷ m
• X-rays: about 10⁻¹³ m to 10⁻⁸ m. γ-rays: shorter than about 10⁻¹⁰ m
• 1 nm = 10⁻⁹ m

Polarisation

• Only transverse waves can be polarised; this is the evidence that light is transverse
• Malus's law: transmitted intensity = incident intensity × cos²θ (I = I₀ cos²θ)
• θ is the angle between the transmission axis and the plane of polarisation of the incoming light
• θ = 0°: I = I₀. θ = 45°: I = ½I₀. θ = 60°: I = ¼I₀. θ = 90°: I = 0
• transmitted amplitude = A₀ cos θ
• For filters in a row, apply Malus's law at each one, using the angle to the previous filter's axis
• Rotating a filter through 360° gives two maxima and two zeros

Superposition

Stationary waves

• Resultant displacement = sum of the individual displacements (for example +3 cm and −1 cm give +2 cm)
• node to next node = antinode to next antinode = λ/2
• node to nearest antinode = λ/4
• String fixed at both ends (node at each end): length = n × λ/2, where n is the number of loops
• Air column closed at one end (node at the closed end, antinode at the open end): fundamental has length = λ/4
• Air column open at both ends (antinode at each end): fundamental has length = λ/2
• Use v = fλ with the wavelength of the progressive waves

Diffraction

• Diffraction: the spreading of a wave as it passes through a gap or round an edge
• Most diffraction when gap width ≈ wavelength
• Gap much wider than the wavelength: very little diffraction
• Wavelength, frequency and speed are unchanged by diffraction
• At the same speed, higher frequency means shorter wavelength (λ = v/f), so less diffraction through the same gap
• Long wavelengths bend round large obstacles: sound (about 1 m) bends round a corner, light (about 5 × 10⁻⁷ m) does not

Interference

• Coherent sources: constant phase difference (so the same frequency)
• Maximum (bright fringe, loud sound): path difference = nλ, where n = 0, 1, 2, ...
• Minimum (dark fringe, quiet sound): path difference = (n + ½)λ
• Double slit: wavelength = slit separation × fringe spacing ÷ slit-to-screen distance (λ = ax/D)
• Fringe spacing x = λD/a: fringes are wider for a longer wavelength, a larger D or a smaller a
• Resultant amplitude: sum of the amplitudes when in phase, difference of the amplitudes when in antiphase
• White light: the central fringe is white, the other fringes are coloured because x depends on λ

The diffraction grating

• slit spacing × sin(angle) = order × wavelength (d sin θ = nλ)
• θ is measured from the straight-through (zero order) direction
• d = 1 ÷ (number of lines per metre), e.g. 500 lines per mm = 5.0 × 10⁵ lines per m, so d = 2.0 × 10⁻⁶ m
• Highest order: the largest whole number n that is not more than d/λ, because sin θ cannot be more than 1
• Total number of maxima = 2 × (highest order) + 1
• A longer wavelength gives a larger angle: in each order red is deviated more than violet
• If the angle between the two first-order maxima is given, θ is half of that angle

Electricity

Electric current

• charge = current × time (Q = It); 1 C = 1 A s
• Charge is quantised: Q = Ne, where N is a whole number and e = 1.60 × 10⁻¹⁹ C
• Number of electrons = total charge ÷ e
• current = area × number density × drift speed × charge of each carrier (I = Anvq)
• n is the number of charge carriers per unit volume, unit m⁻³
• Same metal and same current: drift speed is inversely proportional to cross-sectional area
• 1 mA h = 1 × 10⁻³ A × 3600 s = 3.6 C

Potential difference and power

• potential difference = energy transferred ÷ charge (V = W/Q); 1 V = 1 J C⁻¹
• Energy gained by a charge q moved through a p.d. V: W = qV
• power = p.d. × current (P = VI); 1 W = 1 J s⁻¹
• power = current² × resistance (P = I²R)
• power = p.d.² ÷ resistance (P = V²/R)
• energy transferred = power × time (W = Pt = VIt), with t in seconds
• Fixed R: P is proportional to I² and to V²

Resistance and resistivity

• resistance = p.d. ÷ current (R = V/I), so V = IR; unit Ω
• Ohm's law: the current in a metallic conductor is proportional to the p.d. across it, provided the temperature is constant
• resistance = resistivity × length ÷ cross-sectional area (R = ρL/A); ρ is in Ω m
• A = πd²/4, so twice the diameter gives four times the area and a quarter of the resistance
• I–V graphs (I up, V across): metal at constant temperature, a straight line through the origin; filament lamp, a curve whose gradient falls as V rises
• Diode: almost no current in reverse bias (very large resistance); current rises steeply above about 0.7 V in forward bias
• LDR: resistance falls as light intensity rises. Thermistor (NTC): resistance falls as temperature rises

D.C. circuits

Practical circuits

• e.m.f. = energy transferred from other forms to electrical ÷ charge (E = W/Q), unit V
• p.d. = energy transferred from electrical to other forms ÷ charge
• e.m.f. = current × (external resistance + internal resistance): E = I(R + r)
• terminal p.d. = e.m.f. − current × internal resistance (V = E − Ir)
• When no current flows, terminal p.d. = e.m.f.
• Ammeter in series (very low resistance); voltmeter in parallel (very high resistance)
• Symbols: a rectangle is a resistor; with a diagonal arrow through it, a variable resistor

Kirchhoff’s laws

• First law: sum of currents into a junction = sum of currents out (conservation of charge)
• Second law: sum of e.m.f.s around a closed loop = sum of p.d.s around that loop (conservation of energy)
• Series: R = R₁ + R₂ + ...
• Parallel: 1/R = 1/R₁ + 1/R₂ + ...
• A parallel combination is always smaller than its smallest resistor; n equal resistors R in parallel give R/n
• In parallel the larger current is in the smaller resistor: I₁/I₂ = R₂/R₁
• Cells facing in opposite directions: subtract their e.m.f.s

Potential dividers

• output p.d. = supply p.d. × R₁ ÷ (R₁ + R₂), where the output is across R₁
• V₁/V₂ = R₁/R₂ for two resistors in series
• LDR: brighter light, lower resistance, smaller p.d. across the LDR, larger p.d. across the fixed resistor
• Thermistor (NTC): hotter, lower resistance, smaller p.d. across the thermistor, larger p.d. across the fixed resistor
• Potentiometer: p.d. along a uniform wire is proportional to its length, so E₁/E₂ = L₁/L₂ (balance lengths)
• Null method: at balance there is no current in the test cell, so no lost volts, and the p.d. measured equals its e.m.f.
• A voltmeter of similar resistance connected across one resistor lowers the resistance of that part, so it reads less than expected

Particle physics

Atoms, nuclei and radiation

• Number of neutrons = nucleon number − proton number (A − Z)
• α: helium nucleus (2 protons + 2 neutrons), charge +2e, mass about 4u. α-decay: A falls by 4, Z falls by 2
• β⁻: electron, charge −e, mass about u/1840. β⁻ decay: A the same, Z rises by 1, an electron antineutrino is also emitted
• β⁺: positron, charge +e, same mass as the electron. β⁺ decay: A the same, Z falls by 1, an electron neutrino is also emitted
• γ: electromagnetic wave, no charge, no mass; A and Z do not change
• An antiparticle has the same mass as its particle but the opposite charge; the positron is the antiparticle of the electron
• 1 u = 1/12 of the mass of a carbon-12 atom = 1.66 × 10⁻²⁷ kg

Fundamental particles

• Charge +⅔e: up (u), charm (c), top (t)
• Charge −⅓e: down (d), strange (s), bottom (b)
• An antiquark has the opposite charge, e.g. anti-up −⅔e, anti-down +⅓e
• Proton = uud: charge ⅔ + ⅔ − ⅓ = +1e. Neutron = udd: charge ⅔ − ⅓ − ⅓ = 0
• Hadrons: baryon = three quarks; meson = one quark + one antiquark
• β⁻ decay: a down quark changes to an up quark (d → u), with an electron and an electron antineutrino emitted
• β⁺ decay: an up quark changes to a down quark (u → d), with a positron and an electron neutrino emitted