Formulas

Period from many oscillations

T = (time for n oscillations) ÷ n

T = period, the time for one complete oscillation (s); n = number of complete oscillations timed

Resultant of two vectors at right anglesExtended

resultant = √(a² + b²)

a, b = sizes of two forces (N) or two velocities (m/s) that act at right angles to each other; the direction is found from tan θ = b / a, where θ is the angle between the resultant and a

Speed

v = s / t

v = speed (m/s); s = distance travelled (m); t = time taken (s)

Average speed

average speed = total distance travelled ÷ total time taken

distance in m, time in s, average speed in m/s

Distance from a speed–time graph

distance travelled = area under the speed–time graph

constant speed: area of a rectangle = speed × time; constant acceleration: split the area into a rectangle and a triangle where needed; area of a triangle = ½ × base × height

AccelerationExtended

a = Δv / Δt

a = acceleration (m/s²); Δv = change in velocity (m/s); Δt = time taken for the change (s)

Gravitational field strength

g = W / m

g = gravitational field strength (N/kg); W = weight (N); m = mass (kg)

Density

ρ = m / V

ρ = density (kg/m³ or g/cm³); m = mass (kg or g); V = volume (m³ or cm³)

Spring constantExtended

k = F / x

k = spring constant (N/m or N/cm); F = force, the load on the spring (N); x = extension (m or cm)

Resultant force and accelerationExtended

F = ma

F = resultant force (N); m = mass (kg); a = acceleration (m/s²), in the same direction as F

Moment of a force

moment = force × perpendicular distance from the pivot

force in N; perpendicular distance from the pivot to the line of the force in m; moment in N m

Principle of moments

total clockwise moment = total anticlockwise moment

for an object in equilibrium; all moments taken about the same pivot, in N m

MomentumExtended

p = mv

p = momentum (kg m/s); m = mass (kg); v = velocity (m/s)

ImpulseExtended

impulse = FΔt = Δ(mv)

F = force (N); Δt = time for which the force acts (s); Δ(mv) = change in momentum (kg m/s); impulse in N s

Resultant force and momentumExtended

F = Δp / Δt

F = resultant force (N); Δp = change in momentum (kg m/s); Δt = time taken for the change (s)

Conservation of momentumExtended

total momentum before = total momentum after

applies when no resultant external force acts on the objects; momentum in kg m/s, with a sign for direction

Kinetic energyExtended

E_k = ½mv²

E_k = kinetic energy (J); m = mass (kg); v = speed (m/s)

Change in gravitational potential energyExtended

ΔE_p = mgΔh

ΔE_p = change in gravitational potential energy (J); m = mass (kg); g = gravitational field strength (9.8 N/kg near the Earth's surface); Δh = change in height (m)

Work done

W = Fd = ΔE

W = work done (J); F = force (N); d = distance moved in the direction of the force (m); ΔE = energy transferred (J)

Efficiency (energy)Extended

(%) efficiency = (useful energy output) / (total energy input) (× 100%)

both energies in the same unit (J); multiply by 100% to give a percentage

Efficiency (power)Extended

(%) efficiency = (useful power output) / (total power input) (× 100%)

both powers in the same unit (W); multiply by 100% to give a percentage

Power from work done

P = W / t

P = power (W); W = work done (J); t = time taken (s)

Power from energy transferred

P = ΔE / t

P = power (W); ΔE = energy transferred (J); t = time taken (s)

Pressure

p = F / A

p = pressure (N/m² or N/cm²; Pa in Extended); F = force acting at right angles to the surface (N); A = area (m² or cm²)

Change in pressure beneath the surface of a liquidExtended

Δp = ρgΔh

Δp = change in pressure (Pa); ρ = density of the liquid (kg/m³); g = gravitational field strength (9.8 N/kg); Δh = change in depth (m)

Kelvin and Celsius

T (in K) = θ (in °C) + 273

T = temperature in kelvin (K); θ = temperature in degrees Celsius (°C)

Gas at constant temperatureExtended

pV = constant

p = pressure of the gas (Pa); V = volume of the gas (m³); true for a fixed mass of gas at constant temperature, so p₁V₁ = p₂V₂

Specific heat capacityExtended

c = ΔE / (mΔθ)

c = specific heat capacity (J/(kg °C)); ΔE = energy transferred to the object (J); m = mass (kg); Δθ = rise in temperature (°C). Rearranged: ΔE = mcΔθ

Wave speed

v = fλ

v = wave speed (m/s); f = frequency (Hz); λ = wavelength (m)

Law of reflection

angle of incidence = angle of reflection

i = angle of incidence (°); r = angle of reflection (°); both measured from the normal, so i = r

Refractive index from speedsExtended

n = speed of light in a vacuum ÷ speed of light in the material

n = refractive index of the material (no unit). In general n = speed in region 1 ÷ speed in region 2; here region 1 is a vacuum. The speed of light in air is almost the same as in a vacuum.

Refractive index from anglesExtended

n = sin i / sin r

n = refractive index (no unit); i = angle of incidence in air (°); r = angle of refraction in the material (°)

Critical angleExtended

n = 1 / sin c

n = refractive index of the material (no unit); c = critical angle (°)

Speed of electromagnetic wavesExtended

speed in a vacuum = 3.0 × 10⁸ m/s

the same for every region of the spectrum; approximately the same in air; use it in v = fλ (see 3.1)

Speed of sound

v = s / t

v = speed of sound (m/s); s = distance the sound travels (m); t = time taken (s). For an echo, s is twice the distance to the reflecting surface.

Depth or distance from an echoExtended

depth = v × t ÷ 2

v = speed of sound in the material (m/s); t = time from sending the pulse to receiving the echo (s); divide by 2 because the pulse travels there and back

CurrentExtended

I = Q / t

I = current (A); Q = charge (C); t = time (s)

Electromotive forceExtended

E = W / Q

E = e.m.f. (V); W = work done by the source (J); Q = charge (C)

Potential differenceExtended

V = W / Q

V = p.d. (V); W = work done (J); Q = charge (C)

Resistance

R = V / I

R = resistance (Ω); V = p.d. (V); I = current (A)

Resistance of a metallic wireExtended

R ∝ l and R ∝ 1 / A

R = resistance (Ω); l = length of the wire (m); A = cross-sectional area of the wire (m²); same material at the same temperature

Electrical power

P = IV

P = power (W); I = current (A); V = p.d. (V)

Electrical energy

E = IVt

E = energy transferred (J); I = current (A); V = p.d. (V); t = time (s)

Cost of electricity

cost = energy used in kW h × price of 1 kW h

energy in kW h = power (kW) × time (h); 1 kW h = 3.6 × 10⁶ J

Sources in series

E = E₁ + E₂ + E₃ + …

E = combined e.m.f. (V); E₁, E₂, E₃ = e.m.f. of each source (V); all sources connected the same way round

Resistors in series

R = R₁ + R₂ + R₃ + …

R = combined resistance (Ω); R₁, R₂, R₃ = resistance of each resistor (Ω)

Two resistors in parallelExtended

R = R₁R₂ / (R₁ + R₂)

R = combined resistance (Ω); R₁, R₂ = resistance of each resistor (Ω); the same as 1 / R = 1 / R₁ + 1 / R₂

Currents at a junctionExtended

sum of currents into a junction = sum of currents out of the junction

for a supply current I that splits into two branches: I = I₁ + I₂ (A)

P.d.s in seriesExtended

V = V₁ + V₂ + …

V = total p.d. across the series components (V); V₁, V₂ = p.d. across each component (V)

P.d. across parallel branchesExtended

V = V₁ = V₂

V = p.d. across the parallel arrangement (V); V₁, V₂ = p.d. across each branch (V)

Potential dividerExtended

R₁ / R₂ = V₁ / V₂

R₁, R₂ = the two resistors in series (Ω); V₁, V₂ = p.d. across each of them (V)

Current taken by an appliance

I = P / V

I = current (A); P = power of the appliance (W); V = supply p.d. (V); this is P = IV from 4.2.5, rearranged

Transformer equation

V_p / V_s = N_p / N_s

V = voltage (V); N = number of turns; p = primary coil, s = secondary coil

Transformer at 100% efficiencyExtended

I_pV_p = I_sV_s

I = current (A); V = voltage (V); p = primary coil, s = secondary coil

Power loss in cablesExtended

P = I²R

P = power lost as thermal energy in the cables (W); I = current in the cables (A); R = resistance of the cables (Ω)

Number of neutrons

number of neutrons = A − Z

A = nucleon number (mass number), Z = proton number (atomic number)

Balancing a nuclide equationExtended

total A before = total A after; total Z before = total Z after

A = nucleon number (top number), Z = proton number (bottom number)

Corrected count rateExtended

corrected count rate = measured count rate − background count rate

count rates in counts/s or counts/minute

Amount left after n half-lives

amount left = starting amount ÷ 2ⁿ

amount = number of undecayed nuclei, or the count rate due to the source; n = number of half-lives that have passed

Alpha decayExtended

^A_ZX → ^A−4_Z−2Y + ⁴₂He

A = nucleon number, Z = proton number; X = parent nucleus, Y = new nucleus; ⁴₂He = α-particle

Beta decayExtended

^A_ZX → ^A_Z+1Y + ⁰₋₁e

A = nucleon number, Z = proton number; X = parent nucleus, Y = new nucleus; ⁰₋₁e = β-particle

Gamma emissionExtended

^A_ZX → ^A_ZX + ⁰₀γ

A = nucleon number, Z = proton number; γ = gamma radiation

Average orbital speedExtended

v = 2πr / T

v = average orbital speed (m/s), r = average radius of the orbit (m), T = orbital period (s)

Time for light to travel a distance

t = d / c

t = time (s), d = distance (m), c = speed of light in a vacuum = 3.0 × 10⁸ m/s

One light-yearExtended

1 light-year = 9.5 × 10¹⁵ m

one light-year = the distance light travels in a vacuum in one year

Hubble constantExtended

H₀ = v / d

H₀ = Hubble constant (s⁻¹), current estimate 2.2 × 10⁻¹⁸ per second; v = speed at which the galaxy is moving away from the Earth (m/s); d = distance of the galaxy from the Earth (m)

Age of the UniverseExtended

d / v = 1 / H₀

d / v = estimate of the age of the Universe (s); H₀ = Hubble constant (s⁻¹)