Harold's Physics “Cheat Sheet”

Harold’s Physics “Cheat Sheet” 3 May 2021

Mechanics: Atomic and Mechanics: Fluid Mechanics Angular / Electricity / Nuclear / Linear / Thermo- Rotational Magnetism Waves and Translation dynamics Motion Optics Kinematics Horizontal / 1-D: 10−100 Fluid Mechanics: Waves: 1 2 1 2 = 푔표표푔표푙푡ℎ 1 2 푓(푥, 푡) = 퐴 sin (2휋 풙 = 풙0 + 풗푥0푡 + 풂푡 휽 = 휽0 + 흎0푡 + 휶푡 10−24 = 푦표푐푡표 푃1 + 휌푔푦1 + 휌푣1 푥 2 2 2 (푓푡 − ) + 휙) + 푘 10−21 = 푧푒푝푡표 1 휆 = 푃 + 휌푔푦 + 휌푣2 Vertical: 10−18 = 푎푡푡표 2 2 2 2 −15 Optics: 1 2 10 = 푓푒푚푡표 (Conservation of Mass) 풚 = 풚0 + 풗푦0푡 − 품푡 −12 푣 2 10 = 푝𝑖푐표 푓 = 10−9 = 푛푎푛표 푚 휆 −6 휌 = 풙 = 풙ퟎ + 풗푡 휽 = 휽 + 흎푡 10 = 푚𝑖푐푟표 푉 0 −3 1 1 1 10 = 푚𝑖푙푙𝑖 −2 + = 10 = 푐푒푛푡𝑖 ∆ℓ = 훼ℓ0∆푇 푑표 푑푖 푓 풙 = ∫ 풗 푑푡 휽 = ∫ 흎 푑푡 10−1 = 푑푒푐𝑖 0 Position 10 = 1 Refraction: 1 (m) 푠 = 푟휃 10 = 푑푒푐푎 (bend) 2 푐 (rad) 10 = ℎ푒푐푡표 3 푛 = 10 = 푘𝑖푙표 푣 6 풙(푡) = 퐴 cos(흎푡 + 휙) 10 = 푚푒푔푎 9 Snell’s Law: 10 = 푔𝑖푔푎 1012 = 푡푒푟푎 푛 sin 휃 = 푛 sin 휃 풙(푡) = 퐴 cos(2휋푓푡 + 휙) 1 1 2 2 1015 = 푝푒푡푎 1018 = 푒푥푎 푛1 푣2 21 = 10 = 푧푒푡푡푎 푛2 푣1 1024 = 푦표푡푡푎 10100 = 푔표표푔표푙 Diffraction: 101000 (spread out) = 푔표표푔표푙푝푙푒푥 ∆퐿 = 푑 sin 휃 푚휆 = 푑 sin 휃

Atomic and Fluid Mechanics Mechanics: Mechanics: Electricity / Nuclear / / Thermo- Linear Angular Magnetism Waves and dynamics Optics

푑 ∆풙 푑풙 휃 ∆휃 푑휃 Speed of Light: Fluid Mechanics: Waves and Optics: 풗 = = = 흎 = = = 푚 푡 ∆푡 푑푡 푡 ∆푡 푑푡 푐 ≈ 3.00 × 108 퐴1푣1 = 퐴2푣2 푣 = 푓휆 푠 풗 = 풗0 + 풂푡 흎 = 흎0 + 휶푡 Reflection: 3푅푇 푣푟푚푠 = √ (throw back) 2 2 2 2 푀 풗 = 풗0 + 2풂(푥 − 푥0) 흎 = 흎0 + 2휶(휃 − 휃0) Critical angle: 풗0 + 풗 흎0 + 흎 푛1 풗̅ = 흎̅ = 3푘퐵푇 sin 휃푐 = 2 2 푣푟푚푠 = √ 푛2 휇 풗 = ∫ 풂 푑푡 흎 = ∫ 휶 푑푡 Maxima for a thin film: Velocity 휆 휆 휆 2푑 = , 3 , 5 … (m/s) 2푛 2푛 2푛 1 2휋 푓 = 흎 = = 2휋푓 Angular 푇 푇 Velocity /

Angular 푘 흎 = √ (푠푝푟𝑖푛푔) Frequency 푚 (rad/s)

품 흎 = √ (푝푒푛푑푢푙푢푚) ℓ

풗 = 흎푟 풗 = 흎 × 풓

풗(푡) = −퐴흎 sin(휔푡 + 휙)

풗(푡) = −퐴흎 sin(2휋푓푡 + 휙)

Atomic and Fluid Mechanics Mechanics: Mechanics: Electricity / Nuclear / / Thermo- Linear Angular Magnetism Waves and dynamics Optics

Linear: Angular: Constants: 풗 ∆풗 푑풗 흎 ∆흎 푑흎 푁푚2 풂 = = = 휶 = = = Gravitational Constant 퐺 = 6.67430 × 10−11 푡 ∆푡 푑푡 푡 ∆푡 푑푡 푘푔 푚 푓푡 Gravity Acceleration (Earth) 푔 = 9.80665 2 = 32.1740 2 ∑ 푭 푭푛푒푡 ∑ 흉 흉푛푒푡 푠 푠 풂 = = 휶 = = 푚 푚 푚 퐼 퐼 Speed of Light in Vacuum 푐 = 2.99792458 × 108 푠 Tangential (linear): Centripetal (center): Electron-Volt 1푒푉 = 1.602176462 × 10−19 퐽 풂 = 풂 = 휶 푟 풗2 푡 풂 = = 흎2푟 Charge of an Electron 푒 = −1.602176462 × 10−19 퐶 풄 푟 −31 퐺푟푎푣𝑖푡푦 (품) Mass of an Electron 푚푒 = 9.1093837015 × 10 푘푔

−27 Mass of a Proton 푚푝 = 1.67262192369 × 10 푘푔 Acceleration −27 (m/풔ퟐ) Mass of a Neutron 푚푛 = 1.67492749804 × 10 푘푔 2 (rad/푠2) 1 −12 퐶 Electric Permittivity 휀0 = 2 = 8.85418782 × 10 2 휇0푐 푁푚 푇푚 Magnetic Permeability 휇 = 4휋 × 10−7 0 퐴 1 Avogadro’s Number 푁 = 6.02214076 × 1023 퐴 푚표푙 퐽 Boltzmann Constant 푘 = 1.380649 × 10−23 퐵 퐾 Net: 2 2 2 2 1 9 푁푚 풂 = 풂 + 풂 Coulomb’s Law Constant 푘푒 = = 8.987551 × 10 2 푛푒푡 푡 푐 4휋휀0 퐶

퐶 2 Faraday Constant 퐹 = 9.648533289 × 104 풂 = −퐴흎 cos(휔푡 + 휙) 푚표푙 −34 2 Planck’s Constant ℎ = 6.6267004 × 10 퐽푠 풂(푡) = −퐴흎 cos(2휋푓푡 + 휙) Jerk (Jolt) 풂 ∆풂 푑풂 휶 ∆휶 푑휶 (m/풔ퟑ) 풋⃗ = = = 훇 = = = 3 푡 ∆푡 푑푡 푡 ∆푡 푑푡 (rad/푠 )

Atomic and Fluid Mechanics Mechanics: Mechanics: Electricity / Nuclear / / Thermo- Linear Angular Magnetism Waves and dynamics Optics

Dynamics

2 Magnification: Mass 퐼 = ∑ 푚푟 푑푖 ℎ푖 (kg) |푀| = | | = | | 푑 ℎ / m = actual mass 표 표 퐼 = ∫ 푟2 푑푚 푚 ≈ 푚 NA Moment of I = effective mass 푝 푛 푀푇푒푙푒푠푐표푝푒 Inertia 푓표푏푗푒푐푡푖푣푒 푙푒푛푠 (푘푔 • 푚2) 퐼 = ∫ 풓 • 풓 푑풎 = − 푓푒푦푒푝푖푒푐푒

풑 = 푚풗 푳 = 퐼흎 ∆풑 = 푚∆풗 푳 = 풓푚풗

Conservation of Linear Conservation of Momentum: Angular Momentum: 풑푖 = 풑푓 푳푖 = 푳푓

∑ 푚풗푖 = ∑ 푚풗푓 ∑ 퐼흎푖 = ∑ 퐼흎푓 Momentum Atomic and Nuclear: Fluid Mechanics: (kg•m/s) 풑 = 흎 × 풎 푳 = 풓 × 풑 NA ℎ 훻풑 = 휌품 휆 = (푘푔 • 푚2/푠) 푝 푡2 푳 = 푚풓 × 풗 ∆풑 = 푱 = ∫ 푭 푑푡 푡1 퐿 = ∫ 풓 × 풗 푑푚 Elastic Collision =

bounce off each other ∆푳 = ∫ 흉 푑푡 Inelastic Collision =

stick together

Atomic and Fluid Mechanics Mechanics: Mechanics: Electricity / Nuclear / / Thermo- Linear Angular Magnetism Waves and dynamics Optics

푭 = 푚풂 Electricity: 푭 = 푚품 휏 = 푟퐹 sin 휃 Fluid Mechanics: 푔 Coulomb’s Law: 푞1푞2 퐹 = 푃퐴 푭 = 푘 2 ∑ 푭 = 푭푛푒푡 = 푚풂 ∑ 흉 = 흉푛푒푡 = 퐼휶 풓 퐹 푃 = = 휌ℎ푔 1 푞1푞2 퐴 풑 ∆푝 휏 = 푟푭 = 퐼휶 푭 = 4휋휖 풓2 푭 = = 흉 = 풓 × 푭 0 푡 ∆푡 퐹푏푢표푦 = 휌푉푔 Force (N = 푑풑 푑 푑푳 푑흎 푭 = 푬 푞 푭 = = (푚풗) 흉 = = 퐼 푃푉 = 푛푅푇 = 푁푘퐵푇 kg•m/풔ퟐ) 푑푡 푑푡 푑푡 푑푡 Magnetism: NA / 퐹 = 푞푣퐵 sin 휃 퐽 푭 ≤ 휇푵 푭 = 푚풂 퐵 푅 = 8.31 Torque 푓 풄 (푚표푙 • 퐾)

(J = N•m) 2 퐹퐵 = 퐵퐼ℓ sin 휃 Hooke’s Law: 푚풗 퐽 푭 = −23 푭 = −푘휟풙 푘퐵 = 1.38 × 10 푠 푟 푭 = 푞풗 × 푩 퐾 퐵

퐺푚 푚 푭 = 푚푟흎2 1 2 푭 = 퐼ℓ × 푩 푭퐺 = − 2 퐵 푟

퐺푚 푚 푭 = ∫ 퐼 푑ℓ × 푩 푭 = − 1 2 풓̂ 퐺 푟2

Impulse 푱 = 푭 ∆푡 = ∆풑 = 풎 ∆풗 푯 = 흉 ∆푡 = ∆푳 = 푰 ∆휔 (N•s) NA NA NA (N•m•s) 푱 = ∆풑 = ∫ 푭 푑푡 푯 = ∆푳 = ∫ 흉 푑푡

Yank 푷 = 퐼 훇 풀 = 푚 푱 (N/풔ퟐ) 푷 = 풓 × 풀

NA NA NA / 푭 ∆퐹 푑푭 Rotatum 풀 = = = 흉 ∆휏 푑흉 푡 ∆푡 푑푡 푷 = = = = 퐼훇 (J/s) 푡 ∆푡 푑푡

Atomic and Fluid Mechanics Mechanics: Mechanics: Electricity / Nuclear / / Thermo- Linear Angular Magnetism Waves and dynamics Optics

Energy Thermodynamics: 푊 = 퐹푑 푊 = 흉 ∆휃(휃 − 휃0) 푊 = −푃 ∆푉

Work 푊 = 퐹 ∆푥 cos 휃 푊 = 휏 ∆휃 cos 휃 푊 푾 = 푸푉 푒 = | | NA (J = N•m) 푄퐻

푊 = ∫ 푭 • 푑풓 푊 = ∫ 흉 • 푑휽 푇퐻 − 푇퐶 푒푐 = 푇퐻 Fluid Mechanics: Bernoulli’s Equation: 1 2 푃1 + 휌푔푦1 + 휌푣1 Kinetic Translational: Rotational: 2 1 푒푉 1 2 Atomic and Nuclear: Energy 1 2 1 2 −19 = 푃2 + 휌푔푦2 + 휌푣2 퐾 = 퐾퐸 = 푚풗 퐾 = 퐾퐸 = 퐼흎 = 1.60 × 10 퐽 2 퐾푚푎푥 = ℎ푓 − 휙 (J) 2 2 Thermodynamics: 3 퐾 = 푘 푇 푎푣푔 2 퐵 푈퐸 = 푞푉 Atomic and Nuclear: Fluid Mechanics: 퐸 = ℎ푓 1 푞 푞 푃 = 푃 + 휌푔ℎ 푈 = 1 2 0 ∆푈푔 = 푚푔ℎ 퐸 4휋휖0 푟 퐸 = 푚푐2 Continuity of Mass: ∆퐸 = (∆푚)푐2 Potential 퐺푚1푚2 Coiled Spring: 1 휌 퐴 푣 = 휌 퐴 푣 푈 = − 1 1 1 2 2 2 Energy 퐺 푟 1 푈푐 = 푄푉 푈 = − 푘훥푥2 2 Relativity: (J) 푠 2 Continuity of Volume: 2 −11 푁푚 1 퐴1푣1 = 퐴2푣2 퐺 = 6.67 × 10 2 2 1 2 푈푐 = 퐶푉 퐸 = 푚푐 푘푔 2 2 Thermodynamics: 푣 √1 − 2 1 ∆푈 = 푄 + 푊 ( 푐 ) 푈 = 퐿퐼2 퐿 2

Atomic and Fluid Mechanics Mechanics: Mechanics: Electricity / Nuclear / / Thermo- Linear Angular Magnetism Waves and dynamics Optics

Thermodynamics: 푘퐴∆푇 퐻 = 퐿

푄 = 푚퐻푓 Conservation of Energy: 푄 = 푚퐻푣

퐸푖 = 퐸푓 Heat ∆퐸 = 푄 = 푚퐶∆푇 Energy 푚퐶∆푇 = 푚퐶(푇푓 − 푇푖) NA ∑ 퐸 = ∑ 퐸 (J) 푖 푓 퐽 퐶퐻 푂 = 4180 퐽 퐸 = 푊 + 푄 + 퐾 + ∆푈푔 + 푈퐺 + 푈푠 + 푈퐸 + 푈푐 + 푈퐿 + ⋯ = 푐표푛푠푡푎푛푡 2 푘푔 °퐾

푇푓 푚 퐶 ∆푇 + 푚 퐶 ∆푇 = 1 1 1푖 2 2 2푖 푚1퐶1 + 푚2퐶2

푃 = 푝푄 푊 푊 푃 = = 퐹푣 푃 = = 휏휔 푡 푡 푃 = 퐼푉 where:

푁 ∆퐸 푑퐸 ∆푊 푑푊 푃 = 푝푟푒푠푠푢푟푒 ( ) Power 푃 = = 푃 = = 푃 = 퐼2푅 푚2 NA ∆푡 푑푡 ∆푡 푑푡 (W) 푉2 푄 = 푣표푙푢푚푒푡푟𝑖푐 푃 = 퐹푣 cos 휃 푃 = 휏휔 cos 휃 푃 = 푚3 푅 푓푙표푤 푟푎푡푒 ( ) 푃 = 푭 • 풗 푃 = 흉 • 흎 푠

Atomic and Fluid Mechanics Mechanics: Mechanics: Electricity / Nuclear / / Thermo- Linear Angular Magnetism Waves and dynamics Optics

Engineering Application

1 2휋 푇 = = 푓 휔 2휋 Waves and Optics: 1 푇 = 푣 푇 = 푓 = Period / 푚 푏 휆 푓 푇푠 = 2휋√ NA Frequency 푘 For: 1 Doppler Effect: (Hz) 푦 = 푠𝑖푛(푏휃) 푓 = 풗 ± 풗풓 푇 ℓ 푦 = 푐표푠(푏휃) 풇풓 = 풇풔 ( ) 푇 = 2휋√ 풗 ∓ 풗풔 푝 푔

푚1푥1 + 푚2푥2 푥퐶푀 = 푚1 + 푚2

∑ 푚푖푥푖 푥퐶푀 = ∑ 푚푖 Center ∑ 푚풓 of Mass 푀 풓 = NA NA NA 1 퐶푀 ∑ 푚 (m) 푥̅ = ∫ 푥 푑푚 푀 0

푀 where 푀 = ∫0 푑푚 and 푑푚 = 휌 푑푧 푑푦 푑푥

Conservation of Conservation of Conservation of ∑ 퐹 = ∑ 푚푔 = 0 ∑ 휏 = ∑ 퐹 푥 = 0 Charge Mass Energy Rigid 푦 푦 퐶푀 Bodies (Down = ‘−‘) (CW = ‘−‘) ∑ 푄푖 = ∑ 푄푓 ∑ 푚푖 = ∑ 푚푓 ∑ 퐸푖 = ∑ 퐸푓 (Circuits) (Chemistry) (Physics)

Electricity Electricity

푭 푄 푬 = 퐶 = 푞 푉

휖 퐴 푉 퐶 = 0 퐸 = − 푑 푎푣푔 푑 Electric 휅휖 퐴 Field ∆푉 푑푉 Capacitance 퐶 = 0 퐸 = − = − 푑 (V/m) ∆푟 푑푟 (F) (N/C) Gauss’s Law: 퐶푝 = ∑ 퐶푖 푄 푖 ∮ 푬 • 푑푨 = 휖0 1 1 = ∑ 퐶푠 푖 퐶푖 푬 = 휌푱

푉 푉 = 퐼푅 푅 = 퐼

푄 푉 = 휌ℓ 퐶 푅 = 퐴 Resistance Potential 푞푖 (V) 푉 = 푘 ∑ (Ω) 푖 푟푖 푅푠 = ∑ 푅푖 푖

1 푞푖 1 1 푉 = ∑ = ∑ 4휋휖0 푖 푟푖 푅 푅 푝 푖 푖

Φ 퐿 = 푁 퐼

푉 ∆푄 푑푄 퐿 = 퐿 퐼 = = 푑퐼 ∆푡 푑푡 ⁄푑푡

푉 2 Current 퐼 = Inductance 푁 퐴 퐿 = 휇0 (A) 푅 (H) ℓ

퐼 = 푁푒푣푑퐴 퐿푠 = ∑ 퐿푖 푖 푒 = −1.602176462 × 10−19 퐶 1 1 = ∑ 퐿푝 푖 퐿푖

Magnetism Magnetism 휇 퐼 퐵 = 0 2휋 푟

푆표푙푒푛표𝑖푑: 퐵푠 = 휇0푛퐼

푁 푤ℎ푒푟푒 푛 = 푡푢푟푛푠 푝푒푟 푚푒푡푒푟 퓵

휇 푞풗⃗⃗⃗ × 풓⃗⃗ 푩 = 0 4휋 푟3

Faraday’s Law of Induction: 휇0 푞풗⃗⃗⃗ × 풓̂ 푩 = ∆ϕ퐵 Magnetic 2 4휋 푟 EMF ∮ 푬 • 푑퓵 = − Field ∆푡 (푇 • 푚) (V) (T) 휇 = 4휋 × 10−7 푑퐼 0 퐴 휀 = −퐿 푑푡 Ampere’s Circuit Law: 푑ϕ ∮ 푩 • 푑퓵 = 휇 퐼 + 휇 휀 퐸 표 0 0 푑푡

Gauss’s Law for Magnetism: ∮ 푩 • 푑푨 = 0

휇 퐼 푑퓵 × 풓⃗⃗ 푑퐵 = 0 4휋 푟3

휙푩 = 퐵퐴 cos 휃

Magnetic ϕ퐵 = 푩 • 푨

Flux Gauss’s Law for Magnetism: (Wb) ϕ퐵 = ∫ 푩 • 푑푨

∆ϕ 휀 = − 퐵 푎푣푔 ∆푡

EMF 푑ϕ (V) 휺 = ∮ 푬 • 푑풔 = − 퐵 푑푡

휺 = 푩푙푣