MDCAT Physics formulas

Physics · Vectors and equilibrium

Vectors and components

• Components of A at angle θ to the positive x-axis: A_x = A cos θ, A_y = A sin θ
• Adding by components: R_x = A_x + B_x + ..., R_y = A_y + B_y + ...
• magnitude of resultant: R = √(R_x² + R_y²)
• direction of resultant: tan θ = R_y/R_x, then pick the quadrant from the signs of R_x and R_y
• Two vectors A and B with angle θ between them: R = √(A² + B² + 2AB cos θ)
• Greatest resultant = A + B (same direction); least resultant = A − B (opposite directions)
• Weight mg on a slope at angle θ to the horizontal: mg sin θ down the slope, mg cos θ into the slope

Products of vectors and torque

• scalar product: A · B = AB cos θ = A_xB_x + A_yB_y + A_zB_z; A · A = A²
• vector product: size of A × B = AB sin θ, direction perpendicular to the plane of A and B (right-hand rule); B × A = −(A × B)
• Unit vectors: i · i = 1, i · j = 0, i × i = 0, i × j = k
• tan θ = (size of A × B) ÷ (A · B)
• torque = force × perpendicular distance from the pivot (τ = rF sin θ), unit N m
• torque of a couple = one force × perpendicular distance between the two forces
• Equilibrium: first condition ΣF = 0; second condition Στ = 0 (clockwise torques = anticlockwise torques)

Physics · Force and motion

Kinematics

• average velocity = total displacement ÷ total time; average speed = total distance ÷ total time
• v = u + at
• s = ut + ½at² and s = ½(u + v)t
• v² − u² = 2as
• From rest with uniform acceleration, distances in the 1st, 2nd, 3rd, 4th seconds are in the ratio 1 : 3 : 5 : 7
• Displacement–time graph: slope = velocity. Velocity–time graph: slope = acceleration, area = displacement
• Area below the time axis counts as negative for displacement but is added for distance

Newton's laws and momentum

• First law: no net force means no change in velocity; inertia depends on mass only
• Second law: net force = mass × acceleration (F = ma) = rate of change of momentum (F = Δp/Δt)
• Third law: action and reaction are equal, opposite and act on different bodies, so they never cancel
• momentum = mass × velocity (p = mv), unit kg m s⁻¹ or N s
• impulse = force × time = change in momentum = area under a force–time graph
• Conservation of momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ for an isolated system
• Elastic collision: kinetic energy also conserved, speed of approach = speed of separation. Inelastic collision: some kinetic energy is lost

Projectile motion

• Horizontal: velocity = v cos θ (constant), distance = v cos θ × t
• Vertical: starting velocity = v sin θ, acceleration = g downwards
• time of flight: T = 2v sin θ ÷ g
• maximum height: H = v² sin² θ ÷ 2g
• range on level ground: R = v² sin 2θ ÷ g; greatest at θ = 45°, where R_max = v²/g
• At the top: vertical velocity = 0, speed = v cos θ, acceleration = g, and velocity is at 90° to acceleration
• Air resistance reduces the range and the maximum height

Physics · Work, power and energy

Work and power

• work = force × displacement × cos θ (W = Fd cos θ), unit joule (J) = N m
• θ = 90°: work is zero. θ greater than 90°: work is negative
• Variable force: work = area under the force–displacement graph
• Spring: work to stretch from 0 to x = ½kx²; from x₁ to x₂ = ½k(x₂² − x₁²)
• power = work ÷ time = force × velocity (P = W/t = Fv), unit watt (W) = J s⁻¹
• 1 horsepower = 746 W; 1 kW h = 3.6 × 10⁶ J
• efficiency = useful output ÷ total input × 100%

Energy and its conservation

• kinetic energy = ½ × mass × speed² (KE = ½mv² = p²/2m)
• gravitational potential energy = mgh, with h measured from the chosen reference level
• Work–energy principle: net work = change in KE (Fd = ½mv² − ½mu²)
• No friction: loss of PE = gain in KE, so from rest v = √(2gh)
• With friction: loss of PE = gain in KE + work done against friction (f × distance moved)
• elastic potential energy of a spring = ½kx²

Physics · Circular and rotational motion

Angular motion and centripetal force

• angle in radians = arc length ÷ radius (θ = s/r); 2π rad = 360° = 1 revolution
• angular velocity ω = Δθ/Δt in rad s⁻¹; ω = 2π/T = 2πf
• angular acceleration α = Δω/Δt in rad s⁻²
• Linear and angular: s = rθ, v = rω, a = rα
• ω_f = ω_i + αt; θ = ω_it + ½αt²; θ = ½(ω_i + ω_f)t; ω_f² − ω_i² = 2αθ
• centripetal acceleration = v²/r = rω², directed towards the centre
• centripetal force = mv²/r = mrω²

Rotational dynamics

• moment of inertia I = Σmr², unit kg m²; torque = Iα
• Ring or hoop: I = MR². Disc or solid cylinder: I = ½MR². Solid sphere: I = (2/5)MR². Thin rod about its centre: I = (1/12)ML²
• rotational kinetic energy = ½Iω²
• Rolling without slipping: total KE = ½mv² + ½Iω² (disc: ¾mv²; hoop: mv²)
• angular momentum L = Iω = mvr, unit kg m² s⁻¹ (J s), directed along the axis by the right-hand rule
• Conservation of angular momentum: I₁ω₁ = I₂ω₂ when no external torque acts
• Satellite: orbital speed v = √(GM/r); close to the surface v = √(gR), about 7.9 km s⁻¹ for the Earth; T² ∝ r³

Physics · Fluid dynamics

Viscosity and drag

• Viscosity η measures the internal friction of a fluid; unit N s m⁻² (Pa s)
• Stokes' law: drag force = 6πηrv for a small sphere of radius r moving at speed v
• Terminal velocity: drag = weight, net force = 0, acceleration = 0, speed constant
• terminal velocity v_t = 2gr²ρ/9η, where ρ is the density of the sphere
• v_t ∝ r² and v_t ∝ 1/η: larger drops fall faster, thicker fluids slow the fall
• Laminar flow: steady, in layers, streamlines never cross; streamlines close together mean higher speed
• Turbulent flow: irregular and noisy, begins when the speed passes a critical value

Continuity and Bernoulli

• Continuity: area × speed is constant (A₁v₁ = A₂v₂)
• volume flow rate = Av, in m³ s⁻¹
• Bernoulli: P + ½ρv² + ρgh = constant
• Horizontal pipe: P₁ − P₂ = ½ρ(v₂² − v₁²), so higher speed means lower pressure
• Torricelli's theorem: speed of efflux from a hole at depth h below the surface is v = √(2gh)
• force from a pressure difference = pressure difference × area
• Conversions: 1 cm² = 10⁻⁴ m²; 1 litre = 10⁻³ m³

Physics · Waves and oscillations

Simple harmonic motion

• Condition for SHM: a = −ω²x, with ω = 2πf = 2π/T
• displacement: x = x₀ sin ωt, where x₀ is the amplitude
• speed: v = ω√(x₀² − x²); greatest speed = ωx₀ at the mean position
• greatest acceleration = ω²x₀ at the extreme positions
• Energy: PE = ½kx², KE = ½k(x₀² − x²), total = ½kx₀²
• Mass–spring system: T = 2π√(m/k), does not depend on g or on the amplitude
• Simple pendulum: T = 2π√(l/g), does not depend on the mass or on a small amplitude

Progressive waves

• wave speed = frequency × wavelength (v = fλ), and f = 1/T
• Phase difference = 2π × distance apart ÷ wavelength (Δφ = 2πx/λ), in radians
• Speed of a transverse wave on a string: v = √(T/μ), with tension T in N and mass per unit length μ in kg m⁻¹
• Coherent sources have the same frequency and a constant phase difference
• Sources in phase: maximum where path difference = nλ, minimum where path difference = (n + ½)λ
• Intensity is proportional to amplitude², so I_max/I_min = (A₁ + A₂)²/(A₁ − A₂)²
• A compression and the nearest rarefaction are half a wavelength apart

Stationary waves and sound

• String fixed at both ends, or pipe open at both ends: f_n = nv/(2L), n = 1, 2, 3, …
• Pipe closed at one end: f_n = nv/(4L), n = 1, 3, 5, … only
• String vibrating in n loops: L = nλ/2; its fundamental is f₁ = (1/(2L)) × √(T/μ)
• Speed of sound in a gas: v = √(γP/ρ) = √(γRT/M), so v is proportional to √T (T in kelvin) and to 1/√M
• Beat frequency = difference of the two frequencies (f₁ − f₂)
• Doppler, moving source: f′ = f × v/(v − v_s) when approaching, f × v/(v + v_s) when moving away
• Doppler, moving listener: f′ = f × (v + v_o)/v when approaching, f × (v − v_o)/v when moving away

Physics · Thermodynamics

Kinetic theory and the first law

• Ideal gas: PV = nRT, and P = ⅓ρ⟨v²⟩ = ⅔ × (N/V) × ⟨½mv²⟩
• Average translational kinetic energy of a molecule = (3/2)kT, with T in kelvin; v_rms = √(3RT/M), proportional to √T
• First law: Q = ΔU + W (Q = heat supplied to the gas, W = work done by the gas)
• Isothermal: ΔU = 0, so Q = W
• Adiabatic: Q = 0, so W = −ΔU (compression heats the gas, expansion cools it)
• Isochoric: W = 0, so Q = ΔU
• Isobaric: W = PΔV = nRΔT

Heat engines and entropy

• efficiency = work output ÷ heat input: η = W/Q₁ = 1 − Q₂/Q₁
• Carnot engine: η = 1 − T₂/T₁, and Q₂/Q₁ = T₂/T₁ (T₁ source, T₂ sink, both in kelvin)
• A real engine is always less efficient than a Carnot engine between the same temperatures; 100% would need a sink at 0 K
• Entropy change ΔS = Q/T, unit J K⁻¹; positive for heat gained, negative for heat lost
• Total entropy increases in an irreversible process and stays constant in a reversible one; it never decreases
• Melting and boiling raise the entropy of a substance; freezing and condensing lower it

Physics · Electrostatics

Coulomb's law and electric field

• Coulomb's law in a vacuum: F = kq₁q₂/r², with k = 1/(4πε₀) = 9 × 10⁹ N m² C⁻²
• In a medium: F_medium = F_vacuum ÷ ε_r, so ε_r = F_vacuum/F_medium
• electric field strength = force ÷ charge (E = F/q), unit N C⁻¹ or V m⁻¹
• Field of a point charge: E = kq/r²
• Uniform field between parallel plates: E = V/d
• Field lines start on positive charges and end on negative charges, never cross, and are closer together where the field is stronger

Potential and capacitors

• potential difference = work done ÷ charge (V = W/q); energy gained by a charge = qV
• Point charge: V = kq/r; field strength is the potential gradient, E = −ΔV/Δr, so for a point charge E = V/r
• 1 eV = 1.6 × 10⁻¹⁹ J
• capacitance = charge ÷ potential difference (C = Q/V); 1 F = 1 C V⁻¹
• Parallel-plate capacitor: C = ε₀ε_rA/d
• Parallel: C = C₁ + C₂ + …; series: 1/C = 1/C₁ + 1/C₂ + …
• Energy stored = ½QV = ½CV² = Q²/(2C)

Physics · Current electricity

Resistance and Ohm's law

• current = charge ÷ time (I = Q/t), and Q = ne with e = 1.6 × 10⁻¹⁹ C
• Ohm's law: V = IR, with R constant at constant temperature
• resistance = resistivity × length ÷ area (R = ρL/A); ρ in Ω m; A = πd²/4
• Temperature: R_t = R₀(1 + αΔt); α is positive for metals
• Series: R = R₁ + R₂ + …
• Parallel: 1/R = 1/R₁ + 1/R₂ + …; for two resistors R = R₁R₂/(R₁ + R₂)
• n equal resistors R in parallel give R/n

Circuits, EMF and power

• Terminal p.d. = EMF − lost volts: V = E − Ir
• Current from a cell: I = E/(R + r)
• power = p.d. × current: P = VI = I²R = V²/R
• energy = power × time; 1 kW h = 3.6 × 10⁶ J
• Kirchhoff's first rule: total current into a junction = total current out
• Kirchhoff's second rule: round any closed loop, sum of EMFs = sum of IR drops; EMFs that oppose are subtracted
• Balanced Wheatstone bridge: P/Q = R/S

Physics · Electromagnetism

Magnetic fields and forces

• Long straight wire: B = μ₀I/(2πr), with μ₀ = 4π × 10⁻⁷ T m A⁻¹
• Inside a solenoid: B = μ₀nI, where n = N/L is the number of turns per metre
• force on a wire = B × I × L × sin θ (F = BIL sin θ), θ between the wire and the field
• Direction of the force: Fleming's left-hand rule (first finger field, second finger current, thumb force)
• Parallel wires: force per unit length F/L = μ₀I₁I₂/(2πd), the same size on both wires
• Fields from two wires add as vectors: between wires with opposite currents they add; between wires with the same currents they subtract
• 1 T = 1 N A⁻¹ m⁻¹

Charges in magnetic fields

• force on a moving charge = q × v × B × sin θ (F = qvB sin θ), θ between velocity and field
• Circular path: qvB = mv²/r, so r = mv/(qB) = √(2mE_k)/(qB)
• Period T = 2πm/(qB), which does not depend on the speed or the radius
• Electrons accelerated through V: eV = ½mv², which gives e/m = 2V/(B²r²)
• Direction: Fleming's left-hand rule for a positive charge; reverse it for an electron
• Ammeter: shunt in parallel, R_s = I_gR_g/(I − I_g)
• Voltmeter: series resistance R_h = V/I_g − R_g

Physics · Electromagnetic induction

Faraday's and Lenz's laws

• magnetic flux = B × A × cos θ (Φ = BA cos θ), θ between the field and the normal to the loop; unit weber, 1 Wb = 1 T m²
• Faraday's law: EMF = −N × ΔΦ/Δt
• Motional EMF of a rod cutting the field at right angles: EMF = BLv; induced current I = BLv/R
• Lenz's law: the induced current opposes the change in flux that produces it (the minus sign in Faraday's law)
• A north pole approaching a coil makes the near face a north pole (repels); a north pole moving away makes it a south pole (attracts)
• Seen from outside, a face with anticlockwise current is a north pole and a face with clockwise current is a south pole

Inductance, generators and transformers

• Self-induced EMF = −L × ΔI/Δt; L in henry, 1 H = 1 V s A⁻¹
• Mutual induction: EMF in secondary = −M × ΔI_p/Δt
• Energy stored in an inductor = ½LI²
• Generator: EMF = NBAω sin θ, θ between the field and the normal to the coil; peak EMF E₀ = NBAω, with ω = 2πf
• Transformer turns ratio: V_s/V_p = N_s/N_p
• Ideal transformer: V_pI_p = V_sI_s, so I_s/I_p = N_p/N_s
• Laminated core reduces eddy currents; soft iron reduces hysteresis loss

Physics · Alternating current

AC quantities

• V = V₀ sin ωt, with ω = 2πf and T = 1/f
• rms value = peak value ÷ √2 (V_rms = V₀/√2 = 0.707 V₀; I_rms = I₀/√2)
• peak-to-peak value = 2 × peak value
• average power = rms voltage × rms current × power factor (P = V_rms I_rms cos φ)
• For a resistor: P = I_rms²R = V_rms²/R = V₀²/2R
• A sine wave goes from zero to its peak in a quarter of a period (T/4)
• Mains values such as 220 V are rms values

AC circuits

• inductive reactance = 2π × frequency × inductance (X_L = ωL = 2πfL), in Ω
• capacitive reactance = 1 ÷ (2π × frequency × capacitance) (X_C = 1/ωC = 1/(2πfC)), in Ω
• impedance of a series RLC circuit: Z = √(R² + (X_L − X_C)²), and I_rms = V_rms/Z
• supply voltage in a series circuit: V = √(V_R² + (V_L − V_C)²)
• phase angle: tan φ = (X_L − X_C)/R; current lags if X_L > X_C, leads if X_C > X_L
• resonant frequency: f₀ = 1/(2π√(LC))
• At series resonance: Z = R, current is maximum and in phase with the supply voltage

Physics · Physics of solids and electronics

Properties of solids

• stress = force ÷ area (σ = F/A), in Pa or N m⁻²
• strain = extension ÷ original length (ε = ΔL/L), no unit
• Young's modulus = stress ÷ strain (Y = FL/(AΔL)), in Pa
• Order on the stress-strain curve: proportional limit, elastic limit, yield point, ultimate tensile strength (highest stress), breaking point
• elastic energy stored = ½ × force × extension (within the proportional limit)
• Pentavalent impurity (e.g. arsenic) gives n-type: electrons are majority carriers. Trivalent impurity (e.g. gallium) gives p-type: holes are majority carriers
• A pure semiconductor at 0 K is an insulator; any doped piece is electrically neutral as a whole

Electronics

• Potential barrier: about 0.7 V for silicon, 0.3 V for germanium. Forward current = (supply voltage − barrier) ÷ series resistance
• Half-wave rectifier: one diode, uses half of each cycle. Full-wave bridge: four diodes, two conduct in each half cycle
• emitter current = base current + collector current (I_E = I_B + I_C)
• current gain = collector current ÷ base current (β = I_C/I_B)
• Amplifier biasing: base-emitter junction forward biased, collector-base junction reverse biased
• AND: output 1 only if all inputs are 1. OR: output 1 if any input is 1. NOT: output is the opposite of the input
• NAND = AND followed by NOT; NOR = OR followed by NOT; XOR gives 1 when the two inputs are different; XNOR gives 1 when they are the same

Physics · Dawn of modern physics

Photons and the photoelectric effect

• photon energy = Planck's constant × frequency (E = hf = hc/λ), h = 6.6 × 10⁻³⁴ J s
• Einstein's equation: hf = φ + KE_max, with work function φ = hf₀ = hc/λ₀
• stopping potential: eV₀ = KE_max; 1 eV = 1.6 × 10⁻¹⁹ J
• Graph of KE_max against f: straight line, slope h, intercept on the frequency axis f₀
• Compton shift: Δλ = (h/m₀c)(1 − cos θ); zero at θ = 0°, greatest at θ = 180°
• Pair production needs a photon of at least 1.02 MeV (2 × 0.51 MeV); any extra becomes kinetic energy of the pair
• Annihilation of an electron and a positron at rest gives two 0.51 MeV photons moving in opposite directions

Wave nature of particles

• de Broglie wavelength = Planck's constant ÷ momentum (λ = h/p = h/mv)
• In terms of kinetic energy: λ = h/√(2mE_k); for a charge q accelerated through V, λ = h/√(2mqV)
• Uncertainty principle: Δx Δp ≈ h and ΔE Δt ≈ h
• Time dilation: t = t₀/√(1 − v²/c²) (longer)
• Length contraction: L = L₀√(1 − v²/c²) (shorter, only along the motion)
• Relativistic mass: m = m₀/√(1 − v²/c²)
• Mass-energy: E = mc², so energy released = mass lost × c²

Physics · Atomic spectra

Bohr model and spectra

• angular momentum: mvr = nh/2π, with n = 1, 2, 3, ...
• radius: r_n = n²r₁, with r₁ = 0.053 nm
• energy: E_n = −13.6/n² eV (−13.6, −3.40, −1.51, −0.85 eV for n = 1 to 4); ionisation energy from level n = 13.6/n² eV
• photon energy: hf = E_n − E_p
• 1/λ = R_H(1/p² − 1/n²), with R_H = 1.1 × 10⁷ m⁻¹
• Series by lower level p: Lyman 1 (ultraviolet), Balmer 2 (visible), Paschen 3, Brackett 4, Pfund 5 (all infrared)
• Number of possible lines from level n down to the ground state = n(n − 1)/2

X-rays and lasers

• maximum photon energy = electron charge × tube voltage (eV = hf_max = hc/λ_min), so λ_min = hc/eV
• Higher tube voltage: shorter λ_min, more penetrating X-rays. Larger filament current: more electrons, greater intensity only
• K_α line: an electron drops from the L shell into a vacancy in the K shell; K_β: from the M shell
• Dense materials of high atomic number (bone, lead) absorb X-rays more than soft tissue. Uses: radiography, CT scans, cancer therapy, studying crystals
• Metastable state lasts about 10⁻³ s; an ordinary excited state about 10⁻⁸ s
• Laser light is monochromatic, coherent, intense and in a narrow beam
• Helium-neon laser: excited helium atoms pass energy to neon atoms by collision; neon emits red light of 632.8 nm

Physics · Nuclear physics

Nucleus and radioactivity

• mass defect: Δm = Zm_p + Nm_n − m_nucleus
• binding energy = Δm × c²; 1 u = 931 MeV; binding energy per nucleon = binding energy ÷ A
• α decay: A falls by 4, Z falls by 2. β⁻ decay: A unchanged, Z rises by 1. γ emission: A and Z unchanged
• decay law: N = N₀e^−λt; activity A = λN, in becquerel (1 Bq = 1 decay per second)
• half-life: T_½ = 0.693/λ
• After n half-lives: fraction left = (½)ⁿ, fraction decayed = 1 − (½)ⁿ

Nuclear energy and radiation

• In every nuclear equation the mass numbers and the charge numbers balance on both sides
• energy released = mass lost × c²; 1 u = 931 MeV
• Moderator (graphite, heavy water) slows fast neutrons; control rods (cadmium, boron) absorb neutrons; coolant carries heat out of the core
• Fast breeder reactor: uranium-238 absorbs a neutron and becomes fissile plutonium-239
• Detectors: Geiger-Müller counter (one particle starts an avalanche, giving one pulse), Wilson cloud chamber (tracks of droplets), solid-state detector (reverse-biased p-n junction)
• Units: activity in becquerel (Bq); absorbed dose in gray (1 Gy = 1 J kg⁻¹); equivalent dose in sievert = dose in Gy × RBE
• Somatic effects harm the exposed person; genetic effects appear in their children. Iodine-131 is used for the thyroid, cobalt-60 γ-rays for cancer treatment