Engineering Physics I Formula Cheat Sheets, Cheat Sheet of Engineering Physics

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PHYS 2310 Engineering Physics I Formula Sheets Chapters 1-18 Chapter 1/Important Numbers Chapter 2 Units for SI Base Quantities Quantity Unit Name Unit Symbol Length Meter M Time Second s Mass (not weight) Kilogram kg Common Conversions 1 kg or 1 m 1000 g or m 1 m 1 × 106 𝜇𝑚 1 m 100 cm 1 inch 2.54 cm 1 m 1000 mm 1 day 86400 seconds 1 second 1000 milliseconds 1 hour 3600 seconds 1 m 3.281 ft 360° 2𝜋 rad Important Constants/Measurements Mass of Earth 5.98 × 1024 kg Radius of Earth 6.38 × 106 m 1 u (Atomic Mass Unit) 1.661 × 10−27 kg Density of water 1 𝑔/𝑐𝑚3 or 1000 𝑘𝑔/𝑚3 g (on earth) 9.8 m/s2 Density Common geometric Formulas Circumference 𝐶 = 2𝜋𝑟 Area circle 𝐴 = 𝜋𝑟2 Surface area (sphere) 𝑆𝐴 = 4𝜋𝑟2 Volume (sphere) 𝑉 = 4 3 𝜋𝑟3 Volume (rectangular solid) 𝑉 = 𝑙 ∙ 𝑤 ∙ ℎ 𝑉 = 𝑎𝑟𝑒𝑎 ∙ 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 Velocity Average Velocity 𝑉𝑎𝑣𝑔 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑡𝑖𝑚𝑒 = ∆𝑥 ∆𝑡 2.2 Average Speed 𝑠𝑎𝑣𝑔 = 𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑖𝑚𝑒 2.3 Instantaneous Velocity 𝑣 = lim ∆𝑡→0 ∆?̅? ∆𝑡 = 𝑑𝑥 𝑑𝑡 2.4 Acceleration Average Acceleration 𝑎𝑎𝑣𝑔 = ∆𝑣 ∆𝑡 2.7 Instantaneous Acceleration 𝑎 = 𝑑𝑣 𝑑𝑡 = 𝑑2𝑥 𝑑𝑡2 2.8 2.9 Motion of a particle with constant acceleration 𝑣 = 𝑣0 + 𝑎𝑡 2.11 ∆𝑥 = 1 2 (𝑣0 + 𝑣)𝑡 2.17 ∆𝑥 = 𝑣0𝑡 + 1 2 𝑎𝑡2 2.15 𝑣2 = 𝑣0 2 + 2𝑎∆𝑥 2.16 Chapter 3 Chapter 4 Adding Vectors Geometrically ?⃗? + ?⃗? = ?⃗? + ?⃗? 3.2 Adding Vectors Geometrically (Associative Law) (?⃗? + ?⃗?) + 𝑐 = ?⃗? + (?⃗? + 𝑐) 3.3 Components of Vectors 𝑎𝑥 = 𝑎𝑐𝑜𝑠𝜃 𝑎𝑦 = 𝑎𝑠𝑖𝑛𝜃 3.5 Magnitude of vector |𝑎| = 𝑎 = √𝑎𝑥 2 + 𝑎𝑦 2 3.6 Angle between x axis and vector 𝑡𝑎𝑛𝜃 = 𝑎𝑦 𝑎𝑥 3.6 Unit vector notation ?⃗? = 𝑎𝑥𝑖̂ + 𝑎𝑦𝑗̂ + 𝑎𝑧?̂? 3.7 Adding vectors in Component Form 𝑟𝑥 = 𝑎𝑥 + 𝑏𝑥 𝑟𝑦 = 𝑎𝑦 + 𝑏𝑦 𝑟𝑧 = 𝑎𝑧 + 𝑏𝑧 3.10 3.11 3.12 Scalar (dot product) ?⃗? ∙ ?⃗? = 𝑎𝑏𝑐𝑜𝑠𝜃 3.20 Scalar (dot product) ?⃗? ∙ ?⃗? = (𝑎𝑥𝑖̂ + 𝑎𝑦𝑗̂ + 𝑎𝑧?̂?) ∙ (𝑏𝑥𝑖̂ + 𝑏𝑦𝑗̂ + 𝑏𝑧?̂?) ?⃗? ∙ ?⃗? = 𝑎𝑥𝑏𝑥 + 𝑎𝑦𝑏𝑦 + 𝑎𝑧𝑏𝑧 3.22 Projection of ?⃗? 𝑜𝑛 ?⃗? or component of ?⃗? 𝑜𝑛 ?⃗? ?⃗? ∙ ?⃗? |𝑏| Vector (cross) product magnitude 𝑐 = 𝑎𝑏𝑠𝑖𝑛𝜙 3.24 Vector (cross product) ?⃗?𝑥?⃗? = (𝑎𝑥𝑖̂ + 𝑎𝑦𝑗̂ + 𝑎𝑧?̂?)𝑥(𝑏𝑥𝑖̂ + 𝑏𝑦𝑗̂ + 𝑏𝑧?̂?) = (𝑎𝑦𝑏𝑧 − 𝑏𝑦𝑎𝑧)𝑖̂ + (𝑎𝑧𝑏𝑥 − 𝑏𝑧𝑎𝑥)𝑗̂ + (𝑎𝑥𝑏𝑦 − 𝑏𝑥𝑎𝑦)?̂? or ?⃗?𝑥?⃗? = 𝑑𝑒𝑡 | 𝑖̂ 𝑗 ?̂? 𝑎𝑥 𝑎𝑦 𝑎𝑧 𝑏𝑥 𝑏𝑦 𝑏𝑧 | 3.26 Position vector 𝑟 = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧?̂? 4.4 displacement ∆𝑟 = ∆𝑥𝑖̂ + ∆𝑦𝑗̂ + ∆𝑧?̂? 4.4 Average Velocity ?⃗⃗?𝑎𝑣𝑔 = ∆𝑥 ∆𝑡 4.8 Instantaneous Velocity ?⃗? = 𝑑𝑟 𝑑𝑡 = 𝑣𝑥 ?̂? + 𝑣𝑦𝑗̂ + 𝑣𝑧?̂? 4.10 4.11 Average Acceleration ?⃗?𝑎𝑣𝑔 = ∆?⃗? ∆𝑡 4.15 Instantaneous Acceleration ?⃗? = 𝑑?⃗? 𝑑𝑡 ?⃗? = 𝑎𝑥𝑖̂ + 𝑎𝑦𝑗̂ + 𝑎𝑧?̂? 4.16 4.17 Projectile Motion 𝑣𝑦 = 𝑣0𝑠𝑖𝑛𝜃0 − 𝑔𝑡 4.23 ∆𝑥 = 𝑣0𝑐𝑜𝑠𝜃𝑡 + 1 2 𝑎𝑥𝑡 2 or ∆𝑥 = 𝑣0𝑐𝑜𝑠𝜃𝑡 if 𝑎𝑥=0 4.21 ∆𝑦 = 𝑣0𝑠𝑖𝑛𝜃𝑡 − 1 2 𝑔𝑡2 4.22 𝑣𝑦 2 = (𝑣0𝑠𝑖𝑛𝜃0) 2 − 2𝑔∆y 4.24 𝑣𝑦 = 𝑣0𝑠𝑖𝑛𝜃0 − 𝑔𝑡 4.23 Trajectory 𝑦 = (𝑡𝑎𝑛𝜃0)𝑥 − 𝑔𝑥2 2(𝑣0𝑐𝑜𝑠𝜃0) 2 4.25 Range 𝑅 = 𝑣0 2 𝑔 sin(2𝜃0) 4.26 Relative Motion 𝑣𝐴𝐶⃗⃗ ⃗⃗ ⃗⃗ ⃗ = 𝑣𝐴𝐵⃗⃗ ⃗⃗ ⃗⃗ ⃗ + 𝑣𝐵𝐶⃗⃗ ⃗⃗ ⃗⃗ ⃗ 𝑎𝐴𝐵⃗⃗ ⃗⃗ ⃗⃗ ⃗ = 𝑎𝐵𝐴⃗⃗⃗⃗⃗⃗⃗⃗ 4.44 4.45 Uniform Circular Motion 𝑎 = 𝑣2 𝑟 𝑇 = 2𝜋𝑟 𝑣 4.34 4.35 Chapter 9 Impulse and Momentum Impulse 𝐽 = ∫ ?⃗?(𝑡)𝑑𝑡 𝑡𝑓 𝑡𝑖 𝐽 = 𝐹𝑛𝑒𝑡∆𝑡 9.30 9.35 Linear Momentum ?⃗? = 𝑚?⃗? 9.22 Impulse-Momentum Theorem 𝐽 = Δ?⃗? = ?⃗?𝑓 − ?⃗?𝑖 9.31 9.32 Newton’s 2nd law ?⃗?𝑛𝑒𝑡 = 𝑑?⃗? 𝑑𝑡 9.22 System of Particles ?⃗?𝑛𝑒𝑡 = 𝑚?⃗⃗?𝑐𝑜𝑚 ?⃗? = 𝑀?⃗?𝑐𝑜𝑚 ?⃗?𝑛𝑒𝑡 = 𝑑?⃗⃗⃗? 𝑑𝑡 9.14 9.25 9.27 Collision Final Velocity of 2 objects in a head-on collision where one object is initially at rest 1: moving object 2: object at rest 𝑣1𝑓 = ( 𝑚1 − 𝑚2 𝑚1 + 𝑚2 ) 𝑣1𝑖 𝑣2𝑓 = ( 2𝑚1 𝑚1 + 𝑚2 ) 𝑣1𝑖 9.67 9.68 Conservation of Linear Momentum (in 1D) ?⃗? = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ?⃗?𝑖 = ?⃗?𝑓 9.42 9.43 Elastic Collision ?⃗?1𝑖 + ?⃗?2𝑖 = ?⃗?1𝑓 + ?⃗?2𝑓 𝑚1𝑣𝑖1 + 𝑚2𝑣12 = 𝑚1𝑣𝑓1 + 𝑚2𝑣𝑓2 𝐾1𝑖 + 𝐾2𝑖 = 𝐾1𝑓 + 𝐾2𝑓 9.50 9.51 9.78 Collision continued… Inelastic Collision 𝑚1𝑣01 + 𝑚2𝑣02 = (𝑚1 + 𝑚2)𝑣𝑓 Conservation of Linear Momentum (in 2D) ?⃗?1𝑖 + ?⃗?2𝑖 = ?⃗?1𝑓 + ?⃗?2𝑓 9.77 Average force 𝐹𝑎𝑣𝑔 = − 𝑛 ∆𝑡 ∆𝑝 = − 𝑛 ∆𝑡 𝑚∆𝑣 𝐹𝑎𝑣𝑔 = − ∆𝑚 ∆𝑡 ∆𝑣 9.37 9.40 Center of Mass Center of mass location 𝑟𝑐𝑜𝑚 = 1 𝑀 ∑ 𝑚𝑖 𝑛 𝑖=1 𝑟𝑖 9.8 Center of mass velocity ?⃗?𝑐𝑜𝑚 = 1 𝑀 ∑ 𝑚𝑖 𝑛 𝑖=1 ?⃗?𝑖 Rocket Equations Thrust (Rvrel) 𝑅𝑣𝑟𝑒𝑙 = 𝑀𝑎 9.88 Change in velocity Δ𝑣 = 𝑣𝑟𝑒𝑙𝑙𝑛 𝑀𝑖 𝑀𝑓 9.88 Chapter 10 Angular displacement (in radians 𝜃 = 𝑠 𝑟 Δ𝜃 = 𝜃2 − 𝜃1 10.1 10.4 Average angular velocity 𝜔𝑎𝑣𝑔 = ∆𝜃 ∆𝑡 10.5 Instantaneous Velocity 𝜔 = 𝑑𝜃 𝑑𝑡 10.6 Average angular acceleration 𝛼𝑎𝑣𝑔 = ∆𝜔 ∆𝑡 10.7 Instantaneous angular acceleration 𝛼 = 𝑑𝜔 𝑑𝑡 10.8 Rotational Kinematics 𝜔 = 𝜔0 + 𝛼𝑡 10.12 Δ𝜃 = 𝜔0𝑡 + 1 2 𝛼𝑡2 10.13 𝜔2 = 𝜔0 2 + 2𝛼Δ𝜃 10.14 Δ𝜃 = 1 2 (𝜔 + 𝜔0)𝑡 10.15 Δ𝜃 = 𝜔𝑡 − 1 2 𝛼𝑡2 10.16 Relationship Between Angular and Linear Variables Velocity 𝑣 = 𝜔𝑟 10.18 Tangential Acceleration 𝑎𝑡 = 𝛼𝑟 10.19 Radical component of ?⃗? 𝑎𝑟 = 𝑣2 𝑟 = 𝜔2𝑟 10.23 Period 𝑇 = 2𝜋𝑟 𝑣 = 2𝜋 𝜔 10.19 10.20 Rotation inertia 𝐼 = ∑ 𝑚𝑖𝑟𝑖 2 10.34 Rotation inertia (discrete particle system) 𝐼 = ∫ 𝑟2𝑑𝑚 10.35 Parallel Axis Theorem h=perpendicular distance between two axes 𝐼 = 𝐼𝑐𝑜𝑚 + 𝑀ℎ 2 10.36 Torque 𝜏 = 𝑟𝐹𝑡 = 𝑟⊥𝐹 = 𝑟𝐹𝑠𝑖𝑛𝜃 10.39- 10.41 Newton’s Second Law 𝜏𝑛𝑒𝑡 = 𝐼𝛼 10.45 Rotational work done by a toque 𝑊 = ∫ 𝜏𝑑𝜃 𝜃𝑓 𝜃𝑖 𝑊 = 𝜏∆𝜃 (𝜏 constant) 10.53 10.54 Power in rotational motion 𝑃 = 𝑑𝑊 𝑑𝑡 = 𝜏𝜔 10.55 Rotational Kinetic Energy 𝐾 = 1 2 𝐼𝜔2 10.34 Work-kinetic energy theorem ∆𝐾 = 𝐾𝑓 − 𝐾𝑖 = 1 2 𝐼𝜔𝑓 2 − 1 2 𝐼𝜔𝑖 2 = 𝑊 10.52 Moments of Inertia I for various rigid objects of Mass M Thin walled hollow cylinder or hoop about central axis 𝐼 = 𝑀𝑅2 Annular cylinder (or ring) about central axis 𝐼 = 1 2 𝑀(𝑅1 2 + 𝑅2 2) Solid cylinder or disk about central axis 𝐼 = 1 2 𝑀𝑅2 Solid cylinder or disk about central diameter 𝐼 = 1 4 𝑀𝑅2 + 1 12 𝑀𝐿2 Solid Sphere, axis through center 𝐼 = 2 5 𝑀𝑅2 Solid Sphere, axis tangent to surface 𝐼 = 7 5 𝑀𝑅2 Thin Walled spherical shell, axis through center 𝐼 = 2 3 𝑀𝑅2 Thin rod, axis perpendicular to rod and passing though center 𝐼 = 1 12 𝑀𝐿2 Thin rod, axis perpendicular to rod and passing though end 𝐼 = 1 3 𝑀𝐿2 Thin Rectangular sheet (slab), axis parallel to sheet and passing though center of the other edge 𝐼 = 1 12 𝑀𝐿2 Thin Rectangular sheet (slab_, axis along one edge 𝐼 = 1 3 𝑀𝐿2 Thin rectangular sheet (slab) about perpendicular axis through center 𝐼 = 1 12 𝑀(𝑎2 + 𝑏2) Chapter 14 Chapter 15 Density 𝜌 = ∆𝑚 ∆𝑉 𝜌 = 𝑚 𝑉 14.1 14.2 Pressure 𝑝 = ∆𝐹 ∆𝐴 𝑝 = 𝐹 𝐴 14.3 14.4 Pressure and depth in a static Fluid P1 is higher than P2 𝑝2 = 𝑝1 + 𝜌𝑔(𝑦1 − 𝑦2) 𝑝 = 𝑝0 + 𝜌𝑔ℎ 14.7 14.8 Gauge Pressure 𝜌𝑔ℎ Archimedes’ principle 𝐹𝑏 = 𝑚𝑓𝑔 14.16 Mass Flow Rate 𝑅𝑚 = 𝜌𝑅𝑉 = 𝜌𝐴𝑣 14.25 Volume flow rate 𝑅𝑉 = 𝐴𝑣 14.24 Bernoulli’s Equation 𝑝 + 1 2 𝜌𝑣2 + 𝜌𝑔𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 14.29 Equation of continuity 𝑅𝑚 = 𝜌𝑅𝑉 = 𝜌𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 14.25 Equation of continuity when 𝑅𝑉 = 𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 14.24 Frequency cycles per time 𝑓 = 1 𝑇 15.2 displacement 𝑥 = 𝑥𝑚cos (𝜔𝑡 + 𝜙) 15.3 Angular frequency 𝜔 = 2𝜋 𝑇 = 2𝜋𝑓 15.5 Velocity 𝑣 = −𝜔𝑥𝑚sin(𝜔𝑡 + 𝜙) 15.6 Acceleration 𝑎 = −𝜔2𝑥𝑚cos (𝜔𝑡 + 𝜙) 15.7 Kinetic and Potential Energy 𝐾 = 1 2 𝑚𝑣2 𝑈 = 1 2 𝑘𝑥2 Angular frequency 𝜔 = √ 𝑘 𝑚 15.12 Period 𝑇 = 2𝜋√ 𝑚 𝑘 15.13 Torsion pendulum 𝑇 = 2𝜋√ 𝐼 𝑘 15.23 Simple Pendulum 𝑇 = 2𝜋√ 𝐿 𝑔 15.28 Physical Pendulum 𝑇 = 2𝜋√ 𝐼 𝑚𝑔𝐿 15.29 Damping force ?⃗?𝑑 = −𝑏?⃗? displacement 𝑥(𝑡) = 𝑥𝑚𝑒 − 𝑏𝑡 2𝑚cos (𝜔′𝑡 + 𝜙) 15.42 Angular frequency 𝜔′ = √ 𝑘 𝑚 − 𝑏2 4𝑚2 15.43 Mechanical Energy 𝐸(𝑡) ≈ 1 2 𝑘𝑥𝑚 2 𝑒− 𝑏𝑡 𝑚 15.44 Chapter 16 Sinusoidal Waves Mathematical form (positive direction) 𝑦(𝑥, 𝑡) = 𝑦𝑚sin (𝑘𝑥 − 𝜔𝑡) 16.2 Angular wave number 𝑘 = 2𝜋 𝜆 16.5 Angular frequency 𝜔 = 2𝜋 𝑇 = 2𝜋𝑓 16.9 Wave speed 𝑣 = 𝜔 𝑘 = 𝜆 𝑇 = 𝜆𝑓 16.13 Average Power 𝑃𝑎𝑣𝑔 = 1 2 𝜇𝑣𝜔2𝑦𝑚 2 16.33 Traveling Wave Form 𝑦(𝑥, 𝑡) = ℎ(𝑘𝑥 ± 𝜔𝑡) 16.17 Wave speed on stretched string 𝑣 = √ 𝜏 𝜇 16.26 Resulting wave when 2 waves only differ by phase constant 𝑦′(𝑥, 𝑡) = [2𝑦𝑚 cos ( 1 2 𝜙)] sin (𝑘𝑥 − 𝜔𝑡 + 1 2 𝜙) 16.51 Standing wave 𝑦′(𝑥, 𝑡) = [2𝑦𝑚 sin(𝑘𝑥)]cos (𝜔𝑡) 16.60 Resonant frequency 𝑓 = 𝑣 𝜆 = 𝑛 𝑣 2𝐿 for n=1,2,… 16.66 Chapter 17 Sound Waves Speed of sound wave 𝑣 = √ 𝐵 𝜌 17.3 displacement 𝑠 = 𝑠𝑚cos (𝑘𝑥 − 𝜔𝑡) 17.12 Change in pressure Δ𝑝 = Δ𝑝𝑚 sin(𝑘𝑥 − 𝜔𝑡) 17.13 Pressure amplitude Δ𝑝𝑚 = (𝑣𝜌𝜔)𝑠𝑚 17.14 Interference Phase difference 𝜙 = Δ𝐿 𝜆 2𝜋 17.21 Fully Constructive Interference 𝜙 = 𝑚(2𝜋) for m=0,1,2… Δ𝐿 𝜆 = 0,1,2 17.22 17.23 Full Destructive interference 𝜙 = (2𝑚 + 1)𝜋 for m=0,12 Δ𝐿 𝜆 = .5,1.5,2.5 … 17.24 17.25 Mechanical Energy 𝐸(𝑡) ≈ 1 2 𝑘𝑥𝑚 2 𝑒− 𝑏𝑡 𝑚 15.44 Sound Intensity Intensity 𝐼 = 𝑃 𝐴 𝐼 = 1 2 𝜌𝑣𝜔2𝑠𝑚 2 17.26 17.27 Intensity -uniform in all directions 𝐼 = 𝑃𝑠 4𝜋𝑟2 17.29 Intensity level in decibels 𝛽 = (10𝑑𝐵) log ( 𝐼 𝐼𝑜 ) 17.29 Mechanical Energy 𝐸(𝑡) ≈ 1 2 𝑘𝑥𝑚 2 𝑒− 𝑏𝑡 𝑚 15.44 Standing Waves Patterns in Pipes Standing wave frequency (open at both ends) 𝑓 = 𝑣 𝜆 = 𝑛𝑣 2𝐿 for n=1,2,3 17.39 Standing wave frequency (open at one end) 𝑓 = 𝑣 𝜆 = 𝑛𝑣 4𝐿 for n=1,3,5 17.41 beats 𝑓𝑏𝑒𝑎𝑡 = 𝑓1 − 𝑓2 17.46 Doppler Effect Source Moving toward stationary observer 𝑓′ = 𝑓 𝑣 𝑣 − 𝑣𝑠 17.53 Source Moving away from stationary observer 𝑓′ = 𝑓 𝑣 𝑣 + 𝑣𝑠 17.54 Observer moving toward stationary source 𝑓′ = 𝑓 𝑣 + 𝑣𝐷 𝑣 17.49 Observer moving away from stationary source 𝑓′ = 𝑓 𝑣 − 𝑣𝐷 𝑣 17.51 Shockwave Half-angle 𝜃 of Mach cone 𝑠𝑖𝑛𝜃 = 𝑣 𝑣𝑠 17.57