Physics Formula Sheet

• Chapter 1: Measurements

  Show/Hide Details

  • Uncertainties

    Show/Hide Formulas
    • Absolute Uncertainty: $\delta X$
    • Fractional Uncertainty: $\frac{\delta X}{X}$
    • Percentage Uncertainty: $\frac{\delta X}{X} \times 100\%$
    • Percentage uncertainty = $\frac{L.C}{\text{measured value}} \times 100$
    • Uncertainty in timing experiment = $\frac{\text{L.C}}{\text{no. of vibration}}$

  • Algebra with Significant Figures

    Show/Hide Formulas
    • Division and Multiplication Example: $\frac{4.54 \times 2.324}{1.3365} = 7.89447063 = 7.89$ (retained to 3 significant figures)
    • Addition and Subtraction Example: $4.345 + 23.51 = 27.855 = 27.86$ (retained to 2 decimal places)
    • Addition and Subtraction Example: $101.2401 - 1.0 = 100.2401 = 100.2$ (retained to 1 decimal place)
    • Addition and Subtraction Example: $101.2401 - 1 = 100.2401 = 100$ (retained to no decimal place)

  • Dimensional Analysis

    Show/Hide Formulas
    • Velocity: $[L/T]$
    • Acceleration: $[L/T^{2}]$
    • Force: $[ML/T^{2}]$
    • Work: $[ML^{2}/T^{2}]$
    • Energy: $[ML^{2}/T^{2}]$
    • Power: $[ML^{2}/T^{3}]$

  • Constants and Conversions

    Show/Hide Formulas
    • $1 N = 10^5 \text{ dyne}$
    • $1 \text{ Joule} = 10^7 \text{ erg}$
    • $1 \text{ Watt} = 10^7 \text{ erg/s}$
    • $u_{1}=kx^{a}$ (Example of power factor rule)

• Chapter 2: Vectors and Equilibrium

  Show/Hide Details

  • Vector Properties and Operations

    Show/Hide Formulas
    • If $\vec{a}=(5N)(\text{East})$, $\vec{b}=(5N)(\text{West})$ then $\vec{a}\ne\vec{b}$ but $|\vec{a}|=|\vec{b}|=5N$
    • Vector does not change when rotated through an angle $\theta=2n\pi$ where $n=1,2,3,\dots$
    • $2\pi=360^{\circ}$
    • Unit vector: $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$
    • Position vector in two dimensions: $\vec{r} = a\hat{i} + b\hat{j}$
    • Position vector in three dimensions: $\vec{r} = a\hat{i} + b\hat{j} + c\hat{k}$
    • Multiplication of a Vector with a Scalar: New vector $= n\vec{A}$
    • Magnitude of new vector: $n|\vec{A}|$

  • Rectangular Components of a Vector

    Show/Hide Formulas
    • X-component: $A_x = A\cos\theta$
    • Y-component: $A_y = A\sin\theta$
    • Magnitude from components: $|\vec{A}|^2 = A_x^2 + A_y^2$
    • Trigonometric Identity: $(\cos^2\theta+\sin^2\theta=1)$
    • Direction from components: $\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$
    • When vector A makes angle $\theta$ with y-axis: $A_y = A\sin\theta$
    • When vector A makes angle $\theta$ with y-axis: $A_x = A\cos\theta$
    • When vector A makes angle $\theta$ with y-axis: $\theta = \tan^{-1}\left(\frac{A_x}{A_y}\right)$

  • Addition of Vectors by Rectangular Components

    Show/Hide Formulas
    • Resultant magnitude: $R = \sqrt{(R_x)^2 + (R_y)^2}$
    • X-component of resultant: $R_x = A_x + B_x$
    • Y-component of resultant: $R_y = A_y + B_y$
    • Direction of resultant: $\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)$
    • For multiple vectors (magnitude): $R = \sqrt{(A_x+B_x+\dots)^2 + (A_y+B_y+\dots)^2}$
    • For multiple vectors (direction): $\theta = \tan^{-1}\left(\frac{A_y+B_y+\dots}{A_x+B_x+\dots}\right)$

  • Parallelogram Law

    Show/Hide Formulas
    • Final resultant magnitude: $c^2=a^2+b^2+2ab\cos\theta$
    • Final resultant magnitude (vector notation): $|\vec{c}|=\sqrt{a^2+b^2+2ab\cos\theta}$
    • Maximum resultant: $c_{max} = \sqrt{(a+b)^2} = |a+b|$
    • Minimum resultant: $c_{min} = \sqrt{(a-b)^2} = |a-b|$
    • For subtraction: $|\vec{c}|=\sqrt{a^2+b^2-2ab\cos\theta}$

  • Scalar or Dot Products

    Show/Hide Formulas
    • Definition: $\vec{A} \cdot \vec{B} = AB\cos\theta$
    • Commutative Law: $\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$
    • Associative Law: $\vec{A} \cdot (\vec{B} \cdot \vec{C}) = (\vec{A} \cdot \vec{B}) \cdot \vec{C}$
    • Distributive Law: $\vec{A} \cdot (\vec{B}+\vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$
    • Perpendicular vectors: If $\theta = 90^\circ$, then $\vec{A} \cdot \vec{B} = 0$
    • Parallel vectors: If $\theta = 0^\circ$, then $\vec{A} \cdot \vec{B} = AB$
    • Anti-parallel vectors: If $\theta = 180^\circ$, then $\vec{A} \cdot \vec{B} = -AB$
    • Unit vector dot products (perpendicular): $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$
    • Unit vector dot products (parallel): $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$
    • Angle between two vectors: $\theta = \cos^{-1}\left(\frac{A_x B_x + A_y B_y + A_z B_z}{AB}\right)$
    • Self scalar product: $\vec{A} \cdot \vec{A} = A^2$

  • Vector or Cross Product

    Show/Hide Formulas
    • Definition: $\vec{A} \times \vec{B} = AB\sin\theta\hat{n}$
    • Non-commutative: $\vec{A} \times \vec{B} \ne \vec{B} \times \vec{A}$
    • Parallel/Anti-parallel vectors: $\vec{A} \times \vec{A} = 0$
    • Unit vector cross products (cyclic): $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$
    • Scalar multiplication: $(\vec{m} \vec{A}) \times \vec{B} = \vec{A} \times (m\vec{B})$
    • Parallel/Anti-parallel vectors (zero product): $\vec{A} \times \vec{B} = \vec{0}$
    • Component form: $\vec{A} \times \vec{B} = (A_x \hat{i} + A_y \hat{j} + A_z \hat{k}) \times (B_x \hat{i} + B_y \hat{j} + B_z \hat{k})$
    • Determinant form: $\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$
    • Scalar triple product (expanded): $\vec{A} \cdot (\vec{B} \times \vec{C}) = A_x(B_y C_z - B_z C_y) - A_y(B_x C_z - B_z C_x) + A_z(B_x C_y - B_y C_x)$
    • Scalar triple product (determinant): $\vec{A} \cdot (\vec{B} \times \vec{C}) = \begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{vmatrix}$

  • Torque

    Show/Hide Formulas
    • Vector definition: $\vec{\tau} = \vec{r} \times \vec{F}$
    • Magnitude: $\tau = rF\sin\theta$
    • Magnitude (perpendicular): $\tau = rF$ (if $\theta = 90^\circ$)
    • Torque of a couple: $\tau = Fd$

  • Conditions of Equilibrium

    Show/Hide Formulas
    • First Condition (Translational): $\sum \vec{F} = 0$
    • For coplanar forces: $\sum F_x = 0$ and $\sum F_y = 0$
    • Example X-component sum: $\sum F_x = 5N + 3N - 4N - 7N - 5N = \text{zero}$
    • Example Y-component sum: $\sum F_y = 10N - 5N - 5N = \text{zero}$
    • Second Condition (Rotational): $\sum \vec{\tau} = 0$
    • Example Clockwise torque: $-10N \times 2m = -20~N~m$
    • Example Anticlockwise torque: $10N \times 2m = 20N~m$
    • Example Sum of torques: $\sum \tau = 20~N~m - (20~N~m) = \text{zero}$

• Chapter 3: Motion and Force

  Show/Hide Details

  • Acceleration

    Show/Hide Formulas
    • Rate of change of velocity: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$
    • Slope of velocity-time graph: $\vec{a} = \tan\theta$

  • Average Speed/Acceleration

    Show/Hide Formulas
    • Average speed (equal distance): $v_{av} = \frac{2v_1v_2}{v_1+v_2}$
    • Average speed (equal time): average speed = $\frac{v_1+v_2}{2}$
    • Average acceleration (two intervals): $a = \frac{a_1+a_2}{t_1+t_2}$

  • Equations of Motion (Galileo)

    Show/Hide Formulas
    • First Equation: $v_f = v_i + at$ or $v_f = v_i + gt$
    • Second Equation: $s = v_i t + \frac{1}{2}at^2$ or $s = v_i t + \frac{1}{2}gt^2$
    • Third Equation: $v_f^2 = v_i^2 + 2as$ or $v_f^2 = v_i^2 + 2gs$
    • Distance in nth second: $s_n = v_i + \frac{1}{2}a(2n-1)$

  • Newton's Laws of Motion

    Show/Hide Formulas
    • Second Law (Proportionality): $F \propto \text{Mass}$ x $\text{Change in velocity per second}$
    • Second Law (Standard Form): $F = ma$
    • Third Law: $\vec{F}_{action} = -\vec{F}_{reaction}$

  • Momentum and Impulse

    Show/Hide Formulas
    • Linear Momentum: $\vec{p} = m\vec{v}$
    • Zero Momentum: if $\vec{v} = 0$ then $\vec{p} = \vec{0}$
    • Impulse Definition: $\vec{I} = \Delta \vec{p}$
    • Impulse Definition: $\vec{I} = \vec{F}\Delta t$
    • Law of Conservation of Linear Momentum: $m_1v_1+m_2v_2 = m_1v_1'+m_2v_2'$

  • Collisions (Elastic in One Dimension)

    Show/Hide Formulas
    • Relative velocities: $v_1+v_1' = v_2+v_2'$
    • Relative velocities: $v_1-v_2 = -(v_1'-v_2')$
    • Final velocity of $m_1$: $v_1' = \frac{(m_1-m_2)V_1}{(m_1+m_2)} + \frac{2m_2V_2}{(m_1+m_2)}$
    • Final velocity of $m_2$: $v_2' = \frac{2m_1V_1}{(m_1+m_2)} + \frac{(m_2-m_1)V_2}{(m_1+m_2)}$

  • Force Due to Water Flow

    Show/Hide Formulas
    • Force: $\vec{F} = \frac{m}{t}\vec{v}$

  • Momentum and Explosive Forces

    Show/Hide Formulas
    • Recoil velocity: $v' = \frac{-m}{M}v$

  • Propulsion of Rocket

    Show/Hide Formulas
    • Instantaneous acceleration: $a = mv/M$

  • Friction

    Show/Hide Formulas
    • Coefficient of friction: $\mu = \frac{\text{Limiting friction }(f)}{\text{Normal reaction }(R)}$
    • Coefficient of friction: $\mu = \frac{f}{R}$
    • Limiting friction: $f = \mu R$
    • Force for horizontal surface: $F = mg$
    • Static friction force: $f_s = \mu_s R$
    • Kinetic friction force: $f_k = \mu_k R$
    • Angle of friction: $\tan\theta = \frac{f}{R} = \mu$
    • Angle of static friction: $\tan\theta_s = \frac{f_s}{R} = \mu_s$
    • Angle of kinetic friction: $\tan\theta_k = \frac{f_k}{R} = \mu_k$

  • Masses in Contact

    Show/Hide Formulas
    • Newton's second law for $m_1$: $F_1 - F_r = m_1 a$
    • Newton's second law for $m_2$: $F_r = m_2 a$
    • Force of reaction: $F_r = \frac{m_2 F}{m_1+m_2}$
    • Acceleration: $a = \frac{F}{m_1+m_2}$

  • Masses Connected by Strings

    Show/Hide Formulas
    • Tension (1 string 2 masses): $T = \frac{m_2 F}{m_1+m_2}$
    • Acceleration (1 string 2 masses): $a = \frac{F}{m_1+m_2}$
    • Acceleration (2 string 3 masses): $a = \frac{F}{m_1+m_2+m_3}$
    • Tension string1 (2 string 3 masses): $T_1 = \frac{(m_1+m_2)F}{m_1+m_2+m_3}$
    • Tension string2 (2 string 3 masses): $T_2 = \frac{(m_2+m_3)F}{m_1+m_2+m_3}$

  • Tension and Acceleration in a String (Pulley)

    Show/Hide Formulas
    • Acceleration: $a = \frac{m_1 - m_2}{m_1 + m_2}g$
    • Tension: $T = \frac{2m_1m_2}{m_1+m_2}g$

  • Atwood Machine

    Show/Hide Formulas
    • Acceleration due to gravity: $g = \frac{m_1+m_2}{m_1-m_2}a$
    • Acceleration (one body vertical): $a = \frac{m_1 g}{m_1+m_2}$
    • Tension (one body vertical): $T = \frac{m_1 m_2 g}{m_1+m_2}$

  • Mechanical Advantage

    Show/Hide Formulas
    • Definition: MECHANICAL ADVANTAGE = $\frac{\text{LOAD}}{\text{EFFORT}}$
    • Load: LOAD = MECHANICAL ADVANTAGE $\times$ EFFORT
    • Effort: EFFORT = $\frac{\text{LOAD}}{\text{MECHANICAL ADVANTAGE}}$

  • Projectile Motion

    Show/Hide Formulas
    • Horizontal coordinate: $x = v_o \cos\theta_o t$
    • Vertical coordinate: $y = v_o \sin\theta_o t - \frac{1}{2}gt^2$
    • Time to max height: $t = \frac{v_y \sin\theta_o}{g}$
    • Total time of flight: $T = \frac{2v_y \sin\theta_o}{g}$
    • Maximum height: $H = \frac{v_y^2 \sin^2\theta_o}{2g}$
    • Range (horizontal): $R = \frac{v_o^2 \sin(2\theta_o)}{g}$
    • Maximum horizontal range ($\theta = 45^\circ$): $R_{max} = \frac{v_o^2}{g}$
    • Relation between Range and Height: $R\tan\theta = 4H$

• Chapter 4: Work and Energy

  Show/Hide Details

  • Work

    Show/Hide Formulas
    • Dot product definition: $W = \vec{F} \cdot \vec{d}$
    • Scalar definition: $W = Fd \cos\theta$
    • Work along closed path (conservative field): $W_{total} = 0$
    • Work along closed path (non-conservative field): $W_{(AB)} \ne (W_{(AB)})_r$
    • Spring force: $F = kx$
    • Work done by variable force (integral): Total Area = $\lim_{\Delta d \to 0} \sum F \cos\theta \Delta d = \text{work done}$

  • Power

    Show/Hide Formulas
    • Definition: $P = \text{work/time}$
    • Linear motion: $P = \frac{W}{t}$
    • Linear motion (vector): $P = \vec{F} \cdot \vec{v}$
    • Angular motion: $P = \tau \cdot \omega$
    • Average power: $P = \frac{W}{t}$
    • Instantaneous power: $P_{inst} = \frac{\Delta W}{\Delta t}$
    • Unit conversion: $1~W = 1~J/s$
    • Unit conversion: $1~h.p = 746~watt$
    • Unit conversion: $1~h.p = 550~\text{foot pound/sec}$
    • Unit conversion: $1~kW~h = 3.6 \times 10^6~J$

  • Energy

    Show/Hide Formulas
    • Kinetic Energy: $K.E = \frac{1}{2}mv^2$
    • Kinetic Energy (momentum): $K.E = \frac{p^2}{2m}$
    • Gravitational Potential Energy: $P.E = mgh$
    • Elastic Potential Energy: $P.E = \frac{1}{2}kx^2$
    • Electric Potential Energy: $P.E = q\Delta V$
    • Work-Energy Principle: Work = $(K.E)_f - (K.E)_i$
    • Absolute Gravitational P.E: $U_m = -\frac{GMm}{R}$
    • Escape Velocity: $V_{esc} = \sqrt{\frac{2GM}{R}}$
    • Escape Velocity (g): $V_{esc} = \sqrt{2gR_e}$
    • Escape Velocity (numerical values): $1.2 \times 10^4~m~s^{-1} = 6.96~\text{miles}~s^{-1} = 11.2~\text{kms}^{-1} = 25000~\text{miles}~h^{-1} = 40320~\text{kmh}^{-1}$
    • Interconversion of K.E and P.E: $mgh + \frac{1}{2}mv^2 = \frac{1}{2}mv^2$
    • Velocity from P.E conversion: $v = \sqrt{2gh}$
    • Einstein's Mass Energy: $E = mc^2$

• Chapter 5: Rotational and Circular Motion

  Show/Hide Details

  • Angular Displacement

    Show/Hide Formulas
    • Arc length: $s = r\theta$
    • Degree to radian conversion: $1^\circ = \frac{\pi}{180}~\text{rad}$
    • Numerical conversion: $1^\circ = 0.0174~\text{rad}$
    • Numerical conversion: $1~\text{rad} = 57.3^\circ$

  • Angular Velocity & Acceleration

    Show/Hide Formulas
    • Average angular velocity: $\omega_{avg} = \frac{\Delta\theta}{\Delta t}$
    • Instantaneous angular velocity: $\omega_{ins} = \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t}$
    • Tangential velocity relation: $v = r\omega$
    • Tangential velocity (vector): $\vec{v} = \vec{\omega} \times \vec{r}$
    • Angular acceleration: $\alpha = \frac{\Delta\omega}{\Delta t}$
    • Tangential acceleration (vector): $\vec{a} = \vec{\alpha} \times \vec{r}$

  • Centripetal Force & Acceleration

    Show/Hide Formulas
    • Total acceleration: $\vec{a} = \vec{a}_c + \vec{a}_t$
    • Centripetal force (vector, angular): $\vec{F}_c = -m\omega^2\vec{r}$
    • Centripetal force (vector, linear): $\vec{F}_c = -\left(\frac{mv^2}{r}\right)\hat{r}$
    • Centripetal acceleration (vector, angular): $\vec{a}_c = -\omega^2\vec{r}$
    • Centripetal acceleration (vector, linear): $\vec{a}_c = -\left(\frac{v^2}{r}\right)\hat{r}$

  • Comparison of Linear and Rotatory Motion

    Show/Hide Formulas
    • Linear velocity: $\vec{v} = \frac{\Delta d}{\Delta t}$
    • Angular velocity: $\vec{\omega} = \frac{\Delta\theta}{\Delta t}$
    • Linear displacement: $\vec{d}$
    • Angular displacement: $\Delta\theta$
    • Linear acceleration: $\vec{a} = \frac{\Delta v}{\Delta t}$
    • Angular acceleration: $\vec{\alpha} = \frac{\Delta\omega}{\Delta t}$
    • Mass: $m$
    • Moment of inertia: $I = mr^2$
    • Linear momentum: $\vec{P} = m\vec{v}$
    • Angular momentum: $\vec{L} = I\vec{\omega}$
    • Linear impulse: $\vec{J} = \vec{F} \times \Delta t$
    • Angular impulse: $\vec{\tau} \times \Delta t$
    • Force: $\vec{F} = m\vec{a}$ or $\vec{F} = \frac{\Delta \vec{p}}{\Delta t}$
    • Torque: $\vec{\tau} = I\vec{\alpha}$ or $\vec{\tau} = \frac{\Delta \vec{L}}{\Delta t}$
    • Linear Work: $W = \vec{F} \cdot \vec{d}$
    • Rotational Work: $W_{rot} = \frac{1}{2}I(\omega_f^2 - \omega_i^2)$
    • Linear Kinetic Energy: $K.E = \frac{1}{2}mv^2$
    • Rotational Kinetic Energy: $K.E_{rot} = \frac{1}{2}I\omega^2$
    • Newton's First Law (Linear): If $F=0$ then $v=\text{constant}$
    • Newton's Second Law (Linear): $F=ma$
    • Newton's Third Law (Linear): $F_{12} = -F_{21}$
    • Newton's First Law (Rotational): If $\tau=0$ then $\omega=\text{constant}$
    • Newton's Second Law (Rotational): $\tau=I\alpha$
    • Newton's Third Law (Rotational): $\tau_{12} = -\tau_{21}$
    • Linear Equation of Motion (i): $v_f = v_i + at$
    • Linear Equation of Motion (ii): $s = v_i t + \frac{1}{2}at^2$
    • Linear Equation of Motion (iii): $v_f^2 = v_i^2 + 2as$
    • Rotational Equation of Motion (i): $\omega_f = \omega_i + \alpha t$
    • Rotational Equation of Motion (ii): $\theta = \omega_i t + \frac{1}{2}\alpha t^2$
    • Rotational Equation of Motion (iii): $\omega_f^2 = \omega_i^2 + 2\alpha\theta$
    • Distance in nth second: $s_n = v_i + \frac{a}{2}(2n-1)$
    • Angle in nth second: $\theta_n = \omega_i + \frac{\alpha}{2}(2n-1)$
    • Distance with constant velocity: $s = vt$
    • Angle with constant angular velocity: $\theta = \omega t$
    • Average linear velocity: $<v> = \frac{v_i+v_f}{2}$
    • Average angular velocity: $<\omega> = \frac{\omega_i+\omega_f}{2}$

  • Moment of Inertia

    Show/Hide Formulas
    • Definition: $I = mr^2$
    • Rotational Analogue of Newton's Second Law: $\tau = I\alpha$
    • Hoop or cylindrical shell: $I_c = MR^2$
    • Hollow cylinder: $I_c = \frac{1}{2}M(R_1^2+R_2^2)$
    • Solid cylinder or disk: $I_c = \frac{1}{2}MR^2$
    • Rectangular plate: $I_c = \frac{1}{12}M(a^2+b^2)$
    • Long thin rod: $I_c = \frac{1}{12}ML^2$
    • Solid sphere: $I_c = \frac{2}{5}MR^2$
    • Thin spherical shell: $I_c = \frac{2}{3}MR^2$

  • Angular Momentum

    Show/Hide Formulas
    • Linear momentum relation: $mv = p$
    • Definition (scalar): $L = mvr \sin\theta$
    • Definition (vector): $L = \vec{r} \times \vec{p}$
    • For circular orbit: $L = mvr$
    • Definition (rotational): Angular Momentum $L = \text{Moment of Inertia} \times \text{Angular Velocity}$
    • Conservation Law: $L_1 + L_2 + \sum \vec{\tau}_{ext} = 0$
    • Conservation Law (simplified): $I_1 \omega_1 = I_2 \omega_2$

  • Rotational Kinetic Energy

    Show/Hide Formulas
    • General: $K.E_{rot} = \frac{1}{2}I\omega^2$
    • Disc: $K.E_{rot} = \frac{1}{4}mv^2 = \frac{1}{2}(K.E_{lin})$
    • Hoop: $K.E_{rot} = \frac{1}{2}mv^2 = (K.E_{lin})$
    • Velocity of hoop (inclined plane): $v = \sqrt{gh}$
    • Velocity of disc (inclined plane): $v = \sqrt{\frac{4}{3}gh}$

  • Motion in a Vertical Circle

    Show/Hide Formulas
    • Tension at highest point: $T_A + mg = \frac{mv^2}{r}$
    • Minimum tension at highest point: $T_A = 0$
    • Minimum velocity at highest point: $mg = \frac{mv^2}{r}$
    • Minimum velocity at highest point: $v_A = \sqrt{gr}$
    • Critical speed: $v_c = \sqrt{gr}$
    • Tension at lowest point: $T_B - mg = \frac{mv_B^2}{r}$
    • Tension at lowest point: $T_B = mg + \frac{mv_B^2}{r}$
    • Velocity at lowest point: $v_B = \sqrt{5gr}$
    • Velocity at horizontal points: $v_X = v_Y = \sqrt{3gr}$

  • Banking of Curved Tracks

    Show/Hide Formulas
    • Horizontal component of normal force: $R \sin\theta = \frac{mv^2}{r}$
    • Vertical component of normal force: $R \cos\theta = mg$
    • Angle of banking: $\tan\theta = \frac{v^2}{rg}$
    • Angle of banking (inverse tan): $\theta = \tan^{-1}\left(\frac{v^2}{rg}\right)$
    • Maximum speed without skidding: $V_{max} = \sqrt{\mu rg}$
    • Maximum speed without skidding: $V_{max} = \sqrt{\frac{\mu rg}{2\pi}}$

  • Universal Gravitation

    Show/Hide Formulas
    • Newton's Law: $F = G\frac{m_1m_2}{r^2}$
    • Acceleration due to gravity: $g = \frac{GM}{R^2}$
    • Mass of Earth (density): $M = \frac{4}{3}\pi R^3 \rho$
    • Acceleration due to gravity (density): $G = \frac{4}{3}\pi R^3 \rho$

  • Variations in g

    Show/Hide Formulas
    • Above surface: $g' = GM/(R+h)^2$
    • Above surface (approximate): $g' = g\left[1-\frac{2h}{R}\right]$
    • With depth: $g' = g\left[1-\frac{d}{R}\right]$
    • At equator: $g_e = \frac{GM}{R_e^2}$
    • At poles: $g_p = \frac{GM}{R_p^2}$
    • Equator vs Poles: $g_p = g_e + 0.018~\text{ms}^{-2}$

  • Real and Apparent Weight

    Show/Hide Formulas
    • Stationary/Constant velocity lift: $w' = mg$
    • Accelerating upward lift: $w' = (g+a)m$
    • Accelerating downward lift: $w' = (g-a)m$

  • Artificial Satellites

    Show/Hide Formulas
    • Critical velocity: $v_{crit} = \sqrt{gR} = 7.9~\text{km s}^{-1}$
    • Orbital velocity: $v = \sqrt{\frac{GM}{r}}$
    • Frequency for artificial gravity: $F = \frac{1}{2\pi}\sqrt{\frac{g}{R}}$
    • Time period for artificial gravity: $T = 2\pi\sqrt{\frac{R}{g}}$
    • Radius of geo-stationary orbit: $r = \left(\frac{GMT^2}{4\pi^2}\right)^{1/3}$
    • Kepler's Third Law: $T^2 \propto r^3$

• Chapter 6: Fluid Dynamics

  Show/Hide Details

  • Density and Pressure

    Show/Hide Formulas
    • Density: $\rho = \frac{\text{Mass}(m)}{\text{Volume}(V)}$
    • Pressure (definition): $P = \frac{\text{Force}(F)}{\text{Area}(A)}$
    • Pressure (fluid depth): $P = \rho gh$
    • Pascal's Law: $\frac{F_1}{A_1} = \frac{F_2}{A_2}$
    • Unit conversion: $1~\text{atm} = 1.013 \times 10^5~\text{Pa}$
    • Unit conversion: $1~\text{bar} = 10^5~\text{Pa}$
    • Unit conversion: $1~\text{torr} = 133.3~N~m^{-2}$
    • Unit conversion: $1~\text{torr} = 1~\text{mm of Hg}$
    • Thrust: Thrust = Pressure $\times$ Area

  • Buoyancy and Floatation

    Show/Hide Formulas
    • Buoyant Force (Archimedes' Principle): $B = V\rho g$

  • Viscous Drag and Stokes' Law

    Show/Hide Formulas
    • Stokes' Law: $F_D = 6\pi\eta r v$

  • Terminal Velocity

    Show/Hide Formulas
    • Terminal velocity (general): $v_t = \sqrt{\frac{2\rho g}{6\pi r\eta}}$
    • Terminal velocity (simplified): $v_t = \left(\frac{2\rho g}{9\eta}\right)r^2$

  • Fluid Flow and Equation of Continuity

    Show/Hide Formulas
    • Mass flow rate: $P_1A_1V_1 = P_2A_2V_2$
    • Mass flow rate (simplified): $P~av = \text{mass/time}$
    • Volume flow rate: $A_1V_1 = A_2V_2$
    • Volume flow rate (simplified): $AV = \text{volume/time}$

  • Bernoulli's Equation and Applications

    Show/Hide Formulas
    • Bernoulli's Equation: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
    • Potential energy per unit volume: $\rho gh$
    • Kinetic energy per unit volume: $\frac{1}{2}\rho v^2$
    • Kinetic energy: $\frac{1}{2}mv^2$
    • Pressure energy: $PV$
    • Pressure energy per unit volume: $P$
    • Torricelli's Theorem (Efflux Velocity): $v = \sqrt{2g(h_1-h_2)}$
    • Venturimeter relation: $P_1 - P_2 = \frac{1}{2}\rho v^2$

• Chapter 7: Oscillations

  Show/Hide Details

  • Simple Harmonic Motion (SHM)

    Show/Hide Formulas
    • Angular frequency: $\omega=2\pi/T\rightarrow\omega=2\pi f$
    • Displacement (from positive extreme): $x=x_{0}sin(\omega t+90^{\circ})=x_{0}cos~\omega t$
    • Displacement: $x=x_{o}sin~\omega t$
    • Acceleration: $a=-\omega^{2}x$
    • Time period of projection: $T=\frac{2\pi}{\omega}$
    • Speed of projection: $v=\omega\sqrt{r^{2}-x^{2}}$ where $r=$ radius of the circle = amplitude of S.H.M

  • Mass-Spring System

    Show/Hide Formulas
    • Hooke's Law: $F=k~x$
    • Spring constant: $k=\frac{F}{x}$
    • Angular frequency (spring-mass): $\omega=\sqrt{\frac{k}{m}}$
    • Acceleration of mass-spring system: $\overline{a}=-\frac{k}{m}\vec{x}$
    • Time period (mass-spring): $T=2\pi\sqrt{\frac{m}{k}}$
    • Displacement (mass-spring waveform): $x=x_{o}sin[\frac{2\pi}{T}]t$
    • Springs in series: $\frac{1}{k}=\frac{1}{k_{1}}+\frac{1}{k_{2}}+ ...$
    • Springs in parallel: $k=k_{1}+k_{2}+ ...$
    • Velocity proportionality (mass-spring): $ v_{ins} \propto \sqrt{1-\frac{x^{2}}{x_{o}^{2}}}$
    • Instantaneous velocity (mass-spring): $v_{ins}=v_{max}\sqrt{1-\frac{x^{2}}{x_{o}^{2}}}.$
    • Maximum speed (mass-spring): $v_{max}=x_{o}\sqrt{\frac{k}{m}} = v_{o}$
    • Vertical spring force: $F=mg=kx$
    • Mass-spring ratio (vertical): $\frac{m}{k}=\frac{x}{g}$
    • Time period (vertical spring): $T=2\pi\sqrt{\frac{x}{g}}$

  • Simple Pendulum

    Show/Hide Formulas
    • Restoring force (simple pendulum): $F_{r}=-mg~sin\theta,$
    • Restoring force (small amplitude): $F_{r}=-mg\theta$
    • Angular frequency (simple pendulum): $\omega=\sqrt{\frac{g}{l}}$
    • Acceleration (simple pendulum, small amplitude): $a=-(\frac{g}{l})x$
    • Time period (simple pendulum): $T=2\pi\sqrt{\frac{l}{g}}$
    • Frequency (simple pendulum): $f=\frac{1}{2\pi}\sqrt{\frac{g}{l}}$
    • Acceleration (non-SHM pendulum): $a=-g~sin~\theta$
    • Time period (pendulum in ascending lift): $T=2\pi\sqrt{\frac{l}{g+a}}$
    • Time period (pendulum in descending lift): $T=2\pi\sqrt{\frac{l}{g-a}}$

  • Energy Conservation in SHM

    Show/Hide Formulas
    • Instantaneous Kinetic Energy: $K.E_{inst}=\frac{1}{2}k~x_{o}^{2}(1-\frac{\dot{x}^{2}}{x_{o}^{2}})$
    • Maximum Kinetic Energy: $(K.E)_{max}=\frac{1}{2}k\dot{x}_{o}^{2}$
    • Instantaneous Kinetic Energy (alternative): $K.E_{inst}=(K.E)_{max}(1-\frac{x^{2}}{x_{o}^{2}})$
    • Instantaneous Potential Energy: $P.E_{inst.}=\frac{1}{2}kx^{2}$
    • Maximum Potential Energy: $(P.E)_{max}=\frac{1}{2}kx_{o}^{2}$
    • Total Energy in SHM: $=\frac{1}{2}kx_{o}^{2}$

  • Resonance

    Show/Hide Formulas
    • Electrical Resonance Frequency: $f=\frac{1}{2\pi\sqrt{LC}}$

• Chapter 8: Waves

  Show/Hide Details

  • Wave Properties

    Show/Hide Formulas
    • Wave speed: $v = f\lambda$
    • Particles in phase (path difference): $n\lambda$
    • Particles out of phase (path difference): $(n + 1/2)\lambda$

  • Velocity of Sound in Air

    Show/Hide Formulas
    • General Mechanical Wave: $v = \sqrt{E/\rho}$
    • Newton's Formula (Isothermal): $v = \sqrt{P/\rho}$
    • Laplace's Formula (Adiabatic): $v = \sqrt{\gamma P/\rho}$
    • Temperature Dependence: $v_t = v_o + 0.61t$
    • Ratio of speeds with temperature: $\frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}}$

  • Superposition and Interference

    Show/Hide Formulas
    • Superposition Principle: $Y_{total} = Y_1 + Y_2 + \dots + Y_n$
    • Beat frequency: $f_{beat} = |f_A - f_B|$

  • Waves on Stretched Strings

    Show/Hide Formulas
    • Speed of wave: $v = \sqrt{T/m}$
    • Fundamental frequency (fixed both ends): $f_1 = \frac{1}{2l}\sqrt{\frac{T}{m}}$
    • Harmonics (fixed both ends): $f_n = n f_1$

  • Standing Waves in Air Columns

    Show/Hide Formulas
    • Length of open pipe (harmonics): $L = n\frac{\lambda}{2}$
    • Length of closed pipe (harmonics): $L = (2n-1)\frac{\lambda}{4}$
    • Fundamental frequency (open pipe): $f = v/2L$
    • Fundamental frequency (closed pipe): $f = v/4L$

  • Doppler's Effect

    Show/Hide Formulas
    • Source towards listener: $f' = \left(\frac{v}{v-u_s}\right)f$
    • Source away listener: $f' = \left(\frac{v}{v+u_s}\right)f$
    • Listener towards source: $f' = \left(\frac{v+u_o}{v}\right)f$
    • Listener away source: $f' = \left(\frac{v-u_o}{v}\right)f$
    • General (towards each other): $f' = \left(\frac{v+u_o}{v-u_s}\right)f$
    • General (away from each other): $f' = \left(\frac{v-u_o}{v+u_s}\right)f$

• Chapter 9: Physical Optics

  Show/Hide Details

  • Interference of Light (Young's Double Slit)

    Show/Hide Formulas
    • Optical path: $nd$
    • Bright fringes (constructive): $d \sin\theta = m\lambda$
    • Dark fringes (destructive): $d \sin\theta = (2m+1)\frac{\lambda}{2}$
    • Position of $m^{th}$ bright fringe: $y_m = \frac{m\lambda D}{d}$
    • Position of $m^{th}$ dark fringe: $y_m = \frac{(2m+1)\lambda D}{2d}$
    • Wavelength from bright fringe: $\lambda = \frac{y_m d}{mD}$
    • Wavelength from dark fringe: $\lambda = \frac{2y_m d}{(2m+1)D}$
    • Fringe width: $F.W = \frac{\lambda D}{d}$

  • Diffraction

    Show/Hide Formulas
    • Minima for narrow slit: $d \sin\theta = m\lambda$
    • Diffraction Grating (maxima): $d \sin\theta = m\lambda$
    • Grating element: $d = 1/N$
    • Resolving power (grating): $RP = \frac{\lambda}{\Delta\lambda}$
    • Resolving power (grating): $RP = N \times m$
    • Bragg's Law: $2d \sin\theta = m\lambda$

  • Polarization

    Show/Hide Formulas
    • Malus' Law (Intensity): $I = I_o \cos^2\theta$

• Chapter 10: Optical Instruments

  Show/Hide Details

  • Lenses and Magnification

    Show/Hide Formulas
    • Power of a Lens (meters): $P = \frac{1}{f}$
    • Power of a Lens (cm): $P = \frac{100}{f}$
    • Combination of Lenses (Power): $P = P_1 + P_2 + \dots + P_n$
    • Combination of Lenses (Focal Length): $\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} + \dots + \frac{1}{f_n}$
    • Lens Formula: $\frac{1}{p} + \frac{1}{q} = \frac{1}{f}$
    • Linear magnification: $M = \frac{q}{p}$
    • Linear magnification: $M = \frac{h_i}{h_o}$
    • Angular magnification: $M_a = \frac{\text{Angle formed at aided eye}}{\text{Angle formed at naked eye}}$

  • Resolving Power

    Show/Hide Formulas
    • Angle of Resolution: $\alpha_{min} = 1.22 \frac{\lambda}{D}$
    • Resolving power (lens): $R = \frac{D}{1.22\lambda}$

  • Microscopes

    Show/Hide Formulas
    • Simple Microscope Magnifying Power: $M = 1 + \frac{d}{f}$
    • Compound Microscope Magnification: $M = \frac{L}{f_o}\left(1+\frac{d}{f_e}\right)$

  • Refraction and Speed of Light

    Show/Hide Formulas
    • Refractive index of prism: $n = \frac{\sin\left(\frac{A+D_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$
    • Michelson's Speed of Light: $c = 16fd$
    • Index of Refraction: $n = \frac{c}{v}$
    • Snell's Law: $n_1 \sin\theta_1 = n_2 \sin\theta_2$

• Chapter 11: Heat & Thermodynamics

  Show/Hide Details

  • Temperature and Heat Capacity

    Show/Hide Formulas
    • Heat exchange: $Q = mc\Delta T$
    • Specific Heat: $c = \frac{Q}{m\Delta T}$

  • Temperature Conversions

    Show/Hide Formulas
    • Celsius to Fahrenheit: $T_F = \frac{9}{5}T_C + 32$
    • Fahrenheit to Celsius: $T_C = \frac{5}{9}(T_F - 32)$
    • Celsius to Kelvin: $T_K = T_C + 273$
    • Kelvin to Celsius: $T_C = T_K - 273$
    • Relation between scales: $\frac{T_C - 0}{100} = \frac{T_F - 32}{180} = \frac{T_K - 273}{100}$

  • Kinetic Theory of Gases

    Show/Hide Formulas
    • RMS speed of gas molecules: $v_{rms} = \sqrt{\frac{3KT}{m}}$
    • Pressure from KMT: $P = \frac{2}{3} \frac{N}{V} (K.E)_{avg}$
    • Combined Gas Law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

  • Laws of Thermodynamics

    Show/Hide Formulas
    • Thermodynamic work (constant pressure): $W = P\Delta V$
    • First Law of Thermodynamics: $\Delta Q = \Delta U + \Delta W$
    • Efficiency of Carnot Engine: $\eta_c = \left(1-\frac{T_L}{T_H}\right) \times 100$
    • Change in entropy: $\Delta S = \frac{\Delta Q}{T}$

  • Molar Specific Heats and Processes

    Show/Hide Formulas
    • Heat supplied at constant volume: $\Delta Q_v = n C_v \Delta T$
    • Heat supplied at constant pressure: $\Delta Q_p = n C_p \Delta T$
    • Mayer's relation: $C_p - C_v = R$
    • Adiabatic index: $\gamma = \frac{C_p}{C_v}$
    • Isothermal process equation: $PV=\text{constant}$
    • Adiabatic process equation: $PV^\gamma=\text{constant}$
    • Isochoric process relation: $\frac{P}{T}=\text{constant}$
    • Isobaric process relation: $\frac{V}{T}=\text{constant}$
    • Slope of Isothermal P-V curve: $\frac{\Delta P}{\Delta V} = -\frac{P}{V}$
    • Slope of Adiabatic P-V curve: $\frac{\Delta P}{\Delta V} = -\gamma\frac{P}{V}$
    • Isothermal Elasticity: $E=P$
    • Adiabatic Elasticity: $E=\gamma P$

• Chapter 12: Electrostatics

  Show/Hide Details

  • Charge and Electric Field

    Show/Hide Formulas
    • Charge per unit area: $\Delta Q/\Delta A$
    • Acceleration of charged particle in electric field: $a = qE/m$
    • Elementary charge: $e = 1.602 \times 10^{-19}~C$

  • Coulomb's Law

    Show/Hide Formulas
    • Electrical Force: $F = k \frac{q_1 q_2}{r^2}$
    • Coulomb's constant in vacuum: $k = 9 \times 10^9 N-m^2/C^2$
    • Permittivity of free space relation: $k = 1/(4\pi\epsilon_0)$
    • Permittivity of free space value: $\epsilon_0 = 8.85 \times 10^{-12} C^2/N-m^2$
    • Action-reaction pair: $\vec{F}_{12} = -\vec{F}_{21}$
    • Force in medium: $F' = (1/\epsilon_r)F$
    • Force in medium: $F' = F/\epsilon_r$
    • Relative Permittivity: $\epsilon_r = F/F'$

  • Electric Field Intensity

    Show/Hide Formulas
    • Definition: $E = \frac{F}{q_o}$
    • Due to point charge: $E = k \frac{q}{r^2}$
    • Due to infinite sheet: $E = \frac{\sigma}{2\epsilon_o}$
    • Between parallel plates (vacuum): $E = \frac{\sigma}{\epsilon_o}$
    • Between parallel plates (dielectric): $E = \frac{\sigma}{\epsilon_r \epsilon_o}$
    • Between parallel plates (dielectric): $E = \frac{\sigma}{K \epsilon_o}$

  • Electric Flux and Gauss's Law

    Show/Hide Formulas
    • Definition (vector): $\Phi_E = \vec{E} \cdot \vec{A}$
    • Definition (scalar): $\Phi_E = EA\cos\theta$
    • Maximum flux: $\phi_{max} = EA$ (when $\theta = 0^{\circ}$)
    • Minimum flux: $\phi_{min} = 0$ (when $\theta = 90^{\circ}$)
    • Gauss's Law: $\Phi_E = \frac{Q_{enclosed}}{\epsilon_o}$
    • Total flux: $\phi_{total} = (1/\epsilon_o)Q$

  • Electric Potential

    Show/Hide Formulas
    • Definition: $V = \frac{W}{q_o}$
    • Due to point charge: $V = k \frac{q}{r}$
    • Relation between E and V: $E = -\frac{\Delta V}{\Delta r}$

  • Capacitance

    Show/Hide Formulas
    • Definition: $C = \frac{Q}{V}$
    • Parallel plate (vacuum): $C = \frac{\epsilon_o A}{d}$
    • Parallel plate (dielectric): $C = \frac{\epsilon A}{d}$
    • Parallel plate (dielectric): $C = \frac{K\epsilon_o A}{d}$
    • Energy stored: $U = \frac{1}{2}QV$
    • Energy stored: $U = \frac{1}{2}CV^2$
    • Energy stored: $U = \frac{Q^2}{2C}$
    • Series combination: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$
    • Parallel combination: $C_{eq} = C_1 + C_2 + \dots$
    • RC Time Constant: $\tau = RC$

  • Millikan's Oil Drop Experiment

    Show/Hide Formulas
    • Balanced forces: $qE = mg$
    • Charge on oil drop: $q = \frac{mg}{E}$

• Chapter 13: Current Electricity

  Show/Hide Details

  • Current and Resistance

    Show/Hide Formulas
    • Current: $I = \frac{\Delta Q}{\Delta t}$
    • Ohm's Law: $V = IR$
    • Resistance: $R = \rho \frac{L}{A}$

  • Resistor Combinations

    Show/Hide Formulas
    • Series: $R_{eq} = R_1 + R_2 + \dots$
    • Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots$

  • Power Dissipation

    Show/Hide Formulas
    • Power: $P = VI$
    • Power: $P = I^2R$
    • Power: $P = \frac{V^2}{R}$

  • Kirchhoff's Rules

    Show/Hide Formulas
    • First Rule (Junction Rule): $\sum I = 0$
    • Second Rule (Loop Rule): $\sum V = 0$

  • Wheatstone Bridge and Potentiometer

    Show/Hide Formulas
    • Wheatstone Bridge (balance condition): $\frac{R_1}{R_2} = \frac{R_3}{R_4}$
    • Potentiometer (voltage measurement): $V = kL$

• Chapter 14: Electromagnetism

  Show/Hide Details

  • Magnetic Force

    Show/Hide Formulas
    • On current-carrying conductor (vector): $\vec{F} = I (\vec{L} \times \vec{B})$
    • On current-carrying conductor (magnitude): $F = ILB \sin\theta$
    • On moving charge (Lorentz Force, vector): $\vec{F} = q (\vec{v} \times \vec{B})$
    • On moving charge (Lorentz Force, magnitude): $F = qvB \sin\theta$

  • Motion of Charge in Magnetic Field

    Show/Hide Formulas
    • Radius of circular path: $r = \frac{mv}{qB}$
    • Frequency of revolution (cyclotron): $f = \frac{qB}{2\pi m}$

  • Magnetic Field and Ampere's Law

    Show/Hide Formulas
    • Magnetic Flux (vector): $\Phi_B = \vec{B} \cdot \vec{A}$
    • Magnetic Flux (scalar): $\Phi_B = BA \cos\theta$
    • Ampere's Law: $\oint \vec{B} \cdot d\vec{l} = \mu_o I_{enclosed}$
    • Magnetic field (long straight wire): $B = \frac{\mu_o I}{2\pi r}$
    • Magnetic field (inside solenoid): $B = \mu_o n I$
    • Magnetic field (inside toroid): $B = \frac{\mu_o N I}{2\pi r}$
    • Force per unit length (parallel wires): $\frac{F}{L} = \frac{\mu_o I_1 I_2}{2\pi d}$

  • Torque on Current Loop and Galvanometer

    Show/Hide Formulas
    • Torque on current loop (vector): $\vec{\tau} = \vec{\mu} \times \vec{B}$
    • Torque on current loop (magnitude): $\tau = NIAB \sin\alpha$
    • Magnetic dipole moment: $\vec{\mu} = NI\vec{A}$
    • Shunt resistance (ammeter): $R_s = \frac{I_g R_g}{I - I_g}$
    • Multiplier resistance (voltmeter): $R_m = \frac{V}{I_g} - R_g$

• Chapter 15: Electromagnetic Induction

  Show/Hide Details

  • Magnetic Flux and Faraday's Law

    Show/Hide Formulas
    • Magnetic Flux (vector): $\Phi_B = \vec{B} \cdot \vec{A}$
    • Magnetic Flux (scalar): $\Phi_B = BA \cos\theta$
    • Faraday's Law of Induction: $\mathcal{E} = -N \frac{\Delta\Phi_B}{\Delta t}$
    • Motional EMF: $\mathcal{E} = vBL \sin\theta$

  • Inductance

    Show/Hide Formulas
    • Self-induced EMF: $\mathcal{E}_L = -L \frac{\Delta I}{\Delta t}$
    • Self-inductance (solenoid): $L = \frac{\mu_o N^2 A}{l}$
    • Energy stored in inductor: $U = \frac{1}{2}LI^2$
    • Mutually induced EMF: $\mathcal{E}_s = -M \frac{\Delta I_p}{\Delta t}$
    • Mutual inductance (coaxial solenoids): $M = \frac{\mu_o N_p N_s A}{l}$

  • AC Generators and Transformers

    Show/Hide Formulas
    • AC Generator EMF: $\mathcal{E} = N A B \omega \sin(\omega t)$
    • Maximum AC Generator EMF: $\mathcal{E}_{max} = N A B \omega$
    • Transformer voltage ratio: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$
    • Transformer current ratio: $\frac{N_s}{N_p} = \frac{I_p}{I_s}$
    • Transformer efficiency: $\eta = \frac{\text{Output Power}}{\text{Input Power}} \times 100\%$
    • Transformer efficiency (power): $\eta = \frac{V_s I_s}{V_p I_p} \times 100\%$

• Chapter 16: Alternating Current

  Show/Hide Details

  • AC Quantities

    Show/Hide Formulas
    • Instantaneous AC current: $I = I_o \sin(\omega t + \phi)$
    • Instantaneous AC voltage: $V = V_o \sin(\omega t + \phi)$
    • RMS current: $I_{rms} = \frac{I_o}{\sqrt{2}}$
    • RMS voltage: $V_{rms} = \frac{V_o}{\sqrt{2}}$

  • Reactance and Impedance

    Show/Hide Formulas
    • Ohm's Law for Resistor: $V_R = I_R R$
    • Inductive Reactance: $X_L = \omega L = 2\pi f L$
    • Ohm's Law for Inductor: $V_L = I_L X_L$
    • Capacitive Reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$
    • Ohm's Law for Capacitor: $V_C = I_C X_C$
    • Impedance (RC series): $Z = \sqrt{R^2 + X_C^2}$
    • Phase angle (RC series): $\tan\phi = \frac{X_C}{R}$
    • Impedance (RL series): $Z = \sqrt{R^2 + X_L^2}$
    • Phase angle (RL series): $\tan\phi = \frac{X_L}{R}$
    • Impedance (RLC series): $Z = \sqrt{R^2 + (X_L - X_C)^2}$
    • Phase angle (RLC series): $\tan\phi = \frac{X_L - X_C}{R}$

  • Resonance and Power

    Show/Hide Formulas
    • Resonant frequency (RLC series): $f_r = \frac{1}{2\pi\sqrt{LC}}$
    • Quality Factor (Q-factor): $Q = \frac{\omega_r L}{R}$
    • Quality Factor (Q-factor): $Q = \frac{1}{\omega_r C R}$
    • Quality Factor (Q-factor): $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$
    • Power (Resistor): $P = I_{rms}^2 R$
    • Power (Resistor): $P = V_{rms} I_{rms}$
    • Average Power (AC): $P_{avg} = V_{rms} I_{rms} \cos\phi$
    • Power Factor: $\cos\phi = \frac{R}{Z}$
    • Wattless Current: $I_w = I_{rms} \sin\phi$

• Chapter 17: Physics of Solids

  Show/Hide Details

  • Stress and Strain

    Show/Hide Formulas
    • Stress: $\sigma = \frac{F}{A}$
    • Longitudinal Strain: $\epsilon_L = \frac{\Delta L}{L_o}$
    • Volumetric Strain: $\epsilon_V = \frac{\Delta V}{V_o}$
    • Shear Strain: $\epsilon_S = \tan\theta$
    • Shear Strain (approximate): $\epsilon_S \approx \theta$

  • Moduli of Elasticity

    Show/Hide Formulas
    • Hooke's Law: $\sigma = Y \epsilon$
    • Young's Modulus (definition): $Y = \frac{\text{Tensile Stress}}{\text{Longitudinal Strain}}$
    • Young's Modulus (formula): $Y = \frac{F/A}{\Delta L/L_o}$
    • Young's Modulus (simplified): $Y = \frac{FL_o}{A\Delta L}$
    • Bulk Modulus (definition): $B = \frac{\text{Volumetric Stress}}{\text{Volumetric Strain}}$
    • Bulk Modulus (formula): $B = \frac{-\Delta P}{\Delta V/V_o}$
    • Compressibility: $K = 1/B$
    • Shear Modulus (definition): $G = \frac{\text{Shear Stress}}{\text{Shear Strain}}$
    • Shear Modulus (formula): $G = \frac{F/A}{\theta}$

• Chapter 18: Electronics

  Show/Hide Details

  • Transistors and Amplifiers

    Show/Hide Formulas
    • Current gain ($\beta$ for common emitter): $\beta = \frac{\Delta I_C}{\Delta I_B}$
    • Voltage gain (transistor): $A_V = \beta \frac{R_C}{R_{in}}$
    • Inverting Amplifier Gain: $A_V = -\frac{R_f}{R_{in}}$
    • Non-Inverting Amplifier Gain: $A_V = 1 + \frac{R_f}{R_{in}}$
    • Adder/Summing Amplifier Output: $V_{out} = -R_f \left( \frac{V_1}{R_1} + \frac{V_2}{R_2} + \dots \right)$

  • Logic Gates (Boolean Algebra)

    Show/Hide Formulas
    • AND Gate: $Y = A \cdot B$
    • OR Gate: $Y = A + B$
    • NOT Gate: $Y = \bar{A}$
    • NAND Gate: $Y = \overline{A \cdot B}$
    • NOR Gate: $Y = \overline{A + B}$
    • XOR Gate: $Y = A \oplus B$
    • XOR Gate: $Y = A\bar{B} + \bar{A}B$
    • XNOR Gate: $Y = \overline{A \oplus B}$
    • XNOR Gate: $Y = A B + \bar{A}\bar{B}$

• Chapter 19: Dawn of Modern Physics

  Show/Hide Details

  • Relativity

    Show/Hide Formulas
    • Mass-Energy relation: $E=mc^2$
    • Relativistic Mass: $m=\frac{m_o}{\sqrt{1-\frac{v^2}{c^2}}}$
    • Lorentz Contraction (Length): $l=l_o\sqrt{1-\frac{v^2}{c^2}}$
    • Time Dilation: $t=\frac{t_o}{\sqrt{1-\frac{v^2}{c^2}}}$

  • Black Body Radiation

    Show/Hide Formulas
    • Wien's Displacement Law: $\lambda_{max} T = 2.9 \times 10^{-3}~m K$
    • Stefan-Boltzmann Law: $E = \sigma T^4$

  • Photoelectric Effect

    Show/Hide Formulas
    • Photon energy: $E=hf$
    • Photon energy: $E=\frac{hc}{\lambda}$
    • Photon momentum: $p=\frac{h}{\lambda}$
    • Photon momentum: $p=mc$
    • Photon momentum: $p=\frac{hf}{c}$
    • Einstein's Photoelectric Equation: $K.E_{max} = hf - \phi_o$
    • Einstein's Photoelectric Equation: $K.E_{max} = hf - hf_o$
    • Einstein's Photoelectric Equation: $K.E_{max} = h(f-f_o)$
    • Stopping potential relation: $eV_o = K.E_{max}$
    • Stopping potential relation: $eV_o = \frac{1}{2}mv_{max}^2$
    • Work function: $\phi_o = hf_o$
    • Work function: $\phi_o = \frac{hc}{\lambda_o}$

  • Compton Effect

    Show/Hide Formulas
    • Compton shift: $\Delta\lambda = \lambda_f - \lambda_i$
    • Compton shift: $\Delta\lambda = \frac{h}{m_o c}(1-\cos\theta)$
    • Compton wavelength: $\frac{h}{m_o c} = 2.43 \times 10^{-12}~m$

  • Pair Production and Annihilation

    Show/Hide Formulas
    • Pair production energy: $hf = 2m_o c^2 + K.E_{e^-} + K.E_{e^+}$
    • Rest mass energy of electron/positron: $m_o c^2 = 0.51~MeV$

  • De-Broglie Wavelength and Uncertainty Principle

    Show/Hide Formulas
    • De-Broglie wavelength: $\lambda = \frac{h}{mv}$
    • De-Broglie wavelength: $\lambda = \frac{h}{p}$
    • Davisson and Germer wavelength: $\lambda = \frac{h}{\sqrt{2mVe}}$
    • Heisenberg's Uncertainty Principle (Position-Momentum): $\Delta p \cdot \Delta x \approx h$
    • Heisenberg's Uncertainty Principle (Energy-Time): $\Delta E \cdot \Delta t \approx h$

• Chapter 20: Atomic Spectra

  Show/Hide Details

  • Hydrogen Spectrum

    Show/Hide Formulas
    • General formula: $\frac{1}{\lambda} = R \left(\frac{1}{p^2} - \frac{1}{n^2}\right)$
    • Rydberg's constant: $R = 1.0974 \times 10^7~m^{-1}$
    • Lyman Series ($n=2$ longest wavelength): $\lambda = \frac{4}{3R} = 1216~\text{Å}$
    • Lyman Series ($n=\infty$ shortest wavelength): $\lambda = \frac{1}{R} = 912~\text{Å}$
    • Balmer Series ($n=3$ longest wavelength): $\lambda = \frac{36}{5R} = 6563~\text{Å}$
    • Balmer Series ($n=\infty$ shortest wavelength): $\lambda = \frac{4}{R} = 3648~\text{Å}$
    • Paschen Series ($n=4$ longest wavelength): $\lambda = \frac{144}{7R} = 18761.1~\text{Å}$
    • Paschen Series ($n=\infty$ shortest wavelength): $\lambda = \frac{9}{R} = 8208~\text{Å}$
    • Bracket Series ($n=5$ longest wavelength): $\lambda = \frac{400}{9R} = 40533.3~\text{Å}$
    • Bracket Series ($n=\infty$ shortest wavelength): $\lambda = \frac{16}{R} = 14592~\text{Å}$
    • Pfund Series ($n=6$ longest wavelength): $\lambda = \frac{900}{11R} = 74618.18~\text{Å}$
    • Pfund Series ($n=\infty$ shortest wavelength): $\lambda = \frac{25}{R} = 22800~\text{Å}$

  • Bohr's Model

    Show/Hide Formulas
    • Angular momentum quantization: $mvr_n = \frac{nh}{2\pi}$
    • Radius of $n^{th}$ orbit: $r_n = \frac{n^2 h^2}{4\pi^2 m e^2 k}$
    • Radius of $n^{th}$ orbit: $r_n = r_o n^2$
    • Bohr's Radius ($n=1$): $r_o = 0.53~\text{Å}$
    • Energy of electron in $n^{th}$ orbit: $E_n = KE + PE$
    • Energy of electron in $n^{th}$ orbit: $E_n = -\frac{ke^2}{2r_n}$
    • Energy of electron in $n^{th}$ orbit: $E_n = -\frac{E_o}{n^2}$
    • Ground state energy: $E_o = \frac{2\pi^2 m k^2 e^4}{h^2}$
    • Ground state energy: $E_o = 13.6~eV$
    • Energy transition: $\Delta E = E_p - E_q$
    • Energy transition: $\Delta E = hf$

  • X-Rays

    Show/Hide Formulas
    • Characteristic X-ray energy ($K_\alpha$): $hf_{K\alpha} = E_L - E_K$
    • Characteristic X-ray energy ($M_\alpha$): $hf_{M\alpha} = E_M - E_K$
    • Continuous X-ray maximum frequency: $V_e = hf_{max}$

• Chapter 21: Nuclear Physics

  Show/Hide Details

  • Nuclear Composition and Properties

    Show/Hide Formulas
    • Number of neutrons: $N = A - Z$
    • Mass of electron: $0.00055~\text{a.m.u}$
    • Mass of proton: $1.007276~\text{a.m.u}$
    • Mass of neutron: $1.008665~\text{a.m.u}$
    • Mass of deuteron: $2.014102~\text{a.m.u}$
    • Mass of helium nucleus: $4.002603~\text{a.m.u}$
    • Atomic mass unit to MeV: $1~\text{a.m.u} = 931~MeV$
    • Mass defect: $\Delta m = \sum \text{masses of nucleons} - \text{mass of nucleus}$
    • Mass defect for deuterium: $\Delta m = 0.002388~\text{a.m.u}$
    • Binding Energy: $E = \Delta m \times 931~MeV$
    • Binding Energy (expanded): $E = [Z m_p + (A-Z)m_n - m_{nucleus}] \times 931~MeV$
    • Binding Energy for deuterium: $2.24~MeV$

  • Nuclear Forces

    Show/Hide Formulas
    • Coulomb's repulsive force: $F_c = k \frac{e^2}{r^2}$
    • Newton's attraction force: $F_g = G \frac{m^2}{r^2}$

  • Radioactive Decay

    Show/Hide Formulas
    • $\alpha$-decay: $^A_Z X \to ^{A-4}_{Z-2} Y + ^4_2He$
    • $\beta^-$-decay: $^A_Z X \to ^A_{Z+1} Y + ^0_{-1}e + \bar{\nu}$
    • $\beta^+$-decay: $^A_Z X \to ^A_{Z-1} Y + ^0_{+1}e + \nu$
    • $\gamma$-decay: $^A_Z X^* \to ^A_Z X + \gamma$
    • Radioactive decay law: $\Delta N/\Delta t = -\lambda N$
    • Radioactive decay (activity): $N_t = N_o e^{-\lambda t}$
    • Half-life: $T_{1/2} = 0.693/\lambda$
    • Mean life: $T = 1/\lambda$

  • Radiation Exposure and Units

    Show/Hide Formulas
    • Absorbed Dose: $D = \frac{E}{m}$
    • Equivalent Dose: $H = D \times \text{RBE}$
    • Becquerel (unit of activity): $1~Bq = 1~\text{disintegration per second}$
    • Curie (unit of activity): $1~Ci = 3.7 \times 10^{10}~Bq$

  • Nuclear Fission and Fusion

    Show/Hide Formulas
    • D-D Reaction: $^2_1H + ^2_1H \to ^3_2He + ^1_0n + 3.3~MeV$
    • D-T Reaction: $^2_1H + ^3_1H \to ^4_2He + ^1_0n + 17.6~MeV$
    • Energy from 1 atom of U-235: $200~MeV$
    • Energy from 1 kg of U-235: $5.12 \times 10^{26}~MeV$
    • Time for one fission: $\sim 10^{-8}~s$
    • Time for 60 fissions: $\sim 0.6~\mu s$
    • Efficiency of petrol engine: $\eta \approx 25\% \text{ to } 30\%$
    • Efficiency of diesel engine: $\eta \approx 35\% \text{ to } 40\%$

  • Mass Spectrograph

    Show/Hide Formulas
    • Radius of path: $r = \frac{\sqrt{2Vm}}{B^2e}$