All Class 11 Physics Formulas on One Page (PDF)

1. Units and Measurement

Errors

Absolute error: $\displaystyle \Delta a_i = |a_{\text{mean}} - a_i|$
Relative error: $\displaystyle \delta a = \frac{\Delta a_{\text{mean}}}{a_{\text{mean}}}$
Multiply by 100 for percentage error.
Error in a sum or difference: $\displaystyle \Delta Z = \Delta A + \Delta B$
Absolute errors always add.
Error in a product or quotient: $\displaystyle \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B}$
Relative errors add.
Error in a power: $\displaystyle Z = \frac{A^p B^q}{C^r} \;\Rightarrow\; \frac{\Delta Z}{Z} = p\frac{\Delta A}{A} + q\frac{\Delta B}{B} + r\frac{\Delta C}{C}$

Dimensions

Force: $\displaystyle [F] = [\mathrm{M L T^{-2}}]$
Energy: $\displaystyle [E] = [\mathrm{M L^2 T^{-2}}]$
Power: $\displaystyle [P] = [\mathrm{M L^2 T^{-3}}]$
Pressure: $\displaystyle [p] = [\mathrm{M L^{-1} T^{-2}}]$

2. Motion in a Straight Line

Definitions

Average velocity: $\displaystyle \bar v = \frac{\Delta x}{\Delta t}$
Instantaneous velocity: $\displaystyle v = \frac{dx}{dt}$
Instantaneous acceleration: $\displaystyle a = \frac{dv}{dt} = v\frac{dv}{dx}$

Equations of motion (constant acceleration)

First equation: $\displaystyle v = u + at$
Second equation: $\displaystyle s = ut + \tfrac12 at^2$
Third equation: $\displaystyle v^2 = u^2 + 2as$
Distance in the nth second: $\displaystyle s_n = u + \frac{a}{2}(2n - 1)$

Free fall and graphs

Falling from rest: $\displaystyle v = gt, \quad h = \tfrac12 gt^2, \quad v^2 = 2gh$
Relative velocity: $\displaystyle v_{BA} = v_B - v_A$
Reading graphs: $\displaystyle v = \text{slope of } x\text{-}t, \quad a = \text{slope of } v\text{-}t, \quad \Delta x = \text{area under } v\text{-}t$

3. Motion in a Plane

Vectors

Resultant of two vectors: $\displaystyle R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$
Direction of the resultant: $\displaystyle \tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}$
α is the angle from A.
Components: $\displaystyle A_x = A\cos\theta, \quad A_y = A\sin\theta$

Projectile motion

Time of flight: $\displaystyle T = \frac{2u\sin\theta}{g}$
Maximum height: $\displaystyle H = \frac{u^2\sin^2\theta}{2g}$
Horizontal range: $\displaystyle R = \frac{u^2\sin 2\theta}{g}$
Largest at θ = 45°. Angles θ and 90° − θ give the same range.
Path equation: $\displaystyle y = x\tan\theta - \frac{g x^2}{2u^2\cos^2\theta}$

Circular motion

Angular speed: $\displaystyle \omega = \frac{\Delta\theta}{\Delta t} = \frac{2\pi}{T} = 2\pi\nu$
Linear and angular speed: $\displaystyle v = r\omega$
Centripetal acceleration: $\displaystyle a_c = \frac{v^2}{r} = r\omega^2$

4. Laws of Motion

Newton's laws and momentum

Second law: $\displaystyle \vec F = \frac{d\vec p}{dt} = m\vec a$
The form with a holds when mass is constant.
Momentum: $\displaystyle \vec p = m\vec v$
Impulse: $\displaystyle \vec J = \vec F\,\Delta t = \Delta\vec p$
Conservation of momentum: $\displaystyle m_1\vec u_1 + m_2\vec u_2 = m_1\vec v_1 + m_2\vec v_2$
Holds when net external force is zero.

Friction

Limiting static friction: $\displaystyle f_{s,\max} = \mu_s N$
Kinetic friction: $\displaystyle f_k = \mu_k N$
Block on a rough incline: $\displaystyle a = g(\sin\theta - \mu_k\cos\theta)$
Sliding down. It only slides if tan θ exceeds μₛ.

Circular motion

Centripetal force: $\displaystyle F_c = \frac{mv^2}{r}$
Maximum speed on a level road: $\displaystyle v_{\max} = \sqrt{\mu_s r g}$
Ideal speed on a banked road: $\displaystyle v_0 = \sqrt{r g \tan\theta}$
No friction needed at this speed.
Maximum speed on a banked road: $\displaystyle v_{\max} = \sqrt{r g\,\frac{\mu_s + \tan\theta}{1 - \mu_s\tan\theta}}$

5. Work, Energy and Power

Work and energy

Work by a constant force: $\displaystyle W = \vec F\cdot\vec d = Fd\cos\theta$
Work by a variable force: $\displaystyle W = \int_{x_i}^{x_f} F\,dx$
Kinetic energy: $\displaystyle K = \tfrac12 mv^2 = \frac{p^2}{2m}$
Work-energy theorem: $\displaystyle W_{\text{net}} = \Delta K$
Gravitational potential energy: $\displaystyle U = mgh$
Near Earth's surface.
Spring potential energy: $\displaystyle U = \tfrac12 kx^2$
Spring force: $\displaystyle F = -kx$

Power

Average power: $\displaystyle P_{\text{av}} = \frac{W}{t}$
Instantaneous power: $\displaystyle P = \vec F\cdot\vec v$

Collisions

Coefficient of restitution: $\displaystyle e = \frac{v_2 - v_1}{u_1 - u_2}$
e = 1 is elastic, and e = 0 is perfectly inelastic.
1D elastic collision: $\displaystyle v_1 = \frac{(m_1 - m_2)u_1 + 2m_2u_2}{m_1 + m_2}$
Swap the subscripts to get v₂.
Perfectly inelastic collision: $\displaystyle v = \frac{m_1u_1 + m_2u_2}{m_1 + m_2}$

6. System of Particles and Rotational Motion

Centre of mass

Discrete particles: $\displaystyle \vec r_{\text{cm}} = \frac{\sum m_i\vec r_i}{\sum m_i}$
Continuous body: $\displaystyle \vec r_{\text{cm}} = \frac{1}{M}\int \vec r\,dm$

Rotation

Torque: $\displaystyle \vec\tau = \vec r\times\vec F, \quad \tau = rF\sin\theta$
Angular momentum: $\displaystyle \vec L = \vec r\times\vec p, \quad L = I\omega$
Rotational second law: $\displaystyle \tau = I\alpha = \frac{dL}{dt}$
Rotational kinetic energy: $\displaystyle K = \tfrac12 I\omega^2$
Rotational power: $\displaystyle P = \tau\omega$
Equations of rotation: $\displaystyle \omega = \omega_0 + \alpha t, \quad \theta = \omega_0 t + \tfrac12\alpha t^2, \quad \omega^2 = \omega_0^2 + 2\alpha\theta$

Moment of inertia

Definition: $\displaystyle I = \sum m_i r_i^2$
Parallel axis theorem: $\displaystyle I = I_{\text{cm}} + Md^2$
Perpendicular axis theorem: $\displaystyle I_z = I_x + I_y$
For flat bodies only.
Ring, about its central axis: $\displaystyle I = MR^2$
Disc, about its central axis: $\displaystyle I = \tfrac12 MR^2$
Solid sphere, about a diameter: $\displaystyle I = \tfrac25 MR^2$
Hollow sphere, about a diameter: $\displaystyle I = \tfrac23 MR^2$
Rod, about its centre: $\displaystyle I = \tfrac{1}{12}ML^2$
Rod, about one end: $\displaystyle I = \tfrac13 ML^2$

Rolling without slipping

Condition: $\displaystyle v_{\text{cm}} = R\omega$
Total kinetic energy: $\displaystyle K = \tfrac12 mv^2\left(1 + \frac{k^2}{R^2}\right)$
k is the radius of gyration, from I = mk².
Acceleration down an incline: $\displaystyle a = \frac{g\sin\theta}{1 + k^2/R^2}$

7. Gravitation

Force and acceleration

Newton's law of gravitation: $\displaystyle F = \frac{Gm_1m_2}{r^2}$
Acceleration due to gravity: $\displaystyle g = \frac{GM}{R^2}$
At height h: $\displaystyle g_h = \frac{g}{(1 + h/R)^2} \approx g\left(1 - \frac{2h}{R}\right)$
The approximation needs h much smaller than R.
At depth d: $\displaystyle g_d = g\left(1 - \frac{d}{R}\right)$

Energy and escape

Gravitational potential energy: $\displaystyle U = -\frac{GMm}{r}$
Gravitational potential: $\displaystyle V = -\frac{GM}{r}$
Escape speed: $\displaystyle v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$

Orbits

Orbital speed: $\displaystyle v_o = \sqrt{\frac{GM}{r}}$
Close to the surface, vₑ = √2 × vₒ.
Time period: $\displaystyle T = 2\pi\sqrt{\frac{r^3}{GM}}$
Kepler's third law: $\displaystyle T^2 \propto r^3$
Energy of a satellite: $\displaystyle E = -\frac{GMm}{2r}, \quad K = \frac{GMm}{2r}, \quad U = -\frac{GMm}{r}$

8. Mechanical Properties of Solids

Stress and strain

Stress: $\displaystyle \sigma = \frac{F}{A}$
Longitudinal strain: $\displaystyle \varepsilon = \frac{\Delta L}{L}$
Hooke's law: $\displaystyle \text{stress} = \text{modulus} \times \text{strain}$
Within the elastic limit.

Moduli of elasticity

Young's modulus: $\displaystyle Y = \frac{FL}{A\,\Delta L}$
Shear modulus: $\displaystyle G = \frac{F/A}{\Delta x / L} = \frac{F/A}{\theta}$
Bulk modulus: $\displaystyle B = -\frac{p}{\Delta V / V}$
Compressibility: $\displaystyle k = \frac{1}{B}$
Elastic energy per unit volume: $\displaystyle u = \tfrac12 \times \text{stress} \times \text{strain}$

9. Mechanical Properties of Fluids

Pressure and buoyancy

Pressure at depth: $\displaystyle p = p_0 + \rho g h$
Pascal's law (hydraulic lift): $\displaystyle \frac{F_1}{A_1} = \frac{F_2}{A_2}$
Archimedes' principle: $\displaystyle F_b = \rho_{\text{fluid}}\, V_{\text{displaced}}\, g$

Flowing fluids

Continuity equation: $\displaystyle A_1v_1 = A_2v_2$
Bernoulli's equation: $\displaystyle p + \tfrac12\rho v^2 + \rho g h = \text{constant}$
For steady, incompressible, non-viscous flow.
Torricelli's law: $\displaystyle v = \sqrt{2gh}$

Viscosity

Viscous force: $\displaystyle F = \eta A\frac{dv}{dx}$
Stokes' law: $\displaystyle F = 6\pi\eta r v$
Terminal velocity: $\displaystyle v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$
ρ is the sphere's density and σ the fluid's.
Reynolds number: $\displaystyle R_e = \frac{\rho v d}{\eta}$

Surface tension

Surface tension: $\displaystyle S = \frac{F}{L}$
Excess pressure in a drop: $\displaystyle \Delta p = \frac{2S}{r}$
Excess pressure in a soap bubble: $\displaystyle \Delta p = \frac{4S}{r}$
A bubble has two surfaces.
Capillary rise: $\displaystyle h = \frac{2S\cos\theta}{\rho g r}$

10. Thermal Properties of Matter

Temperature and expansion

Temperature scales: $\displaystyle \frac{T_C}{100} = \frac{T_F - 32}{180}, \quad T_K = T_C + 273.15$
Linear expansion: $\displaystyle \Delta L = \alpha L\,\Delta T$
Area and volume expansion: $\displaystyle \Delta A = \beta A\,\Delta T, \quad \Delta V = \gamma V\,\Delta T$
Relation between coefficients: $\displaystyle \beta = 2\alpha, \quad \gamma = 3\alpha$
For isotropic solids.

Calorimetry

Heat and specific heat: $\displaystyle Q = mc\,\Delta T$
Latent heat: $\displaystyle Q = mL$

Heat transfer

Conduction: $\displaystyle \frac{Q}{t} = \frac{kA(T_1 - T_2)}{L}$
Stefan-Boltzmann law: $\displaystyle P = e\sigma A T^4$
Wien's displacement law: $\displaystyle \lambda_m T = b$
Newton's law of cooling: $\displaystyle \frac{dT}{dt} = -k(T - T_s)$

11. Thermodynamics

First law

First law of thermodynamics: $\displaystyle \Delta Q = \Delta U + \Delta W$
Here W is the work done by the gas.
Work by a gas: $\displaystyle W = \int_{V_1}^{V_2} p\,dV$
Change in internal energy: $\displaystyle \Delta U = nC_V\,\Delta T$
For an ideal gas, in any process.
Mayer's relation: $\displaystyle C_p - C_V = R$
Ratio of specific heats: $\displaystyle \gamma = \frac{C_p}{C_V}$

Processes (ideal gas)

Isothermal work: $\displaystyle W = nRT\ln\frac{V_2}{V_1}$
Adiabatic relation: $\displaystyle pV^\gamma = \text{constant}$
Adiabatic work: $\displaystyle W = \frac{nR(T_1 - T_2)}{\gamma - 1}$
Isobaric work: $\displaystyle W = p\,\Delta V$
Isochoric work: $\displaystyle W = 0$

Engines

Efficiency of a heat engine: $\displaystyle \eta = \frac{W}{Q_1} = 1 - \frac{Q_2}{Q_1}$
Carnot efficiency: $\displaystyle \eta = 1 - \frac{T_2}{T_1}$
Temperatures in kelvin.
Coefficient of performance: $\displaystyle \beta = \frac{Q_2}{W}$
For a refrigerator.

12. Kinetic Theory

Ideal gas

Ideal gas law: $\displaystyle pV = nRT = Nk_BT$
Pressure from kinetic theory: $\displaystyle p = \tfrac13\rho\,\overline{v^2}$
Average kinetic energy of a molecule: $\displaystyle \bar K = \tfrac32 k_BT$

Molecular speeds

RMS speed: $\displaystyle v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$
M is the molar mass in kg/mol.
Mean speed: $\displaystyle v_{\text{mean}} = \sqrt{\frac{8RT}{\pi M}}$
Most probable speed: $\displaystyle v_{\text{mp}} = \sqrt{\frac{2RT}{M}}$

Degrees of freedom

Energy per degree of freedom: $\displaystyle E = \tfrac12 k_BT$
Specific heats from f: $\displaystyle C_V = \frac{f}{2}R, \quad C_p = \left(\frac{f}{2} + 1\right)R, \quad \gamma = 1 + \frac{2}{f}$
f = 3 for monatomic and 5 for a rigid diatomic gas.
Mean free path: $\displaystyle \lambda = \frac{1}{\sqrt2\,n\pi d^2}$

13. Oscillations

Motion

Displacement: $\displaystyle x = A\sin(\omega t + \phi)$
Velocity: $\displaystyle v = \omega\sqrt{A^2 - x^2}$
Largest at the mean position, where v = Aω.
Acceleration: $\displaystyle a = -\omega^2 x$
Largest at the extremes, where |a| = Aω².
Angular frequency: $\displaystyle \omega = 2\pi\nu = \frac{2\pi}{T}$

Energy

Kinetic energy: $\displaystyle K = \tfrac12 m\omega^2(A^2 - x^2)$
Potential energy: $\displaystyle U = \tfrac12 m\omega^2 x^2$
Total energy: $\displaystyle E = \tfrac12 m\omega^2 A^2$
Stays constant.

Common systems

Spring and mass: $\displaystyle T = 2\pi\sqrt{\frac{m}{k}}$
Simple pendulum: $\displaystyle T = 2\pi\sqrt{\frac{L}{g}}$
For small angles.
Springs in series: $\displaystyle \frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2}$
Springs in parallel: $\displaystyle k = k_1 + k_2$

14. Waves

Waves

Wave equation: $\displaystyle y = a\sin(kx - \omega t + \phi)$
Wave number and speed: $\displaystyle k = \frac{2\pi}{\lambda}, \quad v = \nu\lambda = \frac{\omega}{k}$
Speed on a string: $\displaystyle v = \sqrt{\frac{T}{\mu}}$
μ is mass per unit length.
Speed of sound (Laplace): $\displaystyle v = \sqrt{\frac{\gamma P}{\rho}}$

Standing waves

String fixed at both ends: $\displaystyle \nu_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \dots$
Open pipe: $\displaystyle \nu_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \dots$
Closed pipe: $\displaystyle \nu_n = \frac{(2n - 1)v}{4L}, \quad n = 1, 2, 3, \dots$
Only odd harmonics exist.

Beats

Beat frequency: $\displaystyle \nu_{\text{beat}} = |\nu_1 - \nu_2|$

All Class 11 Physics formulas

Errors

Dimensions

Definitions

Equations of motion (constant acceleration)

Free fall and graphs

Vectors

Projectile motion

Circular motion

Newton's laws and momentum

Friction

Circular motion

Work and energy

Power

Collisions

Centre of mass

Rotation

Moment of inertia

Rolling without slipping

Force and acceleration

Energy and escape

Orbits

Stress and strain

Moduli of elasticity

Pressure and buoyancy

Flowing fluids

Viscosity

Surface tension

Temperature and expansion

Calorimetry

Heat transfer

First law

Processes (ideal gas)

Engines

Ideal gas

Molecular speeds

Degrees of freedom

Motion

Energy

Common systems

Waves

Standing waves

Beats

Try it on your weakest chapter.