O Level / IGCSE Physics formulas

Motion, forces and energy

Physical quantities and measurement techniques

• micrometer reading = sleeve reading + (thimble divisions × 0.01 mm); then subtract any zero error
• volume of a solid by displacement = new water level − first water level (1 cm³ = 1 ml)
• period of a pendulum = total time ÷ number of complete oscillations
• thickness of one sheet = thickness of the pile ÷ number of sheets
• Scalars: distance, speed, time, mass, energy, temperature
• Vectors: displacement, velocity, acceleration, force, weight, momentum, electric field strength, gravitational field strength
• Two vectors at right angles: resultant² = a² + b², so resultant = √(a² + b²)

Motion

• speed = distance ÷ time (v = s/t), in m/s
• average speed = total distance travelled ÷ total time taken
• acceleration = change in velocity ÷ time taken (a = Δv/Δt), in m/s²
• Distance-time graph: gradient = speed. Horizontal line = at rest; straight sloping line = constant speed; curve getting steeper = accelerating
• Speed-time graph: gradient = acceleration. Horizontal line = constant speed; straight sloping line = constant acceleration; curve = changing acceleration
• Area under a speed-time graph = distance travelled
• Acceleration of free fall near the Earth's surface: g ≈ 9.8 m/s²

Mass and weight

• gravitational field strength = weight ÷ mass (g = W/m), in N/kg
• weight = mass × gravitational field strength (W = mg)
• On Earth g = 9.8 N/kg, unless the question gives another value
• g in N/kg has the same value as the acceleration of free fall in m/s²
• Mass: kg, a scalar, same everywhere. Weight: N, a vector, depends on g
• A beam balance still balances on the Moon, because both sides get lighter by the same factor

Density

• density = mass ÷ volume (ρ = m/V)
• mass = density × volume; volume = mass ÷ density
• Units: g/cm³ or kg/m³. 1 g/cm³ = 1000 kg/m³
• Volume of a rectangular block = length × width × height
• mass of liquid = mass of container with liquid − mass of empty container
• Density of water = 1.0 g/cm³ = 1000 kg/m³
• An object floats in a liquid if its density is less than the density of the liquid

Forces

• resultant force = mass × acceleration (F = ma), with F in N, m in kg, a in m/s²
• Newton's first law: with no resultant force, an object stays at rest or moves in a straight line at constant speed
• Newton's third law: if A pushes on B, B pushes on A with an equal and opposite force. The two forces are the same type and act on different objects
• spring constant = force ÷ extension (k = F/x); extension = stretched length − original length
• Limit of proportionality: the point beyond which extension is no longer proportional to load (the graph stops being a straight line)
• moment = force × perpendicular distance from the pivot, in N m. In equilibrium: total clockwise moment = total anticlockwise moment
• stopping distance = thinking distance + braking distance; thinking distance = speed × reaction time

Momentum

• momentum = mass × velocity (p = mv), in kg m/s
• impulse = force × time (FΔt) = change in momentum Δ(mv), in N s
• resultant force = change in momentum ÷ time taken (F = Δp/Δt)
• Conservation of momentum: total momentum before = total momentum after, if no external force acts
• If two objects stick together: m₁u₁ + m₂u₂ = (m₁ + m₂)v
• Objects that start at rest and push apart have total momentum zero, so they move in opposite directions with equal sizes of momentum

Energy, work and power

• kinetic energy = ½ × mass × speed² (E_k = ½mv²)
• change in gravitational potential energy = mass × g × change in height (ΔE_p = mgΔh), g = 9.8 N/kg unless told otherwise
• work done = force × distance moved in the direction of the force (W = Fd); 1 J = 1 N m
• power = work done ÷ time taken = energy transferred ÷ time taken (P = W/t = ΔE/t); 1 W = 1 J/s
• efficiency = (useful energy output ÷ total energy input) × 100%; the same with power in place of energy
• Conservation of energy: total energy input = useful energy output + wasted energy
• Doubling the speed makes the kinetic energy four times as large

Pressure

• pressure = force ÷ area (p = F/A)
• Unit: pascal, 1 Pa = 1 N/m²
• change in pressure in a liquid = density × g × change in height (Δp = ρgΔh), with ρ in kg/m³ and Δh in m
• g = 9.8 N/kg unless the question gives another value
• total pressure at a depth in an open liquid = atmospheric pressure + ρgh
• Barometer: atmospheric pressure = ρgh for the liquid column, using its vertical height
• 1 cm² = 0.0001 m² (1 m² = 10 000 cm²)

Thermal physics

Kinetic particle model of matter

• Solid: fixed shape and fixed volume. Liquid: fixed volume, takes the shape of its container. Gas: no fixed shape or volume, fills its container.
• Changes of state: melting (solid to liquid), solidification or freezing (liquid to solid), boiling or evaporation (liquid to gas), condensation (gas to liquid).
• Absolute zero = −273 °C, the lowest possible temperature.
• Constant volume: a hotter gas has faster particles that hit the walls harder and more often, so the pressure rises.
• Constant pressure: a hotter gas expands, so its volume increases.
• Constant temperature: pressure × volume stays the same (p₁V₁ = p₂V₂). Halving the volume doubles the pressure.
• A graph of p against V is a curve. A graph of p against 1/V is a straight line through the origin.

Thermal properties and temperature

• temperature in kelvin = temperature in °C + 273 (T = θ + 273). A change of 1 K is the same size as a change of 1 °C.
• specific heat capacity = energy ÷ (mass × temperature change) (c = ΔE/mΔθ), unit J/(kg °C).
• energy = mass × specific heat capacity × temperature change (ΔE = mcΔθ), with mass in kg.
• Pure water at standard atmospheric pressure melts at 0 °C and boils at 100 °C.
• Boiling: at one fixed temperature, bubbles form throughout the liquid. Evaporation: at any temperature, from the surface only.
• Evaporation is faster with a higher temperature, a larger surface area or moving air over the surface.
• Evaporation cools: the most energetic particles escape, so the average kinetic energy of those left behind falls.

Transfer of thermal energy

• Conduction: lattice vibrations in all solids, plus free (delocalised) electrons in metals.
• Poor conductors are called insulators, for example plastic, wood, glass and air.
• Convection: hot fluid is less dense and rises; cold fluid is denser and sinks.
• Dull black surfaces are the best emitters and the best absorbers of infrared radiation.
• Shiny white or silver surfaces are poor emitters, poor absorbers and good reflectors.
• An object emits radiation at a greater rate if its surface is hotter or its surface area is larger.
• A vacuum stops conduction and convection, but not radiation.

Waves

General properties of waves

• wave speed = frequency × wavelength (v = fλ), with v in m/s, f in Hz and λ in m.
• Frequency: the number of waves passing a point each second. Time for one wave (period) = 1 ÷ frequency.
• Wavelength: the distance between two neighbouring identical points, such as crest to crest.
• Amplitude: the maximum distance from the rest position (not from crest to trough).
• Transverse: electromagnetic waves, water surface waves, seismic S-waves. Longitudinal: sound, seismic P-waves.
• Water waves going from deep to shallow water slow down. The wavelength gets shorter and the frequency stays the same.
• Diffraction through a gap is greatest when the gap is about the same size as the wavelength. Longer wavelengths spread more round an edge.

Light

• Reflection: angle of incidence = angle of reflection (i = r). A plane mirror image is virtual, upright, the same size as the object, and as far behind the mirror as the object is in front.
• refractive index = sin i ÷ sin r (n = sin i/sin r), where i is the angle in air. It has no unit.
• refractive index = 1 ÷ sin c (n = 1/sin c), where c is the critical angle.
• linear magnification = image length ÷ object length. It has no unit.
• Converging lens: an object beyond the principal focus gives a real, inverted image. An object closer than the principal focus gives a virtual, upright, magnified image (magnifying glass).
• Short sight: image forms in front of the retina, corrected by a diverging lens. Long sight: image would form behind the retina, corrected by a converging lens.
• Spectrum in order of increasing frequency: red, orange, yellow, green, blue, indigo, violet. Red has the longest wavelength; violet is refracted the most.

Electromagnetic spectrum

• Order of increasing frequency (decreasing wavelength): radio, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays.
• Speed of all electromagnetic waves in a vacuum = 3.0 × 10⁸ m/s; use v = fλ.
• Radio waves: radio and television broadcasts, astronomy. Microwaves: satellite television, mobile phones, Bluetooth, microwave ovens.
• Infrared: remote controllers, thermal imaging, intruder alarms, optical fibres, electric grills.
• Ultraviolet: security marking, detecting fake bank notes, sterilising water. It can cause skin cancer and cataracts.
• X-rays: medical scanning and security scanners. Gamma rays: sterilising food and medical equipment, detecting and treating cancer.
• Too much microwave or infrared radiation heats body tissue and causes burns.

Sound

• Humans can hear frequencies from about 20 Hz to 20 000 Hz.
• Ultrasound is sound with a frequency above 20 kHz (20 000 Hz).
• Speed of sound in air is about 330 to 350 m/s.
• speed = distance ÷ time. For an echo the sound travels there and back, so distance to the surface = speed × time ÷ 2.
• Amplitude sets the loudness; frequency sets the pitch.
• Two instruments playing the same note have the same frequency but different wave shapes on an oscilloscope. This is the quality (timbre).
• Uses of ultrasound: cleaning, scanning an unborn baby, and sonar to find depth.

Electricity and magnetism

Simple magnetism and magnetic fields

• Like poles repel; unlike poles attract
• Magnetic materials: iron, steel, nickel, cobalt. Copper, aluminium, brass, plastic and wood are non-magnetic
• Direction of the field at a point = direction of the force on a N pole there = the way the N end of a compass needle points
• Outside a magnet, field lines run from the N pole to the S pole; closer lines mean a stronger field
• Induced magnetism: the end of the iron nearest the magnet becomes the opposite pole, so it is attracted
• Soft iron: temporary magnet, used for the core of an electromagnet. Steel: permanent magnet
• Electromagnets can be switched on and off (cranes, relays); permanent magnets are used in compasses, motors and loudspeakers

Electrical quantities

• current = charge ÷ time (I = Q/t); 1 A = 1 C/s
• e.m.f. or p.d. = work done ÷ charge (E = W/Q, V = W/Q); 1 V = 1 J/C
• resistance = p.d. ÷ current (R = V/I), measured in ohms (Ω)
• For a wire: resistance is directly proportional to length and inversely proportional to cross-sectional area
• Ohm's law: the current in a metal conductor is directly proportional to the p.d. across it, if the temperature is constant
• A filament lamp gets hotter as the current rises, so its resistance rises. A diode lets current pass in one direction only
• Cells in series: add the e.m.f.s. Identical cells in parallel: the e.m.f. is that of one cell

Electric circuits

• Series: the current is the same at every point; the p.d.s across the components add up to the supply p.d.
• Parallel: the p.d. across each branch is the same; current into a junction = current out of the junction
• Resistors in series: R = R₁ + R₂ + ...
• Two resistors in parallel: 1/R = 1/R₁ + 1/R₂, which gives R = (R₁ × R₂) ÷ (R₁ + R₂)
• Potential divider: R₁/R₂ = V₁/V₂
• NTC thermistor: resistance falls as temperature rises. LDR: resistance falls as the light gets brighter
• An ammeter is connected in series; a voltmeter is connected in parallel with the component

Practical electricity

• power = current × voltage (P = IV), in watts
• energy = current × voltage × time (E = IVt), in joules with the time in seconds
• energy in kW h = power in kW × time in hours; cost = energy in kW h × price of one kW h
• 1 kW h = 1000 W × 3600 s = 3.6 × 10⁶ J
• Fuse rating: the next value above the normal working current (I = P/V)
• Switches, fuses and circuit breakers go in the live wire. A metal case must be earthed; a plastic (double-insulated) case needs no earth wire
• Lamps in parallel each get the full supply p.d., can be switched separately, and stay on if another lamp fails

Electromagnetic effects

• Induced e.m.f. is larger when the magnet or coil moves faster, the magnet is stronger, or the coil has more turns
• Lenz's law: the induced current flows in the direction that opposes the change producing it
• Field of a straight wire: circles round the wire. Right-hand grip rule: thumb along the current, fingers curl the way the field goes
• Left-hand rule: First finger = Field, seCond finger = Current, thuMb = Motion (force), all at right angles
• Reversing the current or the field reverses the force. More turns, more current or a stronger field gives a motor coil a bigger turning effect
• Transformer: V_p/V_s = N_p/N_s. It works only with a.c.
• High-voltage transmission: the same power at a higher voltage needs a smaller current, so less energy is wasted heating the cables

Uses of an oscilloscope

• p.d. = number of vertical divisions × Y-gain (V per division)
• time = number of horizontal divisions × timebase (time per division)
• Peak p.d. is measured from the centre line to the top of the wave; it is half the trough-to-peak height
• Period = time for one complete wave; frequency = 1 ÷ period, with the period in seconds
• 1 ms = 0.001 s; 1 μs = 0.000 001 s
• A steady d.c. p.d. gives a horizontal line moved up or down from the centre
• With the timebase off, an a.c. p.d. gives a vertical line

Nuclear physics

The nuclear model of the atom

• Proton number Z = number of protons in the nucleus
• Nucleon number A = number of protons + number of neutrons
• Number of neutrons = A − Z
• Nuclide notation: ^A_ZX, with the nucleon number on top and the proton number below
• Neutral atom: electrons = protons. Losing electrons gives a positive ion; gaining electrons gives a negative ion
• Isotopes: same proton number, different nucleon number
• Alpha scattering: most pass straight through (mostly empty space); a very few bounce back (tiny, massive, positive nucleus)

Radioactivity

• Alpha (α): 2 protons + 2 neutrons, charge 2+, most ionising, stopped by paper; pulled towards a negative plate
• Beta (β): fast electron from the nucleus, charge 1−, stopped by a few millimetres of aluminium; pulled towards a positive plate and deflected more than alpha
• Gamma (γ): electromagnetic wave, no charge, least ionising, most penetrating (reduced by thick lead); not deflected by electric or magnetic fields
• Alpha decay: nucleon number goes down by 4, proton number goes down by 2
• Beta decay: nucleon number stays the same, proton number goes up by 1. Gamma emission changes neither
• corrected count rate = measured count rate − background count rate
• After 1, 2, 3 half-lives the fraction left is 1/2, 1/4, 1/8

Space physics

Earth and the Solar System

• Order from the Sun: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune
• average orbital speed = 2 × π × orbit radius ÷ orbital period (v = 2πr/T)
• Use r in metres and T in seconds to get v in m/s
• Light from the Sun takes about 500 s to reach the Earth
• The Earth's orbit is an ellipse that is almost a circle
• Further from the Sun: weaker gravitational field, lower orbital speed, longer orbital period
• The gravitational field strength at the surface of the Sun is greater than at the surface of any planet

Stars and the Universe

• Star like the Sun: protostar → stable star → red giant → planetary nebula with a white dwarf at its centre
• Much more massive star: protostar → stable star → red supergiant → supernova → neutron star or black hole
• A supernova leaves a nebula of hydrogen and new heavier elements, which may form new stars and planets
• One light-year = the distance light travels in a vacuum in one year
• The Milky Way is about 100 000 light-years across
• Redshift = an increase in the observed wavelength of light from a star or galaxy that is moving away
• The further away a galaxy is, the greater its redshift and the faster it is moving away, so the Universe is expanding