Class 11 Physics Formula Sheet 2026-27 (Free PDF), All 14 Chapters

Every formula from all 14 NCERT Class 11 Physics chapters in one place, checked against the NCERT textbook (Reprint 2026-27) and the official CBSE 2026-27 syllabus. Qualitative-only topics in the 2026-27 syllabus are marked separately.

How to use this sheet. Plain boxes cover the CBSE 2026-27 board-exam syllabus. Amber boxes marked ★ Qualitative only are topics the 2026-27 syllabus lists for qualitative treatment or as a statement only, so expect conceptual questions rather than derivations or numericals. Grey boxes marked ◇ Not in 2026-27 syllabus are in the NCERT book but not in the current syllabus. They are kept for reference and JEE/NEET. Constants: $g = 9.8\,\text{m s}^{-2}$, $G = 6.67 \times 10^{-11}\,\text{N m}^2\,\text{kg}^{-2}$, $R = 8.314\,\text{J mol}^{-1}\,\text{K}^{-1}$, $k_B = 1.38 \times 10^{-23}\,\text{J K}^{-1}$, $N_A = 6.022 \times 10^{23}\,\text{mol}^{-1}$, $\sigma = 5.67 \times 10^{-8}\,\text{W m}^{-2}\,\text{K}^{-4}$, $1\,\text{atm} = 1.01 \times 10^{5}\,\text{Pa}$.

Units and Measurement
Motion in a Straight Line
Motion in a Plane
Laws of Motion
Work, Energy and Power
System of Particles and Rotational Motion
Gravitation
Mechanical Properties of Solids
Mechanical Properties of Fluids
Thermal Properties of Matter
Thermodynamics
Kinetic Theory
Oscillations
Waves

1Units and Measurement

SI base quantities and units

Quantity	Unit	Symbol	Dimension
Length	metre	m	[L]
Mass	kilogram	kg	[M]
Time	second	s	[T]
Electric current	ampere	A	[A]
Thermodynamic temperature	kelvin	K	[K]
Amount of substance	mole	mol	[mol]
Luminous intensity	candela	cd	[cd]

Common dimensional formulae

Velocity $[LT^{-1}]$; acceleration $[LT^{-2}]$
Force $[MLT^{-2}]$; momentum, impulse $[MLT^{-1}]$
Work, energy $[ML^2T^{-2}]$; power $[ML^2T^{-3}]$
Pressure, stress, elastic modulus $[ML^{-1}T^{-2}]$
Density $[ML^{-3}]$; $G$: $[M^{-1}L^3T^{-2}]$
Strain, angle, refractive index: dimensionless

Dimensional analysis

Only quantities with the same dimensions can be added, subtracted or equated
Uses: check consistency of an equation, convert units, deduce a relation between quantities
A dimensionally correct equation may still be wrong (dimensionless constants are missed); a dimensionally wrong one is always wrong.

Significant figures

All non-zero digits are significant; zeros between them are significant
Leading zeros (0.0023) are not significant; trailing zeros after the decimal point are
Multiply/divide: keep as many significant figures as the least precise value
Add/subtract: keep as many decimal places as the least precise value
Relative error $= \dfrac{\Delta a}{a} \times 100\%$

2Motion in a Straight Line

Velocity and acceleration

Average velocity $\bar{v} = \dfrac{\Delta x}{\Delta t}$; average speed $=\dfrac{\text{total path length}}{\text{total time}}$
Instantaneous: $v = \lim\limits_{\Delta t \to 0}\dfrac{\Delta x}{\Delta t} = \dfrac{dx}{dt}$
$\bar{a} = \dfrac{\Delta v}{\Delta t}$; $\ a = \dfrac{dv}{dt} = \dfrac{d^2x}{dt^2}$
Average speed $\ge$ |average velocity| over any interval.

Graphs

Slope of $x$-$t$ graph = velocity
Slope of $v$-$t$ graph = acceleration
Area under $v$-$t$ graph = displacement
Uniform acceleration: $x$-$t$ is a parabola, $v$-$t$ a straight line

Kinematic equations (constant $a$)

$$v = v_0 + at$$
$$x = v_0 t + \tfrac{1}{2}at^2$$
$$v^2 = v_0^2 + 2ax$$
$x = \dfrac{v + v_0}{2}\,t$
If the particle starts at $x_0$, replace $x$ by $x - x_0$. Free fall: $a = -g$ (upward positive).

3Motion in a Plane

Vectors

$\vec{A} = A_x\hat{i} + A_y\hat{j}$; $A_x = A\cos\theta$, $A_y = A\sin\theta$
$A = \sqrt{A_x^2 + A_y^2}$, $\ \tan\theta = A_y/A_x$
Unit vector $\hat{n} = \vec{A}/|\vec{A}|$
$\vec{A}+\vec{B} = \vec{B}+\vec{A}$; $(\vec{A}+\vec{B})+\vec{C} = \vec{A}+(\vec{B}+\vec{C})$
$\vec{A}-\vec{B} = \vec{A}+(-\vec{B})$

Addition of two vectors

Analytical: $R_x = A_x + B_x$, $R_y = A_y + B_y$
Law of cosines: $$R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$$
Direction: $\tan\alpha = \dfrac{B\sin\theta}{A + B\cos\theta}$ (angle with $\vec{A}$)

Motion in a plane

$\vec{r} = x\hat{i} + y\hat{j}$; $\ \Delta\vec{r} = \Delta x\,\hat{i} + \Delta y\,\hat{j}$
$\vec{v} = \dfrac{d\vec{r}}{dt} = v_x\hat{i} + v_y\hat{j}$; $\ \vec{a} = \dfrac{d\vec{v}}{dt}$
Constant $\vec{a}$: $\vec{v} = \vec{v}_0 + \vec{a}t$, $\ \vec{r} = \vec{r}_0 + \vec{v}_0t + \tfrac{1}{2}\vec{a}t^2$
Velocity is always tangent to the path.

Projectile motion

$x = (v_0\cos\theta_0)t$, $\ y = (v_0\sin\theta_0)t - \tfrac{1}{2}gt^2$
Path (parabola): $y = (\tan\theta_0)x - \dfrac{g}{2(v_0\cos\theta_0)^2}x^2$
Time of flight: $T_f = \dfrac{2v_0\sin\theta_0}{g}$
Maximum height: $h_m = \dfrac{(v_0\sin\theta_0)^2}{2g}$
Range: $$R = \frac{v_0^2\sin 2\theta_0}{g}$$ maximum at $\theta_0 = 45^\circ$: $R_m = v_0^2/g$

Uniform circular motion

$v = \omega R$; $\ \omega = \dfrac{\Delta\theta}{\Delta t} = 2\pi\nu = \dfrac{2\pi}{T}$
Centripetal acceleration (towards centre): $$a_c = \frac{v^2}{R} = \omega^2 R = 4\pi^2\nu^2 R$$

4Laws of Motion

Newton's laws

First law: if net external force is zero, acceleration is zero
Momentum $\vec{p} = m\vec{v}$
Second law: $$\vec{F} = \frac{d\vec{p}}{dt} = m\vec{a}$$ 1 N = 1 kg m s$^{-2}$
Third law: $\vec{F}_{AB} = -\vec{F}_{BA}$ (act on different bodies)

Impulse and momentum conservation

Impulse $= F\Delta t = \Delta p$ (change in momentum)
Isolated system: total momentum is conserved, $\vec{p}_A + \vec{p}_B = \vec{p}'_A + \vec{p}'_B$
Recoil of a gun: $m_b v_b = -M_g V_g$.

Equilibrium of a particle

$\vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \dots = 0$
Componentwise: $\sum F_x = 0$, $\sum F_y = 0$, $\sum F_z = 0$

Friction

Static: $f_s \le (f_s)_{max} = \mu_s N$
Kinetic: $f_k = \mu_k N$, with $\mu_k < \mu_s$
Independent of the area of contact
Rolling friction is much smaller than sliding friction; lubricants reduce friction.

Circular motion

Centripetal force $f_c = \dfrac{mv^2}{R}$
Level road (friction provides $f_c$): $v_{max} = \sqrt{\mu_s R g}$
Banked road with friction: $$v_{max} = \sqrt{Rg\,\frac{\mu_s + \tan\theta}{1 - \mu_s\tan\theta}}$$
Optimum speed (no friction needed): $v_o = \sqrt{Rg\tan\theta}$

5Work, Energy and Power

Scalar product and work

$\vec{A}\cdot\vec{B} = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z$
$\hat{i}\cdot\hat{i} = \hat{j}\cdot\hat{j} = \hat{k}\cdot\hat{k} = 1$; $\hat{i}\cdot\hat{j} = \hat{j}\cdot\hat{k} = \hat{k}\cdot\hat{i} = 0$
Constant force: $W = \vec{F}\cdot\vec{d} = Fd\cos\theta$
Variable force: $$W = \int_{x_i}^{x_f} F(x)\,dx$$ (area under the $F$-$x$ graph)

Kinetic energy, work-energy theorem

$K = \tfrac{1}{2}mv^2 = \dfrac{p^2}{2m}$
Work-energy theorem: $$K_f - K_i = W_{net}$$

Potential energy and conservation

Conservative force: $F(x) = -\dfrac{dV}{dx}$; work is path-independent and zero round a closed path
Gravitational PE near the surface: $V = mgh$
Spring ($F = -kx$): $V = \tfrac{1}{2}kx^2$
Only conservative forces acting: $K + V = \text{constant}$

Power

$P_{av} = \dfrac{W}{t}$; instantaneous $P = \dfrac{dW}{dt} = \vec{F}\cdot\vec{v}$
1 hp = 746 W; 1 kWh = $3.6 \times 10^{6}$ J

Collisions

Momentum is conserved in every collision; KE is conserved only in an elastic collision
Completely inelastic (bodies stick): $v = \dfrac{m_1 v_{1i}}{m_1 + m_2}$ ($m_2$ at rest), KE lost $= \dfrac{1}{2}\dfrac{m_1 m_2}{m_1+m_2}v_{1i}^2$
Elastic, 1D ($m_2$ at rest): $$v_{1f} = \frac{(m_1 - m_2)}{m_1 + m_2}\,v_{1i}, \qquad v_{2f} = \frac{2m_1}{m_1 + m_2}\,v_{1i}$$
Equal masses in an elastic head-on collision exchange velocities. In 2D, apply momentum conservation along $x$ and $y$ separately.

6System of Particles and Rotational Motion

Centre of mass

Two particles on a line: $X = \dfrac{m_1x_1 + m_2x_2}{m_1 + m_2}$
$n$ particles: $$\vec{R} = \frac{\sum m_i\vec{r}_i}{M}$$
Rigid body: $X = \dfrac{1}{M}\int x\,dm$; uniform rod: CM at its centre
$\vec{V} = \vec{P}/M$; $\ M\vec{A} = \vec{F}_{ext}$
If $\vec{F}_{ext} = 0$, total linear momentum is constant and the CM moves uniformly.

Vector product

$|\vec{a}\times\vec{b}| = ab\sin\theta$, direction by the right-hand rule
$\vec{a}\times\vec{b} = -\vec{b}\times\vec{a}$; $\ \vec{a}\times\vec{a} = 0$
$\hat{i}\times\hat{j} = \hat{k}$, $\hat{j}\times\hat{k} = \hat{i}$, $\hat{k}\times\hat{i} = \hat{j}$

Angular kinematics

$\omega = \dfrac{d\theta}{dt}$, $\ \vec{v} = \vec{\omega}\times\vec{r}$, $\ v = \omega r$
$\alpha = \dfrac{d\omega}{dt}$
Constant $\alpha$: $\omega = \omega_0 + \alpha t$; $\ \theta = \theta_0 + \omega_0t + \tfrac{1}{2}\alpha t^2$; $\ \omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Torque and angular momentum

Torque $\vec{\tau} = \vec{r}\times\vec{F}$, $\ \tau = rF\sin\theta$
Angular momentum $\vec{l} = \vec{r}\times\vec{p}$, $\ l = rp\sin\theta$
$$\frac{d\vec{L}}{dt} = \vec{\tau}_{ext}$$
Fixed axis: $L = I\omega$
Conservation: if $\tau_{ext} = 0$, $I\omega = \text{constant}$

Equilibrium of a rigid body

Translational: $\sum\vec{F}_i = 0$
Rotational: $\sum\vec{\tau}_i = 0$
Principle of moments (lever): load arm $\times$ load = effort arm $\times$ effort
Mechanical advantage $= \dfrac{F_1}{F_2} = \dfrac{d_2}{d_1}$
Couple: equal, opposite forces with different lines of action; net force zero, net torque non-zero

Rotational dynamics

Moment of inertia $I = \sum m_i r_i^2$, SI unit kg m$^2$
Radius of gyration: $I = Mk^2$
Rotational KE $= \tfrac{1}{2}I\omega^2$
$$\tau = I\alpha$$
Work $W = \tau\theta$; power $P = \tau\omega$

Moments of inertia (values only, no derivation)

Body	Axis	$I$
Thin ring, radius $R$	perpendicular to plane, through centre	$MR^2$
Thin ring, radius $R$	diameter	$MR^2/2$
Thin rod, length $L$	perpendicular to rod, through midpoint	$ML^2/12$
Circular disc, radius $R$	perpendicular to disc, through centre	$MR^2/2$
Circular disc, radius $R$	diameter	$MR^2/4$
Hollow cylinder, radius $R$	axis of cylinder	$MR^2$
Solid cylinder, radius $R$	axis of cylinder	$MR^2/2$
Solid sphere, radius $R$	diameter	$2MR^2/5$

Linear vs rotational motion

Linear	Rotational (fixed axis)
Displacement $x$, velocity $v$, acceleration $a$	$\theta$, $\omega$, $\alpha$
Mass $M$	Moment of inertia $I$
$F = Ma$	$\tau = I\alpha$
$p = Mv$; $K = \tfrac{1}{2}Mv^2$	$L = I\omega$; $K = \tfrac{1}{2}I\omega^2$
$W = \int F\,ds$; $P = Fv$	$W = \int\tau\,d\theta$; $P = \tau\omega$

7Gravitation

Kepler's laws

Orbits: planets move in ellipses with the Sun at one focus
Areas: the radius vector sweeps equal areas in equal times (angular momentum is conserved)
Periods: $T^2 \propto a^3$; circular orbit: $$T^2 = \frac{4\pi^2}{GM_s}R^3$$

Universal law of gravitation

$$F = \frac{Gm_1m_2}{r^2}$$
Superposition: $\vec{F}_R = \sum\vec{F}_i$
Outside a uniform shell/sphere: acts as if all mass were at the centre
Inside a uniform shell: force is zero

Acceleration due to gravity

At the surface: $g = \dfrac{GM_E}{R_E^2}$
At height $h$: $g(h) = \dfrac{GM_E}{(R_E + h)^2}$ $\approx g\left(1 - \dfrac{2h}{R_E}\right)$ for $h \ll R_E$
At depth $d$: $$g(d) = g\left(1 - \frac{d}{R_E}\right)$$
$g$ decreases with both height and depth; it is zero at the centre.

Potential energy and escape speed

$V = -\dfrac{Gm_1m_2}{r}$ (zero at infinity)
Gravitational potential $= -\dfrac{GM}{r}$ (PE per unit mass)
Total energy of $m$ near $M$: $E = \tfrac{1}{2}mv^2 - \dfrac{GMm}{r}$
Escape speed: $$v_e = \sqrt{\frac{2GM_E}{R_E}} = \sqrt{2gR_E} \approx 11.2\ \text{km s}^{-1}$$

Earth satellites

Orbital speed at height $h$: $v = \sqrt{\dfrac{GM_E}{R_E + h}}$; close to the surface $v \approx \sqrt{gR_E}$
Period: $T = 2\pi\sqrt{\dfrac{(R_E + h)^3}{GM_E}}$; near the surface $T_0 = 2\pi\sqrt{R_E/g} \approx 85$ min
Energy in a circular orbit of radius $r$: $$K = \frac{GMm}{2r}, \qquad V = -\frac{GMm}{r}, \qquad E = -\frac{GMm}{2r}$$
Total energy is negative for any bound orbit; $K = -E$ and $V = 2E$.

8Mechanical Properties of Solids

Stress, strain, Hooke's law

Stress $= F/A$ (N m$^{-2}$ = Pa)
Longitudinal strain $= \Delta L/L$; volume strain $= \Delta V/V$
Hooke's law: stress $= k \times$ strain (within the elastic limit)

Young's and bulk modulus

$$Y = \frac{F/A}{\Delta L/L} = \frac{FL}{A\,\Delta L}$$
Bulk modulus: $B = -\dfrac{p}{\Delta V/V}$ (negative sign: volume decreases)
Compressibility $k = 1/B$

Shear modulus (qualitative only)

Shear strain $= \Delta x/L = \tan\theta \approx \theta$
$G = \dfrac{F/A}{\Delta x/L}$; for most materials $G \approx Y/3$
Only solids have a shear modulus

Poisson's ratio and elastic energy

Poisson's ratio $= \dfrac{\Delta d/d}{\Delta L/L}$ (lateral strain / longitudinal strain); dimensionless
Energy per unit volume in a stretched wire: $$u = \tfrac{1}{2}\times\text{stress}\times\text{strain}$$
Total $U = \tfrac{1}{2}\,\dfrac{Y A (\Delta L)^2}{L}$

9Mechanical Properties of Fluids

Pressure

$P = F/A$; 1 atm $= 1.01\times10^5$ Pa, 1 bar $= 10^5$ Pa, 1 torr = 1 mm Hg = 133 Pa
At depth $h$: $$P = P_a + \rho g h$$
Gauge pressure $= P - P_a = \rho gh$
Pascal's law, hydraulic lift: $\dfrac{F_1}{A_1} = \dfrac{F_2}{A_2}$

Streamline flow and Bernoulli

Equation of continuity: $A_1v_1 = A_2v_2$ (incompressible, steady flow)
Bernoulli: $$P + \tfrac{1}{2}\rho v^2 + \rho g h = \text{constant}$$
Torricelli's law (open tank): $v_1 = \sqrt{2gh}$
Dynamic lift (aerofoil, spinning ball) comes from the pressure difference that Bernoulli predicts.

Viscosity

$\eta = \dfrac{F/A}{v/l}$ (shear stress / strain rate); SI unit Pa s
Stokes' law: $F = 6\pi\eta a v$
Terminal velocity: $$v_t = \frac{2a^2(\rho - \sigma)g}{9\eta}$$ ($\rho$ sphere, $\sigma$ fluid)

Surface tension

$S = F/l$ = surface energy per unit area (N m$^{-1}$ = J m$^{-2}$)
Excess pressure in a drop or an air bubble in liquid: $P_i - P_o = \dfrac{2S}{r}$
Soap bubble (two surfaces): $P_i - P_o = \dfrac{4S}{r}$
Capillary rise: $$h = \frac{2S\cos\theta}{\rho g a}$$
$\theta < 90^\circ$: liquid wets the surface and rises (water-glass). $\theta > 90^\circ$: it is depressed (mercury-glass).

10Thermal Properties of Matter

Temperature scales and gas law

$t_F = \tfrac{9}{5}t_C + 32$
$T = t_C + 273.15$
Ideal gas: $PV = \mu RT$

Thermal expansion

Linear: $\dfrac{\Delta l}{l} = \alpha_l\,\Delta T$
Area: $\dfrac{\Delta A}{A} = \alpha_a\,\Delta T$, with $\alpha_a = 2\alpha_l$
Volume: $\dfrac{\Delta V}{V} = \alpha_V\,\Delta T$, with $$\alpha_V = 3\alpha_l$$
Ideal gas at constant $P$: $\alpha_V = 1/T$
Water contracts from 0 °C to 4 °C (anomalous expansion); it is densest at 4 °C.

Specific heat and calorimetry

$Q = mc\,\Delta T$; specific heat $s = \dfrac{1}{m}\dfrac{\Delta Q}{\Delta T}$
Molar specific heat $C = \dfrac{1}{\mu}\dfrac{\Delta Q}{\Delta T}$; $C_p$ at constant pressure, $C_v$ at constant volume
Calorimetry: heat lost = heat gained (isolated system)

Change of state

Latent heat: $Q = mL$ (no change in temperature)
Water: $L_f = 3.33\times10^5$ J kg$^{-1}$, $L_v = 22.6\times10^5$ J kg$^{-1}$

Conduction

Rate of heat flow through a bar: $$H = KA\,\frac{T_C - T_D}{L}$$
$K$ = thermal conductivity (W m$^{-1}$ K$^{-1}$)
Convection needs bulk flow of a fluid; radiation needs no medium.

Blackbody radiation (qualitative)

Wien's displacement law: $\lambda_m T = b$, $\ b = 2.9\times10^{-3}$ m K
Stefan-Boltzmann law: $H = A\sigma T^4$; real body $H = eA\sigma T^4$ ($e$ = emissivity)
Net loss to surroundings at $T_s$: $H = e\sigma A(T^4 - T_s^4)$

Newton's law of cooling

$$-\frac{dQ}{dt} = k(T_2 - T_1)$$ ($T_2$ body, $T_1$ surroundings; small differences only)

11Thermodynamics

Zeroth and first laws

Zeroth law: A and B each in thermal equilibrium with C are in equilibrium with each other
First law: $$\Delta Q = \Delta U + \Delta W$$ ($\Delta Q$ supplied to the system, $\Delta W$ done by the system)
$\Delta W = P\,\Delta V$; $\ W = \int P\,dV$
$U$ is a state variable; $Q$ and $W$ are not.

Specific heats

Ideal gas (Mayer's relation): $$C_p - C_v = R$$
$\gamma = C_p/C_v$
1 cal = 4.186 J
Water: $s = 4186$ J kg$^{-1}$ K$^{-1}$

Thermodynamic processes (ideal gas)

Process	Condition	Work done by gas	First law
Isothermal	$T$ constant, $PV$ constant	$W = \mu RT\ln\dfrac{V_2}{V_1}$	$\Delta U = 0$, $Q = W$
Adiabatic	$Q = 0$, $PV^\gamma$ constant	$W = \dfrac{\mu R(T_1 - T_2)}{\gamma - 1}$	$\Delta U = -W$
Isochoric	$V$ constant	$W = 0$	$Q = \Delta U$
Isobaric	$P$ constant	$W = P(V_2 - V_1) = \mu R(T_2 - T_1)$	$Q = \Delta U + W$
Cyclic	returns to initial state	area enclosed on the $P$-$V$ diagram	$\Delta U = 0$, $Q = W$

Second law, reversibility

Kelvin-Planck: no process can absorb heat from a reservoir and convert it completely into work
Clausius: no process can have as its sole result the transfer of heat from a colder to a hotter body
Reversible = quasi-static and non-dissipative

Carnot engine

Two isothermal + two adiabatic reversible steps between $T_1$ (source) and $T_2$ (sink)
$$\eta = \frac{W}{Q_1} = 1 - \frac{T_2}{T_1}$$
$Q_1/Q_2 = T_1/T_2$; no engine between the same two temperatures beats the Carnot engine

12Kinetic Theory

Ideal gas and gas laws

$PV = \mu RT = k_BNT$; $\ k_B = R/N_A$
Boyle's law: $PV$ constant at fixed $T$
Charles' law: $V/T$ constant at fixed $P$
Dalton's law: $P = P_1 + P_2 + \dots$ ($P_i = \mu_i RT/V$)

Pressure and temperature

$$P = \tfrac{1}{3}nm\overline{v^2}$$ ($n$ = number density)
$PV = \tfrac{2}{3}E$, where $E = \tfrac{3}{2}k_BNT$ (translational KE)
Mean KE per molecule: $\tfrac{1}{2}m\overline{v^2} = \tfrac{3}{2}k_BT$
$$v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3k_BT}{m}}$$

Equipartition (statement only)

Each translational or rotational degree of freedom carries $\tfrac{1}{2}k_BT$
Each vibrational frequency carries $2\times\tfrac{1}{2}k_BT = k_BT$

Specific heats of gases (from equipartition)

Gas	$U$ per mole	$C_v$	$C_p$	$\gamma$
Monatomic	$\tfrac{3}{2}RT$	$\tfrac{3}{2}R$	$\tfrac{5}{2}R$	$5/3$
Diatomic (rigid)	$\tfrac{5}{2}RT$	$\tfrac{5}{2}R$	$\tfrac{7}{2}R$	$7/5$
Diatomic (with vibration)	$\tfrac{7}{2}RT$	$\tfrac{7}{2}R$	$\tfrac{9}{2}R$	$9/7$
Polyatomic	$(3 + f)RT$	$(3 + f)R$	$(4 + f)R$	$\dfrac{4 + f}{3 + f}$

$f$ = number of vibrational modes of a polyatomic molecule.

Mean free path

$$l = \frac{1}{\sqrt{2}\,n\pi d^2}$$ ($d$ = molecular diameter)

Specific heat of solids

Equipartition: $C = 3R$ per mole (agrees with experiment at ordinary temperatures)

13Oscillations

Periodic motion

$\nu = 1/T$ (1 Hz = 1 s$^{-1}$)
$\omega = \dfrac{2\pi}{T} = 2\pi\nu$
SHM is the projection of uniform circular motion on a diameter.

Simple harmonic motion

$$x(t) = A\cos(\omega t + \phi)$$ ($\omega t + \phi$ = phase)
$v(t) = -\omega A\sin(\omega t + \phi)$; max speed $v_m = \omega A$
$a(t) = -\omega^2 A\cos(\omega t + \phi) = -\omega^2 x$; max $a_m = \omega^2 A$
Restoring force: $F = -kx$, with $k = m\omega^2$

Energy in SHM

$K = \tfrac{1}{2}mv^2 = \tfrac{1}{2}k A^2\sin^2(\omega t + \phi)$
$U = \tfrac{1}{2}kx^2 = \tfrac{1}{2}kA^2\cos^2(\omega t + \phi)$
Total $$E = K + U = \tfrac{1}{2}kA^2$$ (constant without friction)

Spring and pendulum

Mass on a spring: $\omega = \sqrt{\dfrac{k}{m}}$, $$T = 2\pi\sqrt{\frac{m}{k}}$$
Simple pendulum (small angles): $$T = 2\pi\sqrt{\frac{L}{g}}$$
The pendulum's period does not depend on the bob's mass or (for small swings) the amplitude.

14Waves

Progressive wave

$$y(x, t) = a\sin(kx - \omega t + \phi)$$ (travels in $+x$)
Angular wave number $k = \dfrac{2\pi}{\lambda}$; $\ \omega = \dfrac{2\pi}{T} = 2\pi\nu$
Wave speed: $v = \dfrac{\omega}{k} = \lambda\nu = \dfrac{\lambda}{T}$

Speed of waves

Stretched string (tension $T$, linear mass density $\mu$): $$v = \sqrt{\frac{T}{\mu}}$$
Fluid: $v = \sqrt{B/\rho}$; solid bar: $v = \sqrt{Y/\rho}$
Gas: Newton $v = \sqrt{P/\rho}$; Laplace's correction $v = \sqrt{\gamma P/\rho}$

Superposition and reflection

$y = \sum f_i(x - vt)$
Two waves of phase difference $\phi$: $y = \left[2a\cos\tfrac{\phi}{2}\right]\sin\!\left(kx - \omega t + \tfrac{\phi}{2}\right)$
$\phi = 0, 2\pi, \dots$: constructive; $\phi = \pi$: destructive
Rigid boundary: reflected with phase change $\pi$, $y_r = -a\sin(kx + \omega t)$
Open boundary: no phase change, $y_r = a\sin(kx + \omega t)$

Standing waves

$y(x,t) = [2a\sin kx]\cos\omega t$
Node to node (or antinode to antinode) $= \lambda/2$
String fixed at both ends: $$\nu_n = \frac{nv}{2L},\ n = 1, 2, 3, \dots$$

Organ pipes

Open at both ends: $\nu_n = \dfrac{nv}{2L}$, $n = 1, 2, 3, \dots$ (all harmonics)
Closed at one end: $\nu = \left(n + \tfrac{1}{2}\right)\dfrac{v}{2L}$, $n = 0, 1, 2, \dots$ (odd harmonics only)
Fundamental mode = lowest frequency = first harmonic.

Beats

Beat frequency: $$\nu_{beat} = \nu_1 - \nu_2$$
Heard when two waves of slightly different frequencies superpose

Body	Axis	\(I\)
Thin ring, radius \(R\)	perpendicular to plane, through centre	\(MR^2\)
Thin ring, radius \(R\)	diameter	\(MR^2/2\)
Thin rod, length \(L\)	perpendicular to rod, through midpoint	\(ML^2/12\)
Circular disc, radius \(R\)	perpendicular to disc, through centre	\(MR^2/2\)
Circular disc, radius \(R\)	diameter	\(MR^2/4\)
Hollow cylinder, radius \(R\)	axis of cylinder	\(MR^2\)
Solid cylinder, radius \(R\)	axis of cylinder	\(MR^2/2\)
Solid sphere, radius \(R\)	diameter	\(2MR^2/5\)

Linear	Rotational (fixed axis)
Displacement \(x\), velocity \(v\), acceleration \(a\)	\(\theta\), \(\omega\), \(\alpha\)
Mass \(M\)	Moment of inertia \(I\)
\(F = Ma\)	\(\tau = I\alpha\)
\(p = Mv\); \(K = \tfrac{1}{2}Mv^2\)	\(L = I\omega\); \(K = \tfrac{1}{2}I\omega^2\)
\(W = \int F\,ds\); \(P = Fv\)	\(W = \int\tau\,d\theta\); \(P = \tau\omega\)

Process	Condition	Work done by gas	First law
Isothermal	\(T\) constant, \(PV\) constant	\(W = \mu RT\ln\dfrac{V_2}{V_1}\)	\(\Delta U = 0\), \(Q = W\)
Adiabatic	\(Q = 0\), \(PV^\gamma\) constant	\(W = \dfrac{\mu R(T_1 - T_2)}{\gamma - 1}\)	\(\Delta U = -W\)
Isochoric	\(V\) constant	\(W = 0\)	\(Q = \Delta U\)
Isobaric	\(P\) constant	\(W = P(V_2 - V_1) = \mu R(T_2 - T_1)\)	\(Q = \Delta U + W\)
Cyclic	returns to initial state	area enclosed on the \(P\)-\(V\) diagram	\(\Delta U = 0\), \(Q = W\)

Gas	\(U\) per mole	\(C_v\)	\(C_p\)	\(\gamma\)
Monatomic	\(\tfrac{3}{2}RT\)	\(\tfrac{3}{2}R\)	\(\tfrac{5}{2}R\)	\(5/3\)
Diatomic (rigid)	\(\tfrac{5}{2}RT\)	\(\tfrac{5}{2}R\)	\(\tfrac{7}{2}R\)	\(7/5\)
Diatomic (with vibration)	\(\tfrac{7}{2}RT\)	\(\tfrac{7}{2}R\)	\(\tfrac{9}{2}R\)	\(9/7\)
Polyatomic	\((3 + f)RT\)	\((3 + f)R\)	\((4 + f)R\)	\(\dfrac{4 + f}{3 + f}\)