1
Physical Quantities & Measurement
Significant Figures & Errors
Absolute Error
$$\Delta a = |a_{\text{true}} - a_{\text{measured}}|$$
Mean Absolute Error
$$\overline{\Delta a} = \frac{\sum_{i=1}^{n}|\Delta a_i|}{n}$$
Relative (Fractional) Error
$$\delta a = \frac{\overline{\Delta a}}{\bar{a}}$$
Percentage Error
$$\%\,\text{error} = \frac{\overline{\Delta a}}{\bar{a}} \times 100\%$$
Error in Sum / Difference z = a ± b
$$\Delta z = \Delta a + \Delta b$$
Error in Product / Quotient z = a·b / (a/b)
$$\frac{\Delta z}{z} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$$
Error in Power z = aⁿ
$$\frac{\Delta z}{z} = n\,\frac{\Delta a}{a}$$
2
Vectors
Magnitude & Direction
Magnitude of a Vector
$$|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$$
Direction (angle with x-axis)
$$\theta = \tan^{-1}\!\left(\frac{A_y}{A_x}\right)$$
Unit Vector
$$\hat{A} = \frac{\vec{A}}{|\vec{A}|}$$
Dot & Cross Product
Scalar (Dot) Product
$$\vec{A}\cdot\vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$$
Vector (Cross) Product — Magnitude
$$|\vec{A}\times\vec{B}| = AB\sin\theta$$
Lami's Theorem
$$\frac{P}{\sin\alpha} = \frac{Q}{\sin\beta} = \frac{R}{\sin\gamma}$$
Three concurrent coplanar forces in equilibrium; α, β, γ are angles opposite to P, Q, R.
Resultant of Two Vectors
$$|\vec{R}| = \sqrt{A^2 + B^2 + 2AB\cos\theta}$$
3
Kinematics
Equations of Motion (Uniform Acceleration)
First Equation
$$v = u + at$$
Second Equation
$$s = ut + \tfrac{1}{2}at^2$$
Third Equation
$$v^2 = u^2 + 2as$$
Distance in n-th Second
$$s_n = u + \frac{a(2n-1)}{2}$$
Projectile Motion
Time of Flight
$$T = \frac{2u\sin\theta}{g}$$
Maximum Height
$$H = \frac{u^2\sin^2\theta}{2g}$$
Horizontal Range
$$R = \frac{u^2\sin 2\theta}{g}$$
Maximum Range (θ = 45°)
$$R_{\max} = \frac{u^2}{g}$$
Equation of Trajectory
$$y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}$$
Relative Motion
Relative Velocity
$$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$$
4
Laws of Motion
Newton's Laws
Newton's Second Law
$$\vec{F} = m\vec{a} = \frac{d\vec{p}}{dt}$$
Linear Momentum
$$\vec{p} = m\vec{v}$$
Impulse
$$\vec{J} = \vec{F}\,\Delta t = \Delta\vec{p}$$
Friction
Limiting Friction
$$f_s = \mu_s N$$
Kinetic Friction
$$f_k = \mu_k N$$
Angle of Friction
$$\tan\lambda = \mu_s$$
Angle of Repose
$$\tan\theta_r = \mu_s$$
The minimum angle of incline at which a body just begins to slide.
Connected Bodies
Atwood's Machine — Acceleration
$$a = \frac{(m_1 - m_2)\,g}{m_1 + m_2}$$
Atwood's Machine — Tension
$$T = \frac{2m_1 m_2\,g}{m_1 + m_2}$$
5
Work, Energy & Power
Work & Energy
Work Done
$$W = \vec{F}\cdot\vec{s} = Fs\cos\theta$$
Kinetic Energy
$$KE = \tfrac{1}{2}mv^2 = \frac{p^2}{2m}$$
Gravitational Potential Energy
$$PE = mgh$$
Elastic Potential Energy
$$PE = \tfrac{1}{2}kx^2$$
Work–Energy Theorem
$$W_{\text{net}} = \Delta KE = KE_f - KE_i$$
Power & Collisions
Power
$$P = \frac{W}{t} = \vec{F}\cdot\vec{v}$$
Coefficient of Restitution
$$e = \frac{v_2 - v_1}{u_1 - u_2}$$
e = 1: perfectly elastic; e = 0: perfectly inelastic
6
Circular Motion
Kinematics of Circular Motion
Angular Velocity
$$\omega = \frac{\Delta\theta}{\Delta t} = \frac{2\pi}{T} = 2\pi f$$
Linear & Angular Speed
$$v = r\omega$$
Centripetal Acceleration
$$a_c = \frac{v^2}{r} = r\omega^2 = \frac{4\pi^2 r}{T^2}$$
Centripetal Force
$$F_c = \frac{mv^2}{r} = mr\omega^2$$
Applications
Conical Pendulum — Speed
$$v = \sqrt{rg\tan\theta}$$
Banking of Road (without friction)
$$\tan\theta = \frac{v^2}{rg}$$
Minimum Speed at Top of Vertical Circle
$$v_{\min} = \sqrt{gr}$$
Speed at Bottom of Vertical Circle
$$v_{\text{bottom}} = \sqrt{5gr}$$
7
Gravitation
Newton's Law of Gravitation
Gravitational Force
$$F = \frac{Gm_1 m_2}{r^2}$$
G = 6.674 × 10⁻¹¹ N m² kg⁻²
Acceleration due to Gravity
$$g = \frac{GM}{R^2}$$
Variation with Height
$$g_h = g\!\left(1 - \frac{2h}{R}\right) \quad (h \ll R)$$
Variation with Depth
$$g_d = g\!\left(1 - \frac{d}{R}\right)$$
Orbital & Escape Velocity
Orbital Speed
$$v_o = \sqrt{\frac{GM}{r}} = \sqrt{\frac{gR^2}{R+h}}$$
Escape Velocity
$$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR} \approx 11.2\;\text{km s}^{-1}$$
Kepler's Third Law
$$T^2 \propto r^3 \quad\Longrightarrow\quad \frac{T^2}{r^3} = \frac{4\pi^2}{GM}$$
Gravitational Potential Energy
$$U = -\frac{GMm}{r}$$
8
Elasticity
Stress, Strain & Moduli
Stress
$$\sigma = \frac{F}{A} \quad [\text{Pa} = \text{N m}^{-2}]$$
Longitudinal Strain
$$\varepsilon = \frac{\Delta L}{L}$$
Young's Modulus
$$Y = \frac{\sigma}{\varepsilon} = \frac{FL}{A\,\Delta L}$$
Bulk Modulus
$$K = -\frac{\Delta P}{\Delta V/V} = -V\frac{\Delta P}{\Delta V}$$
Modulus of Rigidity
$$\eta = \frac{\text{Shear stress}}{\text{Shear strain}} = \frac{F/A}{\phi}$$
Poisson's Ratio
$$\mu = -\frac{\text{Lateral strain}}{\text{Longitudinal strain}} = -\frac{\Delta D/D}{\Delta L/L}$$
Elastic Potential Energy per Unit Volume
$$u = \tfrac{1}{2}\sigma\varepsilon = \frac{\sigma^2}{2Y} = \tfrac{1}{2}Y\varepsilon^2$$
9
Fluid Mechanics
Hydrostatics
Pressure at Depth
$$P = P_0 + \rho g h$$
Archimedes' Principle — Buoyant Force
$$F_b = \rho_{\text{fluid}}\, V_{\text{sub}}\, g$$
Pascal's Law (hydraulic press)
$$\frac{F_1}{A_1} = \frac{F_2}{A_2}$$
Viscosity & Surface Tension
Stokes' Law
$$F = 6\pi\eta r v$$
Terminal Velocity
$$v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$$
Surface Tension (force per unit length)
$$T = \frac{F}{l}$$
Excess Pressure — Soap Bubble
$$\Delta P = \frac{4T}{r}$$
Excess Pressure — Liquid Drop
$$\Delta P = \frac{2T}{r}$$
Capillary Rise
$$h = \frac{2T\cos\theta}{\rho g r}$$
Fluid Dynamics
Equation of Continuity
$$A_1 v_1 = A_2 v_2$$
Bernoulli's Equation
$$P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}$$
Torricelli's Theorem (efflux speed)
$$v = \sqrt{2gh}$$
10
Heat & Thermodynamics
Thermal Expansion & Calorimetry
Linear Expansion
$$\Delta L = L_0\,\alpha\,\Delta T$$
Relationship: α, β, γ
$$\gamma = 3\alpha = \tfrac{3}{2}\beta$$
Heat & Specific Heat
$$Q = mc\,\Delta T$$
Latent Heat
$$Q = mL$$
Ideal Gas & Kinetic Theory
Ideal Gas Law
$$PV = nRT = NkT$$
R = 8.314 J mol⁻¹ K⁻¹ ; k = 1.38 × 10⁻²³ J K⁻¹
RMS Speed
$$v_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$$
Average Kinetic Energy per Molecule
$$\bar{E} = \tfrac{3}{2}kT$$
Laws of Thermodynamics
First Law of Thermodynamics
$$\Delta U = Q - W$$
Work Done by Gas
$$W = \int P\,dV = P\Delta V \quad \text{(isobaric)}$$
Efficiency of Heat Engine
$$\eta = 1 - \frac{Q_2}{Q_1} = 1 - \frac{T_2}{T_1}$$
Entropy Change
$$\Delta S = \frac{Q_{\text{rev}}}{T}$$
11
Waves
Wave Parameters
Wave Speed
$$v = f\lambda = \frac{\lambda}{T}$$
Equation of Progressive Wave
$$y = A\sin(\omega t - kx) \quad;\quad k = \frac{2\pi}{\lambda},\;\omega = \frac{2\pi}{T}$$
Speed in String
$$v = \sqrt{\frac{T}{\mu}}$$
T = tension, μ = mass per unit length
Speed in Solid
$$v = \sqrt{\frac{Y}{\rho}}$$
Speed in Gas (Newton–Laplace)
$$v = \sqrt{\frac{\gamma P}{\rho}}$$
Stationary Waves & Doppler
Fundamental Frequency — String
$$f_1 = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$$
Open Pipe — Harmonics
$$f_n = \frac{nv}{2L}, \quad n = 1,2,3,\ldots$$
Closed Pipe — Harmonics
$$f_n = \frac{(2n-1)v}{4L}, \quad n = 1,2,3,\ldots$$
Doppler Effect (general)
$$f' = f\,\frac{v \pm v_o}{v \mp v_s}$$
+ when observer moves toward source; − when moving away. Signs for source are reversed.
Beats
$$f_{\text{beat}} = |f_1 - f_2|$$
12
Electric Charges & Fields
Coulomb's Law & Electric Field
Coulomb's Law
$$F = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}$$
k = 1/(4πε₀) = 9×10⁹ N m² C⁻²; ε₀ = 8.85×10⁻¹² C² N⁻¹ m⁻²
Electric Field
$$\vec{E} = \frac{\vec{F}}{q} = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}$$
Electric Field — Infinite Line Charge
$$E = \frac{\lambda}{2\pi\varepsilon_0 r}$$
Electric Field — Infinite Sheet
$$E = \frac{\sigma}{2\varepsilon_0}$$
Gauss's Law
$$\oint \vec{E}\cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$$
Electric Dipole Moment
$$\vec{p} = q\,\vec{d}$$
Torque on Dipole in Field
$$\vec{\tau} = \vec{p}\times\vec{E}$$
13
Electric Potential & Capacitance
Potential & Potential Energy
Electric Potential
$$V = \frac{W}{q} = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$$
Relation: E and V
$$E = -\frac{dV}{dr}$$
Potential Energy of Two Charges
$$U = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r}$$
Capacitance
Capacitance
$$C = \frac{Q}{V}$$
Parallel Plate Capacitor
$$C = \frac{\varepsilon_0 A}{d} \quad\xrightarrow{\text{dielectric}}\quad C = \frac{K\varepsilon_0 A}{d}$$
Series Combination
$$\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$$
Parallel Combination
$$C_p = C_1 + C_2 + \cdots$$
Energy Stored in Capacitor
$$U = \tfrac{1}{2}CV^2 = \frac{Q^2}{2C} = \tfrac{1}{2}QV$$
14
Current Electricity
Ohm's Law & Resistance
Ohm's Law
$$V = IR$$
Resistance
$$R = \frac{\rho L}{A}$$
Temperature Dependence
$$R_T = R_0(1 + \alpha\,\Delta T)$$
Series Resistance
$$R_s = R_1 + R_2 + \cdots$$
Parallel Resistance
$$\frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$$
Power & EMF
Electric Power
$$P = VI = I^2 R = \frac{V^2}{R}$$
Terminal Voltage
$$V = \mathcal{E} - Ir$$
Kirchhoff's Current Law (KCL)
$$\sum I_{\text{in}} = \sum I_{\text{out}}$$
Kirchhoff's Voltage Law (KVL)
$$\sum V = 0 \quad \text{(around any closed loop)}$$
Wheatstone Bridge (balanced)
$$\frac{P}{Q} = \frac{R}{S}$$
Metre Bridge — Unknown Resistance
$$X = R\,\frac{(100-l)}{l}$$
15
Magnetic Effects of Current
Biot–Savart & Ampere's Law
Biot–Savart Law
$$dB = \frac{\mu_0}{4\pi}\frac{I\,dl\sin\theta}{r^2}$$
Field at Centre of Circular Loop
$$B = \frac{\mu_0 I}{2R}$$
Field Inside Long Solenoid
$$B = \mu_0 nI \quad (n = N/L)$$
Ampere's Circuital Law
$$\oint \vec{B}\cdot d\vec{l} = \mu_0 I_{\text{enc}}$$
Field Due to Long Straight Wire
$$B = \frac{\mu_0 I}{2\pi r}$$
Force on Moving Charges & Loops
Lorentz Force
$$\vec{F} = q(\vec{v}\times\vec{B})$$
Force on Current-Carrying Conductor
$$\vec{F} = I(\vec{L}\times\vec{B})$$
Torque on Current Loop
$$\vec{\tau} = \vec{m}\times\vec{B} \quad;\quad \vec{m} = NIA\hat{n}$$
16
Electromagnetic Induction
Faraday's & Lenz's Law
Magnetic Flux
$$\Phi = \vec{B}\cdot\vec{A} = BA\cos\theta$$
Faraday's Law
$$\mathcal{E} = -N\frac{d\Phi}{dt}$$
Motional EMF
$$\mathcal{E} = Bvl$$
Inductance
Self-Inductance
$$\mathcal{E} = -L\frac{dI}{dt} \quad;\quad L = \frac{\mu_0 N^2 A}{l}$$
Mutual Inductance
$$\mathcal{E}_2 = -M\frac{dI_1}{dt}$$
Energy Stored in Inductor
$$U = \tfrac{1}{2}LI^2$$
17
Alternating Current
RMS Values & Reactance
RMS Voltage / Current
$$V_{\text{rms}} = \frac{V_0}{\sqrt{2}}, \quad I_{\text{rms}} = \frac{I_0}{\sqrt{2}}$$
Inductive Reactance
$$X_L = \omega L = 2\pi f L$$
Capacitive Reactance
$$X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$$
Impedance (series RLC)
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
Phase Angle
$$\tan\phi = \frac{X_L - X_C}{R}$$
Resonant Frequency
$$f_0 = \frac{1}{2\pi\sqrt{LC}}$$
Power Factor
$$\cos\phi = \frac{R}{Z}$$
Average Power
$$P = V_{\text{rms}}\,I_{\text{rms}}\cos\phi$$
Transformer Equation
$$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$
18
Electromagnetic Waves & Optics
Reflection & Refraction
Speed of Light
$$c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 3\times10^8\;\text{m s}^{-1}$$
Snell's Law
$$n_1\sin\theta_1 = n_2\sin\theta_2$$
Refractive Index
$$n = \frac{c}{v} = \frac{\sin\theta_i}{\sin\theta_r}$$
Critical Angle
$$\sin\theta_c = \frac{1}{n} \quad (n_2 < n_1)$$
Mirror Formula
$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f} = \frac{2}{R}$$
Lens Formula
$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$
Lensmaker's Formula
$$\frac{1}{f} = (n-1)\!\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$
Magnification (mirror / lens)
$$m = -\frac{v}{u} \quad \text{(mirror)}, \quad m = \frac{v}{u} \quad \text{(lens)}$$
Power of Lens
$$P = \frac{1}{f(\text{m})} \quad [\text{dioptre, D}]$$
Prism — Angle of Deviation
$$\delta = (i_1 + i_2) - A$$
Prism — Minimum Deviation
$$n = \frac{\sin\!\left(\dfrac{A+\delta_m}{2}\right)}{\sin\!\left(\dfrac{A}{2}\right)}$$
Wave Optics
Young's Double Slit — Fringe Width
$$\beta = \frac{\lambda D}{d}$$
Bright Fringe Condition
$$\Delta x = n\lambda, \quad n = 0,1,2,\ldots$$
Dark Fringe Condition
$$\Delta x = (2n-1)\frac{\lambda}{2}$$
Resolving Power of Telescope
$$\theta_{\min} = \frac{1.22\lambda}{D}$$
19
Modern Physics & Photoelectric Effect
Photons & Photoelectric Effect
Energy of Photon
$$E = hf = \frac{hc}{\lambda}$$
h = 6.626 × 10⁻³⁴ J s (Planck's constant)
Einstein's Photoelectric Equation
$$hf = \phi + \tfrac{1}{2}mv_{\max}^2 = \phi + eV_s$$
Threshold Frequency
$$f_0 = \frac{\phi}{h}$$
de Broglie Wavelength
$$\lambda = \frac{h}{p} = \frac{h}{mv}$$
Heisenberg Uncertainty Principle
$$\Delta x\cdot\Delta p \geq \frac{h}{4\pi}$$
Bohr's Atomic Model
Radius of n-th Orbit
$$r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2} = n^2 \times 0.529\;\text{\AA}$$
Energy of n-th Level (Hydrogen)
$$E_n = -\frac{13.6}{n^2}\;\text{eV}$$
Frequency of Emitted Photon
$$hf = E_{n_2} - E_{n_1}$$
Rydberg Formula
$$\frac{1}{\lambda} = R_H\!\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$
R_H = 1.097 × 10⁷ m⁻¹
20
Nuclear Physics
Nuclear Structure & Binding Energy
Nuclear Radius
$$R = R_0 A^{1/3}, \quad R_0 = 1.2\;\text{fm}$$
Mass Defect
$$\Delta m = Zm_p + Nm_n - M_{\text{nucleus}}$$
Binding Energy
$$E_b = \Delta m\,c^2$$
Mass–Energy Equivalence
$$E = mc^2$$
1 u = 931.5 MeV/c²
Radioactive Decay
Decay Law
$$N = N_0\,e^{-\lambda t}$$
Activity
$$A = \lambda N = A_0\,e^{-\lambda t}$$
Half-Life
$$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$$
Mean Life
$$\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{0.693}$$
Q-value of Nuclear Reaction
$$Q = (m_{\text{reactants}} - m_{\text{products}})\,c^2$$