Formula Sheet
Every key formula of the course on one page, grouped by chapter, with a link back to the section that develops it.
Chapter 1 · Units and Measurement
Value and unit§ 1.1 →
$Q=n\,u$; the same quantity in two units: $n_1u_1=n_2u_2$
Speed conversion§ 1.1 →
$1\ \text{km/h}=\tfrac{5}{18}\ \text{m/s}$; $\ 1\ \text{m/s}=3.6\ \text{km/h}$
Kilowatt-hour§ 1.1 →
$1\ \text{kWh}=3.6\times10^6\ \text{J}$
Plane angle (radian)§ 1.1 →
$\theta=\tfrac{s}{r}$ (rad); full circle $2\pi\ \text{rad}=360^\circ$
Solid angle (steradian)§ 1.1 →
$\Omega=\tfrac{A}{r^2}$ (sr); whole sphere $4\pi\ \text{sr}$
Named derived units in base units§ 1.1 →
$\text{N}=\text{kg m s}^{-2}$, $\ \text{J}=\text{kg m}^2\text{s}^{-2}$, $\ \text{W}=\text{kg m}^2\text{s}^{-3}$, $\ \text{Pa}=\text{kg m}^{-1}\text{s}^{-2}$
× and ÷ with significant figures§ 1.2 →
Keep as many significant figures as the factor with the fewest (exact numbers do not count)
+ and − with significant figures§ 1.2 →
Keep as many decimal places as the term with the fewest (all terms in the same unit)
Rounding off§ 1.2 →
Dropped part $\gt$ half: round up; $\lt$ half: leave; exactly half: make the last kept digit even ($7.385\to7.38$, $7.375\to7.38$)
Combining errors (JEE/NEET, beyond the board syllabus)§ 1.2 →
$Z=A\pm B$: $\Delta Z=\Delta A+\Delta B$; $\ Z=\tfrac{A^pB^q}{C^r}$: $\tfrac{\Delta Z}{Z}=p\tfrac{\Delta A}{A}+q\tfrac{\Delta B}{B}+r\tfrac{\Delta C}{C}$ (maximum error)
Common dimensional formulae§ 1.3 →
$[F]=[\text{M}\,\text{L}\,\text{T}^{-2}]$, $\ [E]=[\text{M}\,\text{L}^2\text{T}^{-2}]$, $\ [P]=[\text{M}\,\text{L}^2\text{T}^{-3}]$, $\ [p]=[\text{M}\,\text{L}^{-1}\text{T}^{-2}]$, $\ [\text{momentum}]=[\text{M}\,\text{L}\,\text{T}^{-1}]$
Dimensional constants§ 1.3 →
$[G]=[\text{M}^{-1}\text{L}^3\text{T}^{-2}]$, $\ [h]=[\text{M}\,\text{L}^2\text{T}^{-1}]$
Principle of homogeneity§ 1.3 →
Every term of a correct equation has the same dimensions; arguments of $\sin$, $\exp$, $\log$ are dimensionless (necessary, not sufficient)
Converting between unit systems§ 1.3 →
For $[\text{M}^a\text{L}^b\text{T}^c]$: $\ n_2=n_1\left[\tfrac{M_1}{M_2}\right]^a\left[\tfrac{L_1}{L_2}\right]^b\left[\tfrac{T_1}{T_2}\right]^c$; $\ 1\ \text{J}=10^7\ \text{erg}$, $1\ \text{N}=10^5\ \text{dyne}$
Pendulum (from dimensions)§ 1.3 →
$T=k\sqrt{l/g}$; $k=2\pi$ from experiment, small swings only
Chapter 2 · Motion in a Straight Line
Displacement and path length§ 2.1 →
$\Delta x=x_2-x_1$ (signed); path length $\ge|\Delta x|$, equal only if the body never turns back
Average velocity and average speed§ 2.1 →
$\bar v=\dfrac{\Delta x}{\Delta t}$ (slope of chord); average speed $=\dfrac{\text{path length}}{\Delta t}\ge|\bar v|$
Instantaneous velocity§ 2.1 →
$v=\displaystyle\lim_{\Delta t\to0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt}$ = slope of tangent to x–t graph; speed $=|v|$
Rates of change of powers§ 2.1 →
$\dfrac{d}{dt}(c)=0$, $\ \dfrac{d}{dt}(t^n)=nt^{n-1}$; $\ x=x_0+bt+ct^2+dt^3\Rightarrow v=b+2ct+3dt^2$
Equal distances at two speeds§ 2.1 →
average speed $=\dfrac{2v_1v_2}{v_1+v_2}$ (equal distances); $\dfrac{v_1+v_2}{2}$ (equal times)
Acceleration§ 2.2 →
$\bar a=\dfrac{\Delta v}{\Delta t}$; $\ a=\dfrac{dv}{dt}=\dfrac{d^2x}{dt^2}$ = slope of v–t graph; unit $\text{m/s}^2$
Speeding up or slowing down§ 2.2 →
$v$ and $a$ same sign: speeding up; opposite signs: slowing down (not just "$a\lt0$")
Area under the v–t graph§ 2.2 →
$x_2-x_1=\displaystyle\int_{t_1}^{t_2}v\,dt$ = signed area (below axis negative); path length = total area counted positive
Area under the a–t graph§ 2.2 →
$v_2-v_1=\displaystyle\int_{t_1}^{t_2}a\,dt$ = signed area under a–t graph
Area under a power of $t$§ 2.2 →
$\displaystyle\int_0^T t^n\,dt=\frac{T^{n+1}}{n+1}$ ($n\ge0$)
Kinematic equations (constant acceleration only)§ 2.3 →
$v=u+at$; $\ x-x_0=ut+\tfrac12at^2$; $\ v^2=u^2+2a(x-x_0)$; $\ x-x_0=\tfrac12(u+v)t$
Displacement in the $n$th second§ 2.3 →
$s_n=u+\tfrac a2(2n-1)$ (constant $a$); from rest: $1:3:5:\ldots$
Stopping distance§ 2.3 →
$d_s=ut_r+\dfrac{u^2}{2|a|}$ (constant deceleration $|a|$ after reaction time $t_r$)
Free fall (up positive, no air resistance)§ 2.3 →
$v=u-gt$, $\ y-y_0=ut-\tfrac12gt^2$, $\ v^2=u^2-2g(y-y_0)$; $\ g=9.8\ \text{m/s}^2$
Vertical throw and drop§ 2.3 →
Up at $u$: $H=\dfrac{u^2}{2g}$, $T=\dfrac{2u}{g}$; dropped from $h$: $t=\sqrt{\dfrac{2h}{g}}$, $v=\sqrt{2gh}$
Relative velocity (1-D, exam extension)§ 2.3 →
$v_{BA}=v_B-v_A$, $\ a_{BA}=a_B-a_A$ (beyond the board syllabus; JEE/NEET)
Chapter 3 · Motion in a Plane
Multiplying a vector by a number§ 3.1 →
$|\lambda\mathbf{A}|=|\lambda|\,A$; along $\mathbf{A}$ if $\lambda\gt0$, opposite if $\lambda\lt0$; $\ -\mathbf{A}=(-1)\mathbf{A}$
Vector addition laws§ 3.1 →
head to tail: $\mathbf{R}=\mathbf{A}+\mathbf{B}$; $\ \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}$; $\ (\mathbf{A}+\mathbf{B})+\mathbf{C}=\mathbf{A}+(\mathbf{B}+\mathbf{C})$
Range of the resultant§ 3.1 →
$|A-B|\le|\mathbf{A}+\mathbf{B}|\le A+B$ (max: same direction; min: opposite)
Perpendicular vectors§ 3.1 →
$R=\sqrt{A^2+B^2}$, at $\alpha$ to $\mathbf{A}$ with $\tan\alpha=B/A$ (only if $\mathbf{A}\perp\mathbf{B}$)
Vector subtraction§ 3.1 →
$\mathbf{A}-\mathbf{B}=\mathbf{A}+(-\mathbf{B})$: tails together, arrow from head of $\mathbf{B}$ to head of $\mathbf{A}$; $\ \Delta\mathbf{v}=\mathbf{v}_2-\mathbf{v}_1$
Unit vector§ 3.2 →
$\hat{\mathbf{A}}=\dfrac{\mathbf{A}}{A}$, $\ |\hat{\mathbf{A}}|=1$ (no unit); $\ \mathbf{A}=A\,\hat{\mathbf{A}}$ ($\mathbf{A}\ne\mathbf{0}$)
Components of a vector§ 3.2 →
$\mathbf{A}=A_x\hat{\mathbf{i}}+A_y\hat{\mathbf{j}}$, $\ A_x=A\cos\theta$, $\ A_y=A\sin\theta$ ($\theta$ anticlockwise from $+x$)
Magnitude and direction§ 3.2 →
$A=\sqrt{A_x^2+A_y^2}$, $\ \tan\theta=A_y/A_x$ (quadrant from the signs of $A_x$, $A_y$)
Vectors in 3-D§ 3.2 →
$A=\sqrt{A_x^2+A_y^2+A_z^2}$; $\ A_x=A\cos\alpha$, $A_y=A\cos\beta$, $A_z=A\cos\gamma$; $\ \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$
Adding by components§ 3.2 →
$R_x=\sum A_x$, $\ R_y=\sum A_y$; $\ \mathbf{A}=\mathbf{B}\iff A_x=B_x,\ A_y=B_y$
Resultant of two vectors at angle θ§ 3.2 →
$R=\sqrt{A^2+B^2+2AB\cos\theta}$, $\ \tan\alpha=\dfrac{B\sin\theta}{A+B\cos\theta}$; equal magnitudes: $R=2A\cos\tfrac\theta2$ ($\theta$ = angle tail to tail)
Velocity and acceleration in a plane§ 3.3 →
$\mathbf{v}=\dfrac{d\mathbf{r}}{dt}=v_x\hat{\mathbf{i}}+v_y\hat{\mathbf{j}}$ (tangent to the path); $\ \mathbf{a}=\dfrac{d\mathbf{v}}{dt}=a_x\hat{\mathbf{i}}+a_y\hat{\mathbf{j}}$
Constant acceleration in a plane§ 3.3 →
$\mathbf{v}=\mathbf{v}_0+\mathbf{a}t$, $\ \mathbf{r}=\mathbf{r}_0+\mathbf{v}_0t+\tfrac12\mathbf{a}t^2$ — each component separately (constant $\mathbf{a}$ only)
Projectile: position and trajectory§ 3.3 →
$x=(u\cos\theta)t$, $\ y=(u\sin\theta)t-\tfrac12gt^2$; $\ y=x\tan\theta-\dfrac{gx^2}{2u^2\cos^2\theta}$ (no air resistance)
Time of flight, height, range§ 3.3 →
$T=\dfrac{2u\sin\theta}{g}$, $\ H=\dfrac{u^2\sin^2\theta}{2g}$, $\ R=\dfrac{u^2\sin2\theta}{g}$; $R_\text{max}=\dfrac{u^2}{g}$ at $45^\circ$; $\theta$, $90^\circ-\theta$ same $R$ (same launch and landing level)
Horizontal projection from height h§ 3.3 →
$t=\sqrt{2h/g}$, $\ x=u\sqrt{2h/g}$, $\ v=\sqrt{u^2+2gh}$
Relative velocity (JEE/NEET extension)§ 3.3 →
$\mathbf{v}_{AB}=\mathbf{v}_A-\mathbf{v}_B$; river: $t_\text{min}=d/v_b$; no drift: $\sin\alpha=v_r/v_b$, $t=d/\sqrt{v_b^2-v_r^2}$
Angular speed, period, frequency§ 3.4 →
$\omega=\dfrac{\Delta\theta}{\Delta t}=\dfrac{2\pi}{T}=2\pi\nu$ ($\theta$ in radians); $\ \nu=\dfrac1T$
Linear and angular speed§ 3.4 →
$v=\omega r=\dfrac{2\pi r}{T}$; arc $s=r\theta$ (radians)
Centripetal acceleration§ 3.4 →
$a_c=\dfrac{v^2}{r}=\omega^2r=4\pi^2\nu^2r=\dfrac{4\pi^2r}{T^2}$, towards the centre, $\perp\mathbf{v}$ (uniform circular motion)
Circular motion in vector form§ 3.4 →
$\mathbf{r}=r(\cos\omega t\,\hat{\mathbf{i}}+\sin\omega t\,\hat{\mathbf{j}})$, $\ \mathbf{a}=-\omega^2\mathbf{r}$
Chapter 4 · Laws of Motion
First law (law of inertia)§ 4.1 →
$\mathbf{F}_\text{net}=\mathbf{0}\iff\mathbf{a}=\mathbf{0}\iff\mathbf{v}$ constant (inertial frame)
Linear momentum§ 4.1 →
$\mathbf{p}=m\mathbf{v}$, unit $\text{kg m/s}=\text{N s}$
Second law§ 4.1 →
$\mathbf{F}_\text{net}=\dfrac{d\mathbf{p}}{dt}$; constant mass: $\mathbf{F}_\text{net}=m\mathbf{a}$, $F_x=ma_x$, $F_y=ma_y$; $1\ \text{N}=1\ \text{kg m/s}^2$ (inertial frame)
Impulse–momentum theorem§ 4.1 →
$\mathbf{J}=\mathbf{F}_\text{avg}\,\Delta t=\Delta\mathbf{p}=m\mathbf{v}-m\mathbf{u}$; $J=$ area under the $F$–$t$ graph
Force from a stream of mass§ 4.1 →
$F=v\dfrac{dm}{dt}$; jet stopped by a wall: $F=\rho Av^2$
Third law§ 4.2 →
$\mathbf{F}_{AB}=-\mathbf{F}_{BA}$ (on different bodies, simultaneous, same kind)
Conservation of linear momentum§ 4.2 →
$m_1\mathbf{u}_1+m_2\mathbf{u}_2=m_1\mathbf{v}_1+m_2\mathbf{v}_2$ (net external force zero; per component if zero along that axis)
Recoil (from rest)§ 4.2 →
$MV=-mv$, $\ V=-\dfrac{m}{M}\,v$
Bodies that stick together§ 4.2 →
$v=\dfrac{m_1u_1+m_2u_2}{m_1+m_2}$ (signed velocities)
Rocket thrust§ 4.2 →
$F=v_\text{rel}\dfrac{dm}{dt}$; $\ a=\dfrac{F}{m}$ (deep space), $\dfrac{F}{m}-g$ (vertical lift-off)
Equilibrium of a particle§ 4.3 →
$\sum F_x=0,\ \sum F_y=0,\ \sum F_z=0$; three forces form a closed triangle
Block on an incline§ 4.3 →
$N=mg\cos\theta$; along the slope $mg\sin\theta$; smooth: $a=g\sin\theta$
Laws of friction§ 4.3 →
$f_s\le\mu_sN$ (self-adjusting), $\ f_k=\mu_kN$, $\ \mu_k\lt\mu_s$ (dry surfaces; independent of contact area)
Angle of repose; sliding down a rough incline§ 4.3 →
$\tan\theta_r=\mu_s$; $\ a=g(\sin\theta-\mu_k\cos\theta)$ for $\theta\gt\theta_r$; constant velocity if $\tan\theta=\mu_k$
Pulling at angle α above horizontal§ 4.3 →
$N=mg-T\sin\alpha$, $\ ma=T\cos\alpha-\mu_kN$
Centripetal force§ 4.4 →
$F_c=\dfrac{mv^2}{r}=m\omega^2r$ (net force towards the centre; uniform circular motion)
Car on a level curve§ 4.4 →
$v_\text{max}=\sqrt{\mu_s\,r\,g}$ (independent of mass)
Banked curve (angle θ)§ 4.4 →
$v_0=\sqrt{rg\tan\theta}$; $\ v_\text{max}=\sqrt{rg\,\dfrac{\mu_s+\tan\theta}{1-\mu_s\tan\theta}}$; $\ v_\text{min}=\sqrt{rg\,\dfrac{\tan\theta-\mu_s}{1+\mu_s\tan\theta}}$ ($0$ if $\tan\theta\le\mu_s$)
Atwood machine§ 4.4 →
$a=\dfrac{m_1-m_2}{m_1+m_2}\,g$, $\ T=\dfrac{2m_1m_2}{m_1+m_2}\,g$ (ideal string and pulley)
Table block $m_1$ + hanging $m_2$§ 4.4 →
$a=\dfrac{m_2-\mu_km_1}{m_1+m_2}\,g$, $\ T=m_2(g-a)$ (moves if $m_2\gt\mu_sm_1$)
Apparent weight in a lift§ 4.4 →
$N=m(g+a)$, $a$ upward positive; free fall: $N=0$
Chapter 5 · Work, Energy and Power
Scalar (dot) product§ 5.1 →
$\mathbf{A}\cdot\mathbf{B}=AB\cos\theta=A_xB_x+A_yB_y+A_zB_z$; $\ \cos\theta=\dfrac{\mathbf{A}\cdot\mathbf{B}}{AB}$; $\ \mathbf{A}\perp\mathbf{B}\iff\mathbf{A}\cdot\mathbf{B}=0$ (non-zero vectors)
Work by a constant force§ 5.1 →
$W=\mathbf{F}\cdot\mathbf{d}=Fd\cos\theta$ (constant $\mathbf{F}$); $1\ \text{J}=1\ \text{N m}$
Kinetic energy§ 5.1 →
$K=\tfrac12mv^2=\dfrac{p^2}{2m}$
Work–energy theorem§ 5.1 →
$W_\text{net}=K_f-K_i$ (work of all forces; constant or variable)
Work by a variable force (1-D)§ 5.1 →
$W=\displaystyle\int_{x_i}^{x_f}F(x)\,dx$ = signed area under the $F$–$x$ graph
Potential energy§ 5.2 →
$\Delta U=-W_\text{c}$; $\ F=-\dfrac{dU}{dx}$ (conservative force, 1-D)
Gravitational PE near the Earth§ 5.2 →
$U=mgh$ ($h$ above the reference level; $h\ll R_E$)
Spring force and elastic PE§ 5.2 →
$F_s=-kx$, $\ U=\tfrac12kx^2$ (ideal spring, $x$ from natural length)
Mechanical energy§ 5.2 →
$K_i+U_i=K_f+U_f$ (only conservative forces do work); otherwise $W_\text{nc}=E_f-E_i$
Vertical circle (bob on a string)§ 5.2 →
$v_\text{top}\ge\sqrt{gL}$, $\ v_\text{bottom}\ge\sqrt{5gL}$; then $T_\text{bottom}=6mg$
Equilibrium from U(x) (JEE extension)§ 5.2 →
$\dfrac{dU}{dx}=0$; minimum of $U$: stable; maximum: unstable
Power§ 5.3 →
$P_\text{av}=\dfrac Wt$, $\ P=\dfrac{dW}{dt}=\mathbf{F}\cdot\mathbf{v}$; $1\ \text{hp}=746\ \text{W}$
Kilowatt-hour§ 5.3 →
$1\ \text{kWh}=3.6\times10^6\ \text{J}$ (energy, not power)
Perfectly inelastic collision (1-D)§ 5.3 →
$v=\dfrac{m_1u_1+m_2u_2}{m_1+m_2}$; $\ \Delta K=\dfrac{m_1m_2(u_1-u_2)^2}{2(m_1+m_2)}$
Elastic collision (1-D, head-on)§ 5.3 →
$v_1=\dfrac{(m_1-m_2)u_1+2m_2u_2}{m_1+m_2}$, $\ v_2=\dfrac{(m_2-m_1)u_2+2m_1u_1}{m_1+m_2}$; $\ v_2-v_1=u_1-u_2$
2-D elastic, equal masses§ 5.3 →
One at rest, not head-on: they move off at $90^\circ$ to each other
Coefficient of restitution (extension)§ 5.3 →
$e=\dfrac{v_2-v_1}{u_1-u_2}$; bounce height $h_n=e^{2n}h$
Chapter 6 · System of Particles and Rotational Motion
Centre of mass of two particles§ 6.1 →
$X=\dfrac{m_1x_1+m_2x_2}{m_1+m_2}$; distances from it: $m_1d_1=m_2d_2$
Centre of mass of n particles§ 6.1 →
$\mathbf{R}_\text{cm}=\dfrac{\sum m_i\mathbf{r}_i}{M}$, $\ X=\dfrac{\sum m_ix_i}{M}$, $\ Y=\dfrac{\sum m_iy_i}{M}$, $\ M=\sum m_i$
Uniform and composite bodies§ 6.1 →
Uniform body: c.m. on every axis/plane of symmetry (rod: midpoint; ring, disc, sphere: centre; triangle: centroid). Composite: each part as a particle at its own c.m.; a hole counts as negative mass.
Motion of the centre of mass§ 6.1 →
$M\mathbf{a}_\text{cm}=\mathbf{F}_\text{ext}$ (constant total mass; internal forces cancel)
Momentum of a system§ 6.1 →
$\mathbf{P}=\sum m_i\mathbf{v}_i=M\mathbf{v}_\text{cm}$; if $\mathbf{F}_\text{ext}=0$, $\mathbf{v}_\text{cm}$ is constant
Vector product§ 6.2 →
$|\mathbf{A}\times\mathbf{B}|=AB\sin\theta$, ⟂ to both (right-hand rule); $\mathbf{B}\times\mathbf{A}=-\mathbf{A}\times\mathbf{B}$; $(\mathbf{A}\times\mathbf{B})_z=A_xB_y-A_yB_x$ (cyclic for $x$, $y$)
Angular velocity and acceleration§ 6.2 →
$\omega=\dfrac{d\theta}{dt}$, $\alpha=\dfrac{d\omega}{dt}$; $v=\omega r_\perp$, $a_t=\alpha r_\perp$, $a_c=\omega^2r_\perp$ (fixed axis)
Torque§ 6.2 →
$\boldsymbol\tau=\mathbf{r}\times\mathbf{F}$, $\ \tau=rF\sin\theta=r_\perp F$ (N m)
Angular momentum§ 6.2 →
$\mathbf{l}=\mathbf{r}\times\mathbf{p}$, $\ l=r_\perp p$; $\ \dfrac{d\mathbf{L}}{dt}=\boldsymbol\tau_\text{ext}$ (about a fixed point)
Equilibrium of a rigid body§ 6.2 →
$\sum\mathbf{F}_\text{ext}=0$ and $\sum\boldsymbol\tau_\text{ext}=0$ (about any point); lever: $F_1d_1=F_2d_2$; couple: $\tau=Fd$
Rolling without slipping (🚀 JEE Advanced only)§ 6.2 →
$v_\text{cm}=R\omega$; contact point at rest, top point $2v_\text{cm}$
Rotation with constant α§ 6.3 →
$\omega=\omega_0+\alpha t$, $\ \theta=\omega_0t+\tfrac12\alpha t^2$, $\ \omega^2=\omega_0^2+2\alpha\theta$ (constant $\alpha$, radians)
Moment of inertia and rotational KE§ 6.3 →
$I=\sum m_ir_i^2=Mk^2$ ($r_i$ ⟂ distance from the axis); $\ K=\tfrac12I\omega^2$
Standard moments of inertia (uniform bodies)§ 6.3 →
Ring (central axis) $MR^2$; disc, solid cylinder $\tfrac12MR^2$; rod (centre) $\tfrac1{12}ML^2$, (end) $\tfrac13ML^2$; solid sphere $\tfrac25MR^2$; spherical shell $\tfrac23MR^2$
Dynamics of rotation§ 6.3 →
$\tau=I\alpha$ (fixed axis); $\ W=\tau\theta$, $\ P=\tau\omega$
Angular momentum about a fixed axis§ 6.3 →
$L=I\omega$; if $\tau_\text{ext}=0$: $I_1\omega_1=I_2\omega_2$
Axes theorems (🚀 JEE/NEET) and rolling (🚀 JEE Advanced)§ 6.3 →
$I=I_\text{cm}+Md^2$; lamina $I_z=I_x+I_y$; rolling from height $h$: $v=\sqrt{\dfrac{2gh}{1+k^2/R^2}}$
Chapter 7 · Gravitation
Elliptical orbit§ 7.1 →
$r_p=a(1-e)$, $\ r_a=a(1+e)$, $\ a=\tfrac12(r_p+r_a)$
Law of areas§ 7.1 →
$\dfrac{\Delta A}{\Delta t}=\dfrac{L}{2m}=\text{const}$; $\ v_pr_p=v_ar_a$, $\ \dfrac{v_p}{v_a}=\dfrac{1+e}{1-e}$ (any central force)
Kepler's third law§ 7.1 →
$T^2\propto a^3$; $\ T^2=a^3$ (years, AU, planets of the Sun)
Law of gravitation§ 7.1 →
$F=\dfrac{Gm_1m_2}{r^2}$, $\ G=6.67\times10^{-11}\ \text{N m}^2\,\text{kg}^{-2}$ ($r$ between centres)
Circular orbit about mass M§ 7.1 →
$v=\sqrt{\dfrac{GM}{r}}$, $\ T^2=\dfrac{4\pi^2}{GM}r^3$, $\ M=\dfrac{4\pi^2r^3}{GT^2}$ ($M\gg$ orbiting mass)
Spherical shell§ 7.1 →
Outside: as if all mass at centre; inside: net force $0$ (uniform shell)
$g$ at the surface§ 7.2 →
$g=\dfrac{GM}{R^2}=\tfrac43\pi G\rho R$ (spherically symmetric body)
Mass of the Earth§ 7.2 →
$M_E=\dfrac{gR_E^2}{G}\approx6.0\times10^{24}\ \text{kg}$
$g$ at height $h$§ 7.2 →
$g(h)=\dfrac{g}{(1+h/R_E)^2}$ (any $h$); $\ g(h)\approx g\left(1-\dfrac{2h}{R_E}\right)$ (only $h\ll R_E$)
$g$ at depth $d$§ 7.2 →
$g(d)=g\left(1-\dfrac{d}{R_E}\right)$ (uniform Earth, $0\le d\le R_E$); $0$ at the centre
Height vs depth§ 7.2 →
Same $g$ at height $h$ and depth $d=2h$ ($h,d\ll R_E$)
Gravitational PE§ 7.3 →
$U=-\dfrac{GMm}{r}$ ($U=0$ at infinity; sum over all pairs)
Gravitational potential§ 7.3 →
$V=\dfrac{U}{m}=-\dfrac{GM}{r}$ (outside a spherical mass), unit $\text{J/kg}$
Raising a body by $h$§ 7.3 →
$\Delta U=GMm\left(\dfrac1{R}-\dfrac1{R+h}\right)=\dfrac{mgh}{1+h/R}\approx mgh$ ($h\ll R$)
Escape speed§ 7.3 →
$v_e=\sqrt{\dfrac{2GM}{R}}=\sqrt{2gR}\approx11.2\ \text{km/s}$ (Earth)
Satellite in a circular orbit§ 7.3 →
$v_o=\sqrt{\dfrac{GM}{R+h}}$, $\ T=2\pi\sqrt{\dfrac{(R+h)^3}{GM}}$; near the surface $v_o=\sqrt{gR}$, $v_e=\sqrt2\,v_o$
Energy of a satellite§ 7.3 →
$K=\dfrac{GMm}{2r}$, $\ U=-\dfrac{GMm}{r}$, $\ E=-\dfrac{GMm}{2r}=-K=\tfrac12U$ (circular orbit)
Chapter 8 · Mechanical Properties of Solids
Stress§ 8.1 →
$\sigma=\dfrac{F}{A}$ (Pa $=$ N/m²); wire: $A=\dfrac{\pi d^2}{4}$
Strain (no unit)§ 8.1 →
longitudinal $\dfrac{\Delta L}{L}$; shearing $\dfrac{\Delta x}{L}=\tan\theta\approx\theta$ (small $\theta$, rad); volume $\dfrac{\Delta V}{V}$
Hooke's law§ 8.1 →
stress $=$ modulus $\times$ strain (within the proportional limit)
A wire as a spring§ 8.1 →
$F=\dfrac{YA}{L}\,\Delta L$, $\ k=\dfrac{YA}{L}$ (Hooke's-law region)
Breaking load; working stress§ 8.1 →
$F_\text{max}=\sigma_uA$ (independent of length, own weight neglected); working stress $=\sigma_u/(\text{factor of safety})$
Young's modulus; extension of a wire§ 8.2 →
$Y=\dfrac{FL}{A\,\Delta L}$; $\ \Delta L=\dfrac{FL}{AY}=\dfrac{4FL}{\pi d^2Y}$ (within the proportional limit, own weight neglected)
$Y$ from a load–extension graph§ 8.2 →
$Y=\dfrac{L}{A}\times\text{slope of }F\text{ vs }\Delta L$
Shear modulus§ 8.2 →
$G=\dfrac{F/A}{\Delta x/L}=\dfrac{F}{A\theta}$ (small $\theta$, solids only)
Bulk modulus; compressibility§ 8.2 →
$B=-\dfrac{p}{\Delta V/V}$, $\ k=\dfrac1B$
Poisson's ratio; volume change on stretching§ 8.2 →
$\nu=-\dfrac{\Delta d/d}{\Delta L/L}$; $\ \dfrac{\Delta V}{V}=\varepsilon(1-2\nu)$ (small strains; $\nu\le0.5$)
Elastic potential energy of a wire§ 8.2 →
$U=\tfrac12F\,\Delta L$; per unit volume $u=\tfrac12\sigma\varepsilon=\dfrac{\sigma^2}{2Y}$ (within the proportional limit)
Chapter 9 · Mechanical Properties of Fluids
Pressure and density§ 9.1 →
$p=\dfrac{F_\perp}{A}$ ($1\ \text{Pa}=1\ \text{N/m}^2$), $\ \rho=\dfrac{m}{V}$; $\ 1\ \text{atm}=1.013\times10^5\ \text{Pa}$
Pressure at a depth§ 9.1 →
$p=p_a+\rho gh$, $\ p_2-p_1=\rho gh$ (liquid at rest, constant $\rho$)
Gauge pressure, barometer, manometer§ 9.1 →
$p_\text{gauge}=p-p_a=\rho gh$; $\ p_a=\rho_\text{Hg}gh$ ($h\approx0.76\ \text{m}$)
Hydraulic machine (Pascal's principle)§ 9.1 →
$\dfrac{F_1}{A_1}=\dfrac{F_2}{A_2}$, $\ A_1d_1=A_2d_2$, $\ F_1d_1=F_2d_2$ (pistons level, incompressible liquid, no friction)
Continuity and flow rate§ 9.2 →
$Q=Av$, $\ A_1v_1=A_2v_2$ (steady flow, incompressible fluid)
Bernoulli's principle§ 9.2 →
$p+\tfrac12\rho v^2+\rho gh=\text{constant}$ along a streamline (streamline, incompressible, non-viscous flow)
Speed of efflux (Torricelli)§ 9.2 →
$v=\sqrt{2gh}$ (wide open tank, small hole, depth $h$); horizontal jet from height $y$: $R=2\sqrt{hy}$
Venturi meter§ 9.2 →
$v_1=\sqrt{\dfrac{2(p_1-p_2)}{\rho[(A_1/A_2)^2-1]}}$, $\ p_1-p_2=\rho g\,\Delta h$ (horizontal, ideal fluid)
Lift from a pressure difference§ 9.2 →
$F=\tfrac12\rho\left(v_\text{top}^2-v_\text{bottom}^2\right)A$ (Bernoulli picture, thin wing)
Time to empty a tank (extension)§ 9.2 →
$t=\dfrac{A}{a}\sqrt{\dfrac{2H}{g}}$ (hole area $a\ll A$, ideal flow)
Viscous force§ 9.3 →
$F=\eta A\dfrac{dv}{dx}$ (laminar flow); plates a distance $l$ apart: $F=\eta A\dfrac{v}{l}$; $\ 1\ \text{poise}=0.1\ \text{Pa s}$
Reynolds number; critical velocity§ 9.3 →
$Re=\dfrac{\rho vd}{\eta}$; pipe flow laminar for $Re\lesssim1000$, turbulent for $Re\gtrsim2000$; $\ v_c\approx\dfrac{2000\,\eta}{\rho d}$
Stokes' law§ 9.3 →
$F=6\pi\eta rv$ (small sphere, slow laminar flow)
Terminal velocity§ 9.3 →
$v_t=\dfrac{2r^2(\rho-\sigma)g}{9\eta}$ (Stokes' law holds; $\rho$ sphere, $\sigma$ fluid)
$n$ drops merging§ 9.3 →
$R=n^{1/3}r$, $\ v_t'=n^{2/3}v_t$ (Stokes regime)
Approach to $v_t$ (extension)§ 9.3 →
$v=v_t\left(1-e^{-t/\tau}\right)$, $\ \tau=\dfrac{2r^2\rho}{9\eta}$
Surface tension and surface energy§ 9.4 →
$S=\dfrac{F}{l}$, $\ W=S\,\Delta A$ (constant temperature); film on a wire: $F=2Sl$
Excess pressure§ 9.4 →
Drop, or air bubble in a liquid: $\Delta p=\dfrac{2S}{r}$; $\ $ soap bubble (two surfaces): $\Delta p=\dfrac{4S}{r}$
Capillary rise§ 9.4 →
$h=\dfrac{2S\cos\theta}{\rho gr}$ (narrow vertical tube; $\theta\gt90^\circ$ gives a depression)
Angle of contact§ 9.4 →
$\cos\theta=\dfrac{S_{sa}-S_{sl}}{S_{la}}$ (drop at rest on a solid)
Splitting a drop into $n$ droplets§ 9.4 →
$r=\dfrac{R}{n^{1/3}}$, $\ W=4\pi R^2S\left(n^{1/3}-1\right)$
Chapter 10 · Thermal Properties of Matter
Temperature scales§ 10.1 →
$\dfrac{t_F-32}{180}=\dfrac{t_C}{100}$; $\ T=t_C+273.15$; differences: $\Delta t_F=1.8\,\Delta t_C$, $\Delta T=\Delta t_C$
Any linear scale (fixed points)§ 10.1 →
$\dfrac{X-X_\text{ice}}{X_\text{steam}-X_\text{ice}}=\dfrac{t_C}{100}$ (same fraction on every scale)
Ideal-gas equation; gas thermometer§ 10.1 →
$pV=nRT$, $\ \dfrac{p_1V_1}{T_1}=\dfrac{p_2V_2}{T_2}$ ($T$ in K, low density); $\ T=273.16\ \text{K}\times\dfrac{p}{p_\text{tp}}$ (constant volume)
Thermal expansion§ 10.1 →
$\Delta L=\alpha L\,\Delta T$, $\ \Delta A=\beta A\,\Delta T$, $\ \Delta V=\gamma V\,\Delta T$; isotropic solid: $\beta=2\alpha$, $\gamma=3\alpha$ ($\alpha\Delta T\ll1$); ideal gas at constant $p$: $\gamma=1/T$
Density after expansion§ 10.1 →
$\rho=\dfrac{\rho_0}{1+\gamma\,\Delta T}\approx\rho_0(1-\gamma\,\Delta T)$ (not water between $0$ and $4\ ^\circ\text{C}$)
Thermal stress in a clamped rod§ 10.1 →
$\sigma=Y\alpha\,\Delta T$, $\ F=YA\alpha\,\Delta T$ (rigid supports, within the proportional limit; independent of length)
Heat for a temperature change§ 10.2 →
$Q=mc\,\Delta T=S\,\Delta T$, $\ S=mc$ (no change of state; $c$ constant over the range)
Molar specific heat capacity§ 10.2 →
$C=\dfrac{1}{n}\dfrac{\Delta Q}{\Delta T}=Mc$ ($M$ in kg/mol; metals: $C\approx3R$)
Method of mixtures§ 10.2 →
$\sum m_ic_i(T_f-T_i)=0$, so $T_f=\dfrac{\sum m_ic_iT_i}{\sum m_ic_i}$ (isolated system, no change of state); water equivalent $W=\dfrac{m_\text{cal}c_\text{cal}}{c_\text{w}}$
Latent heat§ 10.2 →
$Q=mL$ at constant temperature; water: $L_f=3.33\times10^5\ \text{J/kg}$, $L_v=2.26\times10^6\ \text{J/kg}$
Ice in a mixture§ 10.2 →
If $Q_\text{avail}$ (warm bodies cooling to $0\ ^\circ\text{C}$) $\lt Q_\text{need}$ (ice to $0\ ^\circ\text{C}$ and fully melted): $T_f=0\ ^\circ\text{C}$ and $m_\text{melted}=\dfrac{Q_\text{avail}-Q_\text{warm ice}}{L_f}$ ($Q_\text{warm ice}$: ice up to $0\ ^\circ\text{C}$)
Constant-power heating curve§ 10.2 →
Slope $\dfrac{dT}{dt}=\dfrac{P}{mc}$; plateau length $t=\dfrac{mL}{P}$
Rate of conduction§ 10.3 →
$H=KA\,\dfrac{T_1-T_2}{L}$ (steady state, uniform slab, lagged sides)
Thermal resistance; slabs in series§ 10.3 →
$R=\dfrac{L}{KA}$, $\ H=\dfrac{\Delta T}{R_1+R_2+\cdots}$; drop across slab $i$ is $HR_i$; equal thicknesses: $K_\text{eq}=\dfrac{2K_1K_2}{K_1+K_2}$ (steady state, same $A$)
Stefan–Boltzmann law§ 10.3 →
$P=e\sigma AT^4$; $\ P_\text{net}=e\sigma A(T^4-T_s^4)$; $\ \sigma=5.67\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$ ($T$ in K)
Wien's displacement law§ 10.3 →
$\lambda_mT=b=2.9\times10^{-3}\ \text{m K}$ (blackbody)
Newton's law of cooling§ 10.3 →
$\dfrac{dT}{dt}=-k(T-T_s)$; $\ T-T_s=(T_0-T_s)e^{-kt}$; average form $\dfrac{T_1-T_2}{t}\approx k\left(\dfrac{T_1+T_2}{2}-T_s\right)$ (small excess $T-T_s$)
Parallel slabs; ice growth (JEE/NEET extension)§ 10.3 →
$\dfrac1R=\dfrac1{R_1}+\dfrac1{R_2}$; $\ t=\dfrac{\rho L_f}{2K\theta}\left(x_2^2-x_1^2\right)$
Chapter 11 · Thermodynamics
Zeroth law§ 11.1 →
A in equilibrium with C and B in equilibrium with C $\Rightarrow$ A in equilibrium with B; thermal equilibrium $\Leftrightarrow$ $T_A=T_B$
First law of thermodynamics§ 11.1 →
$\Delta U=Q-W$ ($Q$ = heat supplied to the system, $W$ = work done by it; $\Delta U$ depends only on the end states)
Work done by a gas§ 11.1 →
$W=\displaystyle\int_{V_1}^{V_2}p\,dV$ = area under the $p$–$V$ path (quasi-static); at constant pressure $W=p\,\Delta V$; $(10^5\ \text{Pa})(1\ \text{L})=100\ \text{J}$
Internal energy of an ideal gas§ 11.1 →
$U=U(T)$ only; monatomic ideal gas $U=\tfrac32nRT=\tfrac32pV$
Mayer's relation and $\gamma$§ 11.2 →
$C_p-C_v=R$, $\gamma=\dfrac{C_p}{C_v}$ (ideal gas); monatomic $C_v=\tfrac32R$, $\gamma=\tfrac53$; diatomic $C_v=\tfrac52R$, $\gamma=\tfrac75$
$\Delta U$ of an ideal gas§ 11.2 →
$\Delta U=nC_v\,\Delta T$ for every process of an ideal gas
Isothermal process§ 11.2 →
$pV=$ const, $\Delta U=0$, $Q=W=nRT\ln\dfrac{V_2}{V_1}=nRT\ln\dfrac{p_1}{p_2}$ (ideal gas, quasi-static)
Adiabatic process§ 11.2 →
$Q=0$; $pV^\gamma=$ const, $TV^{\gamma-1}=$ const; $W=\dfrac{p_1V_1-p_2V_2}{\gamma-1}=nC_v(T_1-T_2)$ (ideal gas, quasi-static)
Isochoric and isobaric processes§ 11.2 →
Isochoric: $W=0$, $Q=nC_v\Delta T$. Isobaric: $W=p\Delta V=nR\Delta T$, $Q=nC_p\Delta T$ (ideal gas)
Cyclic process§ 11.2 →
$\Delta U=0$, $Q_\text{net}=W_\text{net}$ = area enclosed (positive if clockwise on $p$–$V$)
Efficiency of an engine§ 11.3 →
$\eta=\dfrac{W}{Q_1}=1-\dfrac{Q_2}{Q_1}$, with $W=Q_1-Q_2$ per cycle
Carnot engine§ 11.3 →
$\eta_\text{C}=1-\dfrac{T_2}{T_1}$, $\dfrac{Q_2}{Q_1}=\dfrac{T_2}{T_1}$ (reversible; $T$ in kelvin)
Carnot's theorem§ 11.3 →
Any engine between $T_1$ and $T_2$: $\eta\le1-\dfrac{T_2}{T_1}$, i.e. $\dfrac{Q_2}{T_2}\ge\dfrac{Q_1}{T_1}$ (equality only for a reversible engine)
Refrigerator and heat pump (🚀 JEE Advanced only)§ 11.3 →
$\beta=\dfrac{Q_2}{W}\le\dfrac{T_2}{T_1-T_2}$; heat pump $\dfrac{Q_1}{W}\le\dfrac{T_1}{T_1-T_2}$; reversible: $\beta=\dfrac{1-\eta}{\eta}$
Chapter 12 · Kinetic Theory
Moles, molecules and molar mass§ 12.1 →
$n=\dfrac{N}{N_A}=\dfrac{M_s}{M}$, $\ m=\dfrac{M}{N_A}$, $\ N_A=6.02\times10^{23}\ \text{mol}^{-1}$
Ideal-gas equation (molecular form)§ 12.1 →
$pV=nRT=Nk_BT$, $\ k_B=\dfrac{R}{N_A}=1.38\times10^{-23}\ \text{J/K}$; $\ p=n_Vk_BT=\dfrac{\rho RT}{M}$ (dilute gas, $T$ in kelvin, absolute $p$)
Boyle, Charles and the combined law§ 12.1 →
$\dfrac{p_1V_1}{T_1}=\dfrac{p_2V_2}{T_2}$ (fixed amount of ideal gas); $pV=$ const at fixed $T$; $V\propto T$ at fixed $p$
Molar volume at STP§ 12.1 →
$V_m=\dfrac{RT}{p}=22.4\ \text{L}$, $\ n_V=2.69\times10^{25}\ \text{m}^{-3}$ (any ideal gas, $273\ \text{K}$, $1\ \text{atm}$)
Dalton's law of partial pressures§ 12.1 →
$p=p_1+p_2+\dots$, $\ p_i=\dfrac{n_i}{n_\text{total}}\,p$ (mixture of ideal gases)
Pressure from kinetic theory§ 12.2 →
$p=\tfrac13n_Vm\overline{v^2}=\tfrac13\rho v_\text{rms}^2$, $\ pV=\tfrac23E_\text{trans}$ (ideal gas, random motion, elastic collisions)
Temperature and kinetic energy§ 12.2 →
$\tfrac12m\overline{v^2}=\tfrac32k_BT$; $\ E_\text{trans}=\tfrac32nRT$ (ideal gas, $T$ in kelvin)
rms speed§ 12.2 →
$v_\text{rms}=\sqrt{\dfrac{3RT}{M}}=\sqrt{\dfrac{3k_BT}{m}}=\sqrt{\dfrac{3p}{\rho}}$ ($M$ in kg/mol)
Comparing rms speeds§ 12.2 →
$\dfrac{v_1}{v_2}=\sqrt{\dfrac{T_1M_2}{T_2M_1}}$ (ideal gases)
Characteristic speeds (🚀 JEE Advanced only)§ 12.2 →
$v_\text{mp}=\sqrt{\tfrac{2RT}{M}}$, $\ \bar v=\sqrt{\tfrac{8RT}{\pi M}}$, $\ v_\text{rms}=\sqrt{\tfrac{3RT}{M}}$; ratio $1:1.13:1.22$ (Maxwell distribution, beyond NCERT)
Equipartition of energy§ 12.3 →
$\tfrac12k_BT$ per squared term ($k_BT$ per vibrational mode); $\ U=\tfrac f2nRT$ (classical, thermal equilibrium)
Heat capacities from degrees of freedom§ 12.3 →
$C_v=\tfrac f2R$, $\ C_p=\left(\tfrac f2+1\right)R$, $\ \gamma=1+\tfrac2f$; $f=3,5,6,7$ for monatomic, rigid diatomic, rigid non-linear, vibrating diatomic (ideal gas)
Gas mixture§ 12.3 →
$C_v=\dfrac{n_1C_{v1}+n_2C_{v2}}{n_1+n_2}$, $\ C_p=C_v+R$, $\ \gamma=\dfrac{C_p}{C_v}$ (ideal gases; never average $\gamma$)
Dulong–Petit law§ 12.3 →
$C\approx3R\approx25\ \text{J mol}^{-1}\text{K}^{-1}$, $\ c=\dfrac{3R}{M}$ (solid elements, ordinary temperatures)
Mean free path§ 12.3 →
$\lambda=\dfrac{1}{\sqrt2\,\pi d^2n_V}=\dfrac{k_BT}{\sqrt2\,\pi d^2p}$, $\ \nu\approx\dfrac v\lambda$ (ideal gas, hard-sphere molecules of diameter $d$)
Chapter 13 · Oscillations
Period and frequency§ 13.1 →
$\nu=\dfrac1T$ (unit $\text{Hz}=\text{s}^{-1}$); $T$ is the smallest repeat time, $f(t+T)=f(t)$
Displacement in SHM§ 13.1 →
$x=A\cos(\omega t+\phi)$; amplitude $A$, phase $\omega t+\phi$ (radians), phase constant $\phi$, $x(0)=A\cos\phi$
Angular frequency§ 13.1 →
$\omega=\dfrac{2\pi}{T}=2\pi\nu$ (period independent of $A$ and $\phi$); distance in one period $=4A$
Reference circle timing§ 13.1 →
$\Delta t=\dfrac{\Delta\theta}{2\pi}T$; e.g. $0\to\tfrac A2$: $\tfrac T{12}$, $\ \tfrac A2\to A$: $\tfrac T6$, $\ 0\to\tfrac A{\sqrt2}$: $\tfrac T8$
Combining sine and cosine§ 13.1 →
$a\sin\omega t+b\cos\omega t=\sqrt{a^2+b^2}\,\cos(\omega t+\phi)$ with $\cos\phi=\dfrac{b}{\sqrt{a^2+b^2}}$, $\sin\phi=\dfrac{-a}{\sqrt{a^2+b^2}}$
Adding two SHMs, same $\omega$ (extension)§ 13.1 →
$A=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\delta}$, $\ \tan\varepsilon=\dfrac{A_2\sin\delta}{A_1+A_2\cos\delta}$ (same line, same angular frequency)
Velocity in SHM§ 13.2 →
$v=-\omega A\sin(\omega t+\phi)$, $\ |v|=\omega\sqrt{A^2-x^2}$, $\ v_\text{max}=\omega A$ at $x=0$
Acceleration in SHM§ 13.2 →
$a=-\omega^2x$, $\ a_\text{max}=\omega^2A$ at $x=\pm A$; $v$ leads $x$ by $\tfrac\pi2$, $a$ leads $x$ by $\pi$
Force law for SHM§ 13.2 →
$F=-kx$ (net force) $\iff$ SHM with $\omega=\sqrt{\dfrac km}$, $\ T=2\pi\sqrt{\dfrac mk}$ (any amplitude)
Energy in SHM§ 13.2 →
$K=\tfrac12k(A^2-x^2)$, $\ U=\tfrac12kx^2$, $\ E=\tfrac12kA^2=\tfrac12m\omega^2A^2$ (constant); $K$, $U$ have period $\tfrac T2$, averages $\tfrac E2$
Period from two snapshots§ 13.2 →
$T=2\pi\sqrt{\dfrac{x_2^2-x_1^2}{v_1^2-v_2^2}}$ (speeds $v_1$, $v_2$ at $x_1$, $x_2$)
U-tube and Earth tunnel (extension)§ 13.2 →
U-tube, column length $L$: $T=2\pi\sqrt{\dfrac{L}{2g}}$; tunnel through a uniform Earth: $T=2\pi\sqrt{\dfrac{R_E}{g}}\approx84$ min
Spring–block oscillator§ 13.3 →
$T=2\pi\sqrt{\dfrac mk}$, $\ \nu=\dfrac1{2\pi}\sqrt{\dfrac km}$ (light spring, Hooke's law, no friction; any amplitude)
Vertical spring§ 13.3 →
$kl_0=mg$; $\ T=2\pi\sqrt{\dfrac mk}=2\pi\sqrt{\dfrac{l_0}{g}}$ (SHM about the stretched equilibrium)
Springs in combination (🚀 extension, beyond CBSE)§ 13.3 →
parallel (or one each side): $k_\text{p}=k_1+k_2$; series: $\dfrac1{k_\text{s}}=\dfrac1{k_1}+\dfrac1{k_2}$; a piece $\tfrac1n$ of the length has constant $nk$ (light springs)
Simple pendulum§ 13.3 →
$T=2\pi\sqrt{\dfrac lg}$ (small oscillations, $\theta_0\lesssim10^\circ$; independent of mass and amplitude)
Seconds pendulum; measuring $g$§ 13.3 →
$T=2\ \text{s}$: $l=\dfrac{g}{\pi^2}\approx0.993\ \text{m}$; $\ T^2=\dfrac{4\pi^2}{g}l$, so $g=\dfrac{4\pi^2}{\text{slope}}$
Damping and resonance (🚀 JEE Advanced only)§ 13.3 →
damping force $-bv$: $A(t)=A_0e^{-bt/2m}$ (light damping); forced oscillation: largest amplitude when $\omega_\text{d}\approx\omega_0=\sqrt{k/m}$
Chapter 14 · Waves
Progressive sinusoidal wave§ 14.1 →
$y=a\sin(kx-\omega t+\phi)$ towards $+x$; $\ y=a\sin(kx+\omega t+\phi)$ towards $-x$
Wave number and angular frequency§ 14.1 →
$k=\dfrac{2\pi}{\lambda}$, $\ \omega=\dfrac{2\pi}{T}=2\pi\nu$
Wave speed§ 14.1 →
$v=\dfrac\omega k=\dfrac\lambda T=\nu\lambda$ (speed set by the medium)
Phase difference along a wave§ 14.1 →
$\Delta\phi=\dfrac{2\pi}{\lambda}\Delta x$ (same instant); $\ \Delta\phi=\dfrac{2\pi}{T}\Delta t$ (same place)
Velocity of an element of the medium§ 14.1 →
$v_y=-\omega a\cos(kx-\omega t+\phi)$, $\ a_y=-\omega^2y$; $\ \dfrac{(v_y)_\text{max}}{v}=ka=\dfrac{2\pi a}{\lambda}$
Waves on a stretched string§ 14.2 →
$v=\sqrt{\dfrac T\mu}=\sqrt{\dfrac\sigma\rho}$, $\ \mu=\dfrac mL=\rho A$ (flexible string, small amplitude)
Speed of longitudinal waves§ 14.2 →
Fluid: $v=\sqrt{\dfrac B\rho}$; $\ $ thin rod: $v=\sqrt{\dfrac Y\rho}$
Speed of sound in an ideal gas§ 14.2 →
Newton (isothermal, wrong): $v=\sqrt{\dfrac p\rho}$; $\ $ Laplace (adiabatic): $v=\sqrt{\dfrac{\gamma p}{\rho}}=\sqrt{\dfrac{\gamma RT}{M}}$
What changes the speed of sound in a gas§ 14.2 →
$v\propto\sqrt T$ ($T$ in K); independent of $p$ at fixed $T$; $\ \dfrac{v_1}{v_2}=\sqrt{\dfrac{\gamma_1M_2}{\gamma_2M_1}}$ at equal $T$
Interference of two equal waves§ 14.3 →
$a\sin(kx-\omega t)+a\sin(kx-\omega t+\phi)=2a\cos\tfrac\phi2\sin\left(kx-\omega t+\tfrac\phi2\right)$; $\ A=2a\left|\cos\tfrac\phi2\right|$
Reflection of waves§ 14.3 →
Rigid end: phase change $\pi$ (pulse inverted); $\ $ free end: no phase change
Standing wave§ 14.3 →
$y=2a\sin kx\cos\omega t$; nodes at $x=0,\tfrac\lambda2,\lambda,\dots$; antinodes at $x=\tfrac\lambda4,\tfrac{3\lambda}4,\dots$
String fixed at both ends§ 14.3 →
$\lambda_n=\dfrac{2L}n$, $\ \nu_n=\dfrac{n}{2L}\sqrt{\dfrac T\mu}$, $\ n=1,2,3,\dots$
Air columns (no end correction)§ 14.3 →
Closed at one end: $\nu=\dfrac{(2n-1)v}{4L}$ (odd harmonics); $\ $ open both ends: $\nu=\dfrac{nv}{2L}$ (all)
Superposition giving beats§ 14.4 →
$a\cos\omega_1t+a\cos\omega_2t=2a\cos\left(\tfrac{\omega_1-\omega_2}{2}t\right)\cos\left(\tfrac{\omega_1+\omega_2}{2}t\right)$ (nearly equal frequencies)
Beat frequency§ 14.4 →
$\nu_\text{beat}=|\nu_1-\nu_2|$, $\ T_\text{beat}=\dfrac1{|\nu_1-\nu_2|}$; tone heard $\dfrac{\nu_1+\nu_2}2$
Finding which note is higher§ 14.4 →
Wax lowers $\nu$, filing raises it; tighter or shorter string raises $\nu$. Beats slow $\Rightarrow$ $\nu$ moved towards the reference
Doppler effect (beyond NCERT)§ 14.4 →
$\nu'=\nu\,\dfrac{v+v_o}{v-v_s}$ ($v_o$, $v_s$ positive towards each other; air at rest)