Physical Quantities and the International System (SI)
[ v ] = L ⋅ T − 1 = m / s [v] = L \cdot T^{-1} = \mathrm{m/s} [ v ] = L ⋅ T − 1 = m/s [ a ] = L ⋅ T − 2 = m / s 2 [a] = L \cdot T^{-2} = \mathrm{m/s^2} [ a ] = L ⋅ T − 2 = m/ s 2 [ F ] = M ⋅ L ⋅ T − 2 = N [F] = M \cdot L \cdot T^{-2} = \mathrm{N} [ F ] = M ⋅ L ⋅ T − 2 = N [ W ] = M ⋅ L 2 ⋅ T − 2 = J [W] = M \cdot L^2 \cdot T^{-2} = \mathrm{J} [ W ] = M ⋅ L 2 ⋅ T − 2 = J Position, Displacement, and Velocity
v ⃗ m = Δ r ⃗ Δ t \vec{v}_m = \dfrac{\Delta\vec{r}}{\Delta t} v m = Δ t Δ r v ⃗ = d r ⃗ d t \vec{v} = \dfrac{d\vec{r}}{dt} v = d t d r ∣ v ⃗ ∣ = d s d t |\vec{v}| = \dfrac{ds}{dt} ∣ v ∣ = d t d s Δ r ⃗ = r ⃗ 2 − r ⃗ 1 \Delta\vec{r} = \vec{r}_2 - \vec{r}_1 Δ r = r 2 − r 1 Acceleration — Tangential and Centripetal Components
a ⃗ = d v ⃗ d t = d 2 r ⃗ d t 2 \vec{a} = \dfrac{d\vec{v}}{dt} = \dfrac{d^2\vec{r}}{dt^2} a = d t d v = d t 2 d 2 r a t = d v d t a_t = \dfrac{dv}{dt} a t = d t d v a n = v 2 R a_n = \dfrac{v^2}{R} a n = R v 2 ∣ a ⃗ ∣ 2 = a t 2 + a n 2 |\vec{a}|^2 = a_t^2 + a_n^2 ∣ a ∣ 2 = a t 2 + a n 2 Uniform Linear Motion (ULM)
x ( t ) = x 0 + v ⋅ t x(t) = x_0 + v \cdot t x ( t ) = x 0 + v ⋅ t v = c o n s t , a = 0 v = \mathrm{const},\quad a = 0 v = const , a = 0 Δ x = v ⋅ Δ t \Delta x = v \cdot \Delta t Δ x = v ⋅ Δ t x ( t ) is a straight line of slope v x(t) \text{ is a straight line of slope } v x ( t ) is a straight line of slope v Uniformly Accelerated Motion (UAM)
v ( t ) = v 0 + a t v(t) = v_0 + a\,t v ( t ) = v 0 + a t x ( t ) = x 0 + v 0 t + 1 2 a t 2 x(t) = x_0 + v_0\,t + \tfrac{1}{2}a\,t^2 x ( t ) = x 0 + v 0 t + 2 1 a t 2 v 2 = v 0 2 + 2 a ( x − x 0 ) v^2 = v_0^2 + 2a\,(x - x_0) v 2 = v 0 2 + 2 a ( x − x 0 ) Free fall: y ( t ) = y 0 + v 0 t − 1 2 g t 2 \text{Free fall: } y(t) = y_0 + v_0 t - \tfrac{1}{2}g\,t^2 Free fall: y ( t ) = y 0 + v 0 t − 2 1 g t 2 Projectile Motion (Parabolic)
x ( t ) = v 0 cos θ ⋅ t x(t) = v_0\cos\theta\cdot t x ( t ) = v 0 cos θ ⋅ t y ( t ) = v 0 sin θ ⋅ t − 1 2 g t 2 y(t) = v_0\sin\theta\cdot t - \tfrac{1}{2}g\,t^2 y ( t ) = v 0 sin θ ⋅ t − 2 1 g t 2 R = v 0 2 sin ( 2 θ ) g R = \dfrac{v_0^2\sin(2\theta)}{g} R = g v 0 2 sin ( 2 θ ) h m a x = v 0 2 sin 2 θ 2 g h_{max} = \dfrac{v_0^2\sin^2\theta}{2g} h ma x = 2 g v 0 2 sin 2 θ Uniform Circular Motion (UCM)
v = ω r , a c = v 2 / r = ω 2 r v = \omega\,r,\quad a_c = v^2/r = \omega^2 r v = ω r , a c = v 2 / r = ω 2 r T = 2 π / ω , f = ω / ( 2 π ) T = 2\pi/\omega,\quad f = \omega/(2\pi) T = 2 π / ω , f = ω / ( 2 π ) Non-uniform: ω ( t ) = ω 0 + α t , θ ( t ) = θ 0 + ω 0 t + 1 2 α t 2 \text{Non-uniform: }\omega(t)=\omega_0+\alpha t,\;\theta(t)=\theta_0+\omega_0 t+\tfrac{1}{2}\alpha t^2 Non-uniform: ω ( t ) = ω 0 + α t , θ ( t ) = θ 0 + ω 0 t + 2 1 α t 2 Dynamics and Conservation Laws
Theory First Law — Inertia and Inertial Frames
F ⃗ n e t = 0 ⟺ v ⃗ = c o n s t \vec{F}_{net} = 0 \iff \vec{v} = \mathrm{const} F n e t = 0 ⟺ v = const Second Law — F = ma
F ⃗ n e t = m a ⃗ \vec{F}_{net} = m\,\vec{a} F n e t = m a ∑ F x = m a x , ∑ F y = m a y \sum F_x = m\,a_x,\quad \sum F_y = m\,a_y ∑ F x = m a x , ∑ F y = m a y Third Law — Action and Reaction
F ⃗ A B = − F ⃗ B A \vec{F}_{AB} = -\vec{F}_{BA} F A B = − F B A ∣ F ⃗ A B ∣ = ∣ F ⃗ B A ∣ |\vec{F}_{AB}| = |\vec{F}_{BA}| ∣ F A B ∣ = ∣ F B A ∣ Fundamental Forces — Weight, Normal, Friction
P = m g , N = m g cos θ , f = μ N P = mg,\quad N = mg\cos\theta,\quad f = \mu N P = m g , N = m g cos θ , f = μ N P ∥ = m g sin θ , P ⊥ = m g cos θ P_\parallel = mg\sin\theta,\; P_\perp = mg\cos\theta P ∥ = m g sin θ , P ⊥ = m g cos θ Work and the Work-Energy Theorem
W = F d cos θ = ∫ F ⃗ ⋅ d s ⃗ W = F\,d\cos\theta = \int\vec{F}\cdot d\vec{s} W = F d cos θ = ∫ F ⋅ d s W n e t = Δ K E = 1 2 m v 2 − 1 2 m v 0 2 W_{net} = \Delta KE = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2 W n e t = Δ K E = 2 1 m v 2 − 2 1 m v 0 2 P = d W / d t = F ⃗ ⋅ v ⃗ P = dW/dt = \vec{F}\cdot\vec{v} P = d W / d t = F ⋅ v Potential Energy and Conservation
U g = m g h , U e = 1 2 k x 2 U_g = mgh,\quad U_e = \tfrac{1}{2}k\,x^2 U g = m g h , U e = 2 1 k x 2 E m e c h = 1 2 m v 2 + U = c o n s t E_{mech} = \tfrac{1}{2}mv^2 + U = \mathrm{const} E m ec h = 2 1 m v 2 + U = const F s p r i n g = − k x F_{spring} = -k\,x F s p r in g = − k x Two Reference Frames — Vector Geometry
r ⃗ A = r ⃗ O ′ + r ⃗ A / S ′ \vec{r}_A = \vec{r}_{O'} + \vec{r}_{A/S'} r A = r O ′ + r A / S ′ r ⃗ A / S ′ = position of A measured in S’ \vec{r}_{A/S'} = \text{position of A \emph{measured in} S'} r A / S ′ = position of A measured in S’ Velocity Composition — Pure Translation
v ⃗ A = v ⃗ A / S ′ + v ⃗ S ′ / S \vec{v}_A = \vec{v}_{A/S'} + \vec{v}_{S'/S} v A = v A / S ′ + v S ′ / S v ⃗ a b s = v ⃗ r e l + v ⃗ t r a n s p o r t \vec{v}_{abs} = \vec{v}_{rel} + \vec{v}_{transport} v ab s = v r e l + v t r an s p or t v ⃗ A / S ′ = ( d r ⃗ A / S ′ d t ) S ′ \vec{v}_{A/S'} = \left(\frac{d\vec{r}_{A/S'}}{dt}\right)_{S'} v A / S ′ = ( d t d r A / S ′ ) S ′ Acceleration Composition — Pure Translation
a ⃗ A = a ⃗ A / S ′ + a ⃗ S ′ / S \vec{a}_A = \vec{a}_{A/S'} + \vec{a}_{S'/S} a A = a A / S ′ + a S ′ / S F ⃗ f i c t i t i o u s = − m a ⃗ S ′ / S \vec{F}_{fictitious} = -m\,\vec{a}_{S'/S} F f i c t i t i o u s = − m a S ′ / S S ′ inertial ⟺ a ⃗ S ′ / S = 0 ⟹ a ⃗ A = a ⃗ A / S ′ S'\text{ inertial }\iff \vec{a}_{S'/S}=0 \implies \vec{a}_A = \vec{a}_{A/S'} S ′ inertial ⟺ a S ′ / S = 0 ⟹ a A = a A / S ′ Rotating Frames — Relative Derivative Theorem
( d A ⃗ d t ) S = ( d A ⃗ d t ) S ′ + ω ⃗ × A ⃗ \left(\frac{d\vec{A}}{dt}\right)_S = \left(\frac{d\vec{A}}{dt}\right)_{S'} + \vec{\omega}\times\vec{A} ( d t d A ) S = ( d t d A ) S ′ + ω × A e ^ ˙ i = ω ⃗ × e ^ i \dot{\hat{e}}_i = \vec{\omega}\times\hat{e}_i e ^ ˙ i = ω × e ^ i Velocity and Acceleration in Rotating Frame — Fictitious Forces
v ⃗ A = v ⃗ A / S ′ + ω ⃗ × r ⃗ A / S ′ + v ⃗ O ′ / S \vec{v}_A = \vec{v}_{A/S'} + \vec{\omega}\times\vec{r}_{A/S'} + \vec{v}_{O'/S} v A = v A / S ′ + ω × r A / S ′ + v O ′ / S F ⃗ C o r = − 2 m ω ⃗ × v ⃗ A / S ′ \vec{F}_{Cor} = -2m\,\vec{\omega}\times\vec{v}_{A/S'} F C or = − 2 m ω × v A / S ′ F ⃗ c f = m ω 2 r ⊥ r ^ ⊥ \vec{F}_{cf} = m\omega^2 r_\perp \hat{r}_\perp F c f = m ω 2 r ⊥ r ^ ⊥ a ⃗ A = a ⃗ A / S ′ + 2 ω ⃗ × v ⃗ A / S ′ − ω 2 r ⊥ r ^ ⊥ \vec{a}_A = \vec{a}_{A/S'} + 2\vec{\omega}\times\vec{v}_{A/S'} - \omega^2 r_\perp\hat{r}_\perp a A = a A / S ′ + 2 ω × v A / S ′ − ω 2 r ⊥ r ^ ⊥ Moment of Inertia — Resistance to Rotation
I = ∫ r 2 d m I = \int r^2\,dm I = ∫ r 2 d m I d i s k = 1 2 m R 2 , I r o d c e n t e r = 1 12 m L 2 I_{disk} = \tfrac{1}{2}mR^2, \quad I_{rod\,center} = \tfrac{1}{12}mL^2 I d i s k = 2 1 m R 2 , I r o d ce n t er = 12 1 m L 2 I s o l i d s p h e r e = 2 5 m R 2 , I h o l l o w s p h e r e = 2 3 m R 2 I_{solid\,sphere} = \tfrac{2}{5}mR^2, \quad I_{hollow\,sphere} = \tfrac{2}{3}mR^2 I so l i d s p h er e = 5 2 m R 2 , I h o l l o w s p h er e = 3 2 m R 2 I r i n g = m R 2 , I p l a t e = 1 12 m ( a 2 + b 2 ) I_{ring} = mR^2, \quad I_{plate} = \tfrac{1}{12}m(a^2+b^2) I r in g = m R 2 , I pl a t e = 12 1 m ( a 2 + b 2 ) Parallel Axis Theorem (Steiner's Theorem)
I = I C M + M d 2 I = I_{CM} + Md^2 I = I C M + M d 2 I r o d , e n d = 1 3 m L 2 I_{rod,\,end} = \tfrac{1}{3}mL^2 I r o d , e n d = 3 1 m L 2 I d i s k , r i m = 3 2 m R 2 I_{disk,\,rim} = \tfrac{3}{2}mR^2 I d i s k , r im = 2 3 m R 2 Newton's Second Law for Rotation
τ = I α ( analogue of F = m a ) \tau = I\alpha \quad (\text{analogue of } F=ma) τ = I α ( analogue of F = ma ) τ = r F sin θ = r ⊥ F \tau = r F \sin\theta = r_\perp F τ = r F sin θ = r ⊥ F K r o t = 1 2 I ω 2 K_{rot} = \tfrac{1}{2}I\omega^2 K r o t = 2 1 I ω 2 W = τ Δ θ , P = τ ω W = \tau\Delta\theta, \quad P = \tau\omega W = τ Δ θ , P = τ ω Pure Rolling — Without Slipping
v C M = ω R , a C M = α R v_{CM} = \omega R, \quad a_{CM} = \alpha R v C M = ω R , a C M = α R K t o t = 1 2 m v C M 2 + 1 2 I C M ω 2 K_{tot} = \tfrac{1}{2}mv_{CM}^2 + \tfrac{1}{2}I_{CM}\omega^2 K t o t = 2 1 m v C M 2 + 2 1 I C M ω 2 K t o t = 1 2 m v C M 2 ( 1 + I C M m R 2 ) K_{tot} = \tfrac{1}{2}mv_{CM}^2\!\left(1 + \dfrac{I_{CM}}{mR^2}\right) K t o t = 2 1 m v C M 2 ( 1 + m R 2 I C M ) a C M = g sin θ 1 + I C M / ( m R 2 ) a_{CM} = \dfrac{g\sin\theta}{1 + I_{CM}/(mR^2)} a C M = 1 + I C M / ( m R 2 ) g sin θ Conservation of Angular Momentum
L = I ω ( for a rigid body ) L = I\omega \quad (\text{for a rigid body}) L = I ω ( for a rigid body ) τ = d L d t ( analogue of F = d p / d t ) \tau = \dfrac{dL}{dt} \quad (\text{analogue of } F = dp/dt) τ = d t d L ( analogue of F = d p / d t ) I 1 ω 1 = I 2 ω 2 ( if τ e x t = 0 ) I_1\omega_1 = I_2\omega_2 \quad (\text{if } \tau_{ext}=0) I 1 ω 1 = I 2 ω 2 ( if τ e x t = 0 ) Ω = τ / L ( gyroscope precession ) \Omega = \tau/L \quad (\text{gyroscope precession}) Ω = τ / L ( gyroscope precession ) Static Equilibrium of a Rigid Body
∑ F x = 0 , ∑ F y = 0 \sum F_x = 0, \quad \sum F_y = 0 ∑ F x = 0 , ∑ F y = 0 ∑ τ O = 0 ( ∀ O ) \sum \tau_O = 0 \quad (\forall\, O) ∑ τ O = 0 ( ∀ O ) U s t a b l e ≈ 1 2 k θ 2 ( for small oscillations ) U_{stable} \approx \tfrac{1}{2}k\theta^2 \quad (\text{for small oscillations}) U s t ab l e ≈ 2 1 k θ 2 ( for small oscillations ) Oscillations and Harmonic Motion
Theory Simple Harmonic Motion (SHM) — Differential Equation and Solution
m x ¨ + k x = 0 ( equation of motion ) m\ddot{x} + kx = 0 \quad (\text{equation of motion}) m x ¨ + k x = 0 ( equation of motion ) x ( t ) = A cos ( ω 0 t + ϕ ) x(t) = A\cos(\omega_0 t + \phi) x ( t ) = A cos ( ω 0 t + ϕ ) ω 0 = k / m [ r a d / s ] \omega_0 = \sqrt{k/m} \quad [\mathrm{rad/s}] ω 0 = k / m [ rad/s ] T = 2 π ω 0 = 2 π m k , f = 1 T = ω 0 2 π T = \dfrac{2\pi}{\omega_0} = 2\pi\sqrt{\dfrac{m}{k}}, \quad f = \dfrac{1}{T} = \dfrac{\omega_0}{2\pi} T = ω 0 2 π = 2 π k m , f = T 1 = 2 π ω 0 A = x 0 2 + ( v 0 / ω 0 ) 2 , tan ϕ = − v 0 ω 0 x 0 A = \sqrt{x_0^2 + (v_0/\omega_0)^2}, \quad \tan\phi = -\dfrac{v_0}{\omega_0 x_0} A = x 0 2 + ( v 0 / ω 0 ) 2 , tan ϕ = − ω 0 x 0 v 0 Velocity, Acceleration, and Energy in SHM
v ( t ) = − A ω 0 sin ( ω 0 t + ϕ ) v(t) = -A\omega_0\sin(\omega_0 t+\phi) v ( t ) = − A ω 0 sin ( ω 0 t + ϕ ) a ( t ) = − A ω 0 2 cos ( ω 0 t + ϕ ) = − ω 0 2 x a(t) = -A\omega_0^2\cos(\omega_0 t+\phi) = -\omega_0^2\,x a ( t ) = − A ω 0 2 cos ( ω 0 t + ϕ ) = − ω 0 2 x E = 1 2 m v 2 + 1 2 k x 2 = 1 2 k A 2 = c o n s t E = \tfrac{1}{2}mv^2 + \tfrac{1}{2}kx^2 = \tfrac{1}{2}kA^2 = \mathrm{const} E = 2 1 m v 2 + 2 1 k x 2 = 2 1 k A 2 = const v m a x = ω 0 A = A k / m v_{max} = \omega_0 A = A\sqrt{k/m} v ma x = ω 0 A = A k / m v ( x ) = ± ω 0 A 2 − x 2 v(x) = \pm\omega_0\sqrt{A^2 - x^2} v ( x ) = ± ω 0 A 2 − x 2 Simple Pendulum — Small Angle Approximation
ω 0 = g / L ( simple pendulum ) \omega_0 = \sqrt{g/L} \quad (\text{simple pendulum}) ω 0 = g / L ( simple pendulum ) T = 2 π L / g ( independent of m and A for small angles ) T = 2\pi\sqrt{L/g} \quad (\text{independent of } m \text{ and } A \text{ for small angles}) T = 2 π L / g ( independent of m and A for small angles ) T = 2 π I / ( M g d ) ( physical pendulum ) T = 2\pi\sqrt{I/(Mgd)} \quad (\text{physical pendulum}) T = 2 π I / ( M g d ) ( physical pendulum ) L e q = I / ( M d ) ( equivalent length ) L_{eq} = I/(Md) \quad (\text{equivalent length}) L e q = I / ( M d ) ( equivalent length ) Damped Harmonic Oscillator
m x ¨ + b x ˙ + k x = 0 m\ddot{x} + b\dot{x} + kx = 0 m x ¨ + b x ˙ + k x = 0 x ( t ) = A 0 e − γ t cos ( ω 1 t + ϕ ) ( underdamped ) x(t) = A_0 e^{-\gamma t}\cos(\omega_1 t + \phi) \quad (\text{underdamped}) x ( t ) = A 0 e − γ t cos ( ω 1 t + ϕ ) ( underdamped ) ω 1 = ω 0 2 − γ 2 , γ = b 2 m \omega_1 = \sqrt{\omega_0^2 - \gamma^2}, \quad \gamma = \dfrac{b}{2m} ω 1 = ω 0 2 − γ 2 , γ = 2 m b τ = 1 / γ ( characteristic decay time ) \tau = 1/\gamma \quad (\text{characteristic decay time}) τ = 1/ γ ( characteristic decay time ) Q = ω 0 2 γ = ω 0 m b Q = \dfrac{\omega_0}{2\gamma} = \dfrac{\omega_0 m}{b} Q = 2 γ ω 0 = b ω 0 m Forced Oscillations and Resonance
m x ¨ + b x ˙ + k x = F 0 cos ( ω t ) m\ddot{x} + b\dot{x} + kx = F_0\cos(\omega t) m x ¨ + b x ˙ + k x = F 0 cos ( ω t ) A ( ω ) = F 0 / m ( ω 0 2 − ω 2 ) 2 + 4 γ 2 ω 2 A(\omega) = \dfrac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2 + 4\gamma^2\omega^2}} A ( ω ) = ( ω 0 2 − ω 2 ) 2 + 4 γ 2 ω 2 F 0 / m ω r e s = ω 0 2 − 2 γ 2 ≈ ω 0 ( γ ≪ ω 0 ) \omega_{res} = \sqrt{\omega_0^2 - 2\gamma^2} \approx \omega_0 \quad (\gamma \ll \omega_0) ω r es = ω 0 2 − 2 γ 2 ≈ ω 0 ( γ ≪ ω 0 ) Δ ω = ω 0 Q = 2 γ \Delta\omega = \dfrac{\omega_0}{Q} = 2\gamma Δ ω = Q ω 0 = 2 γ Ideal Gases, Real Gases, and Kinetic Theory
Theory Zeroth Law — Temperature and Thermometric Scales
T ( K ) = T ( ° C ) + 273.15 T(K) = T(°C) + 273.15 T ( K ) = T ( ° C ) + 273.15 Δ L = α L 0 Δ T \Delta L = \alpha\,L_0\,\Delta T Δ L = α L 0 Δ T Δ V = β V 0 Δ T \Delta V = \beta\,V_0\,\Delta T Δ V = β V 0 Δ T Ideal Gas Equation of State
R = 8.314 J / ( m o l ⋅ K ) R = 8.314\,\mathrm{J/(mol\cdot K)} R = 8.314 J/ ( mol ⋅ K ) P V = N k B T , k B = 1.381 × 10 − 23 J / K PV = Nk_BT,\quad k_B = 1.381\times10^{-23}\,\mathrm{J/K} P V = N k B T , k B = 1.381 × 1 0 − 23 J/K Kinetic Theory of Gases
⟨ K E ⟩ = 3 2 k B T \langle KE\rangle = \tfrac{3}{2}k_B T ⟨ K E ⟩ = 2 3 k B T v r m s = 3 R T / M v_{rms} = \sqrt{3RT/M} v r m s = 3 R T / M C v = f R / 2 , C p = ( f + 2 ) R / 2 C_v = fR/2,\quad C_p = (f+2)R/2 C v = f R /2 , C p = ( f + 2 ) R /2 Real Gas — Van der Waals Equation
( P + a / V 2 ) ( V − b ) = R T (P + a/V^2)(V-b) = RT ( P + a / V 2 ) ( V − b ) = R T T c = 8 a / ( 27 R b ) , P c = a / ( 27 b 2 ) T_c = 8a/(27Rb),\quad P_c = a/(27b^2) T c = 8 a / ( 27 R b ) , P c = a / ( 27 b 2 ) Thermodynamic Processes and First Law
Theory Work in Thermodynamic Processes
W = ∫ P d V W = \int P\,dV W = ∫ P d V W i s o b a r = P Δ V = n R Δ T W_{isobar} = P\Delta V = nR\Delta T W i so ba r = P Δ V = n R Δ T W i s o t h = n R T ln ( V 2 / V 1 ) W_{isoth} = nRT\ln(V_2/V_1) W i so t h = n R T ln ( V 2 / V 1 ) W a d = − Δ U = − n C v Δ T W_{ad} = -\Delta U = -nC_v\Delta T W a d = − Δ U = − n C v Δ T First Law of Thermodynamics — Conservation of Energy
Δ U = Q − W \Delta U = Q - W Δ U = Q − W Δ U = n C v Δ T \Delta U = nC_v\Delta T Δ U = n C v Δ T Cycle: Δ U = 0 \text{Cycle: }\Delta U = 0 Cycle: Δ U = 0 The Four Fundamental Gas Processes
γ = C p / C v \gamma = C_p/C_v γ = C p / C v γ m o n o = 5 / 3 , γ d i = 1.4 \gamma_{mono}=5/3,\; \gamma_{di}=1.4 γ m o n o = 5/3 , γ d i = 1.4 P V γ = c o n s t , T V γ − 1 = c o n s t PV^\gamma=\mathrm{const},\; TV^{\gamma-1}=\mathrm{const} P V γ = const , T V γ − 1 = const C p − C v = R ( Mayer ) C_p-C_v=R \;(\text{Mayer}) C p − C v = R ( Mayer ) Specific Heat and Latent Heat — Heat Transfer
Q = m c Δ T Q = mc\,\Delta T Q = m c Δ T P r a d = ε σ A T 4 P_{rad} = \varepsilon\sigma A T^4 P r a d = ε σ A T 4 Heat Engines — Carnot and Otto Cycles
Theory Heat Engines — Layout and Efficiency
W = Q H − Q C W = Q_H - Q_C W = Q H − Q C η = W / Q H = 1 − Q C / Q H < 1 \eta = W/Q_H = 1 - Q_C/Q_H < 1 η = W / Q H = 1 − Q C / Q H < 1 C O P r e f = Q C / W COP_{ref} = Q_C/W C O P r e f = Q C / W Carnot Cycle — Maximum Theoretical Efficiency
η C a r n o t = 1 − T C / T H \eta_{Carnot} = 1 - T_C/T_H η C a r n o t = 1 − T C / T H Q H / T H = Q C / T C Q_H/T_H = Q_C/T_C Q H / T H = Q C / T C η r e a l ≤ η C a r n o t \eta_{real} \leq \eta_{Carnot} η r e a l ≤ η C a r n o t Otto Cycle — The Petrol Engine
η O t t o = 1 − 1 / r γ − 1 \eta_{Otto} = 1 - 1/r^{\gamma-1} η O tt o = 1 − 1/ r γ − 1 r = V m a x / V m i n r = V_{max}/V_{min} r = V ma x / V min r = 8 ÷ 12 ⇒ η ≈ 40 ÷ 55 % r=8\div12 \Rightarrow \eta\approx40\div55\% r = 8 ÷ 12 ⇒ η ≈ 40 ÷ 55% Kelvin and Clausius Postulates — The Second Law Statements
C O P r e f , m a x = T C / ( T H − T C ) COP_{ref,max} = T_C/(T_H-T_C) C O P r e f , ma x = T C / ( T H − T C ) C O P h p , m a x = T H / ( T H − T C ) COP_{hp,max} = T_H/(T_H-T_C) C O P h p , ma x = T H / ( T H − T C ) What is a Wave — Fundamental Concepts
v = λ f = λ / T v = \lambda f = \lambda/T v = λ f = λ / T f = 1 / T , ω = 2 π f , k = 2 π / λ f = 1/T, \quad \omega = 2\pi f, \quad k = 2\pi/\lambda f = 1/ T , ω = 2 π f , k = 2 π / λ E ∝ A 2 — energy carried E \propto A^2 \text{ — energy carried} E ∝ A 2 — energy carried Wave Equation — Mathematical Description
y ( x , t ) = A cos ( k x − ω t + ϕ 0 ) y(x,t) = A\cos(kx - \omega t + \phi_0) y ( x , t ) = A cos ( k x − ω t + ϕ 0 ) v = ω / k = λ f v = \omega/k = \lambda f v = ω / k = λ f ∂ 2 y / ∂ t 2 = v 2 ∂ 2 y / ∂ x 2 \partial^2 y/\partial t^2 = v^2\,\partial^2 y/\partial x^2 ∂ 2 y / ∂ t 2 = v 2 ∂ 2 y / ∂ x 2 Wave Speed — Dependence on Medium
v s t r i n g = F T / μ v_{string} = \sqrt{F_T/\mu} v s t r in g = F T / μ v s o l i d = E / ρ v_{solid} = \sqrt{E/\rho} v so l i d = E / ρ v s o u n d = γ R T / M ≈ 331 T / 273 m / s v_{sound} = \sqrt{\gamma RT/M} \approx 331\sqrt{T/273}\;\mathrm{m/s} v so u n d = γ R T / M ≈ 331 T /273 m/s Superposition Principle and Interference
y 1 + y 2 = 2 A cos ( Δ ϕ / 2 ) cos ( k x − ω t + Δ ϕ / 2 ) y_1+y_2 = 2A\cos(\Delta\phi/2)\cos(kx-\omega t + \Delta\phi/2) y 1 + y 2 = 2 A cos ( Δ ϕ /2 ) cos ( k x − ω t + Δ ϕ /2 ) Constr.: Δ r = m λ \text{Constr.: } \Delta r = m\lambda Constr.: Δ r = mλ Destr.: Δ r = ( m + 1 2 ) λ \text{Destr.: } \Delta r = (m+\tfrac{1}{2})\lambda Destr.: Δ r = ( m + 2 1 ) λ f b e a t = ∣ f 2 − f 1 ∣ f_{beat} = |f_2 - f_1| f b e a t = ∣ f 2 − f 1 ∣ Standing Waves — Resonance
f n = n v / ( 2 L ) ( n = 1 , 2 , 3 , … ) — string / o-o pipe f_n = n\,v/(2L) \quad (n=1,2,3,\ldots) \text{ — string / o-o pipe} f n = n v / ( 2 L ) ( n = 1 , 2 , 3 , … ) — string / o-o pipe f n = ( 2 n − 1 ) v / ( 4 L ) ( n = 1 , 2 , 3 , … ) — o-c pipe f_n = (2n-1)\,v/(4L) \quad (n=1,2,3,\ldots) \text{ — o-c pipe} f n = ( 2 n − 1 ) v / ( 4 L ) ( n = 1 , 2 , 3 , … ) — o-c pipe λ n = 2 L / n (string) , λ n = 4 L / ( 2 n − 1 ) (o-c) \lambda_n = 2L/n \;\;\text{(string)}, \quad \lambda_n = 4L/(2n-1) \;\;\text{(o-c)} λ n = 2 L / n (string) , λ n = 4 L / ( 2 n − 1 ) (o-c) Sound — Intensity and Sound Level
I = P / ( 4 π r 2 ) I = P/(4\pi r^2) I = P / ( 4 π r 2 ) β = 10 log 10 ( I / I 0 ) [ d B ] \beta = 10\log_{10}(I/I_0) \;\mathrm{[dB]} β = 10 log 10 ( I / I 0 ) [ dB ] I 0 = 10 − 12 W / m 2 I_0 = 10^{-12} \;\mathrm{W/m^2} I 0 = 1 0 − 12 W/ m 2 I ∝ 1 / r 2 I \propto 1/r^2 I ∝ 1/ r 2 Doppler Effect
f ′ = f 0 ( v ± v o ) / ( v ∓ v s ) f' = f_0\,(v \pm v_o)/(v \mp v_s) f ′ = f 0 ( v ± v o ) / ( v ∓ v s ) f ′ > f 0 (approach) , f ′ < f 0 (recede) f' > f_0 \text{ (approach)}, \quad f' < f_0 \text{ (recede)} f ′ > f 0 (approach) , f ′ < f 0 (recede) Mach cone: sin θ = v / v s ( v s > v ) \text{Mach cone: } \sin\theta = v/v_s \;(v_s > v) Mach cone: sin θ = v / v s ( v s > v ) 
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