Math Formulas

PHYS 2310 Engineering Physics I Formula Sheets Chapters 1-18 Chapter 1/Important Numbers Chapter 2 Units for SI Base Quantities Velocity Quantity Unit Name Unit Symbol 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 ∆𝑥 Length Meter M Average Velocity 𝑉 = = 2.2 𝑎𝑣𝑔 𝑡𝑖𝑚𝑒 ∆𝑡 Time Second s Mass (not weight) Kilogram kg 𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 Average Speed 𝑠𝑎𝑣𝑔 = 2.3 𝑡𝑖𝑚𝑒 ∆𝑥̅ 𝑑𝑥 Common Conversions Instantaneous Velocity 𝑣 = lim = 2.4 1 kg or 1 m 1000 g or m 1 m 1 × 106 𝜇𝑚 ∆𝑡→0 ∆𝑡 𝑑𝑡 1 m 100 cm 1 inch 2.54 cm 1 m 1000 mm 1 day 86400 seconds Acceleration 1 second 1000 milliseconds 1 hour 3600 seconds 1 m 3.281 ft 360° 2𝜋 rad ∆𝑣 Average Acceleration 𝑎 = 2.7 𝑎𝑣𝑔 ∆𝑡 Important Constants/Measurements 𝑑𝑣 𝑑2𝑥 24 Instantaneous 2.8 Mass of Earth 5.98 × 10 kg 𝑎 = = 2 Acceleration 𝑑𝑡 𝑑𝑡 2.9 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 Motion of a particle with constant acceleration g (on earth) 9.8 m/s2 𝑣 = 𝑣0 + 𝑎𝑡 2.11 Density 1 ∆𝑥 = (𝑣 + 𝑣)𝑡 2.17 Common geometric Formulas 2 0 2 Circumference 𝐶 = 2𝜋𝑟 Area circle 𝐴 = 𝜋𝑟 1 2 ∆𝑥 = 𝑣0𝑡 + 𝑎𝑡 2.15 Surface area 4 3 2 𝑆𝐴 = 4𝜋𝑟2 Volume (sphere) 𝑉 = 𝜋𝑟 (sphere) 3 2 2 𝑣 = 𝑣0 + 2𝑎∆𝑥 2.16 𝑉 = 𝑙 ∙ 𝑤 ∙ ℎ Volume (rectangular solid) 𝑉 = 𝑎𝑟𝑒𝑎 ∙ 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 Chapter 3 Chapter 4 Adding Vectors ⃗⃗ ⃗⃗ 3.2 Position vector 𝑟⃗ = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧𝑘̂ 4.4 Geometrically 𝑎⃗ + 𝑏 = 𝑏 + 𝑎⃗ Adding Vectors displacement ∆𝑟⃗ = ∆𝑥𝑖̂ + ∆𝑦𝑗̂ + ∆𝑧𝑘̂ 4.4 Geometrically (𝑎⃗ + 𝑏⃗⃗) + 𝑐⃗ = 𝑎⃗ + (𝑏⃗⃗ + 𝑐⃗) 3.3 (Associative Law) ∆𝑥 Average Velocity 𝑉⃗⃗𝑎𝑣𝑔 = 4.8 𝑎 = 𝑎𝑐𝑜𝑠𝜃 ∆𝑡 Components of Vectors 𝑥 3.5 𝑎 = 𝑎𝑠𝑖𝑛𝜃 𝑑𝑟⃗ 4.10 𝑦 Instantaneous Velocity 𝑣⃗ = = 𝑣 𝑖̂ + 𝑣 𝑗̂ + 𝑣 𝑘̂ 𝑑𝑡 𝑥 𝑦 𝑧 4.11 Magnitude of vector |𝑎| = 𝑎 = √𝑎2 + 𝑎2 3.6 ∆𝑣⃗ 𝑥 𝑦 Average Acceleration 𝑎⃗ = 4.15 𝑎𝑣𝑔 ∆𝑡 Angle between x axis 𝑎𝑦 𝑡𝑎𝑛𝜃 = 3.6 𝑑𝑣⃗ and vector 𝑎 Instantaneous 𝑎⃗ = 4.16 𝑥 𝑑𝑡 Acceleration ̂ 4.17 Unit vector notation 𝑎⃗ = 𝑎𝑥𝑖̂ + 𝑎𝑦𝑗̂ + 𝑎𝑧𝑘̂ 3.7 𝑎⃗ = 𝑎𝑥𝑖̂ + 𝑎𝑦𝑗̂ + 𝑎𝑧𝑘 𝑟 = 𝑎 + 𝑏 3.10 Adding vectors in 𝑥 𝑥 𝑥 Projectile Motion 𝑟𝑦 = 𝑎𝑦 + 𝑏𝑦 3.11 Component Form 𝑣𝑦 = 𝑣0𝑠𝑖𝑛𝜃0 − 𝑔𝑡 4.23 𝑟𝑧 = 𝑎𝑧 + 𝑏𝑧 3.12 1 ∆𝑥 = 𝑣 𝑐𝑜𝑠𝜃𝑡 + 𝑎 𝑡2 4.21 Scalar (dot product) 𝑎⃗ ∙ 𝑏⃗⃗ = 𝑎𝑏𝑐𝑜𝑠𝜃 3.20 0 2 𝑥 or ∆𝑥 = 𝑣0𝑐𝑜𝑠𝜃𝑡 if 𝑎𝑥=0 𝑎⃗ ∙ 𝑏⃗⃗ = (𝑎𝑥𝑖̂ + 𝑎𝑦𝑗̂ + 𝑎𝑧𝑘̂) ∙ (𝑏𝑥𝑖̂ + 𝑏𝑦𝑗̂ + 𝑏𝑧𝑘̂) Scalar (dot product) 3.22 1 2 𝑎⃗ ∙ 𝑏⃗⃗ = 𝑎 𝑏 + 𝑎 𝑏 + 𝑎 𝑏 ∆𝑦 = 𝑣0𝑠𝑖𝑛𝜃𝑡 − 𝑔𝑡 4.22 𝑥 𝑥 𝑦 𝑦 𝑧 𝑧 2 2 2 Projection of 𝑎⃗ 𝑜𝑛 𝑏⃗⃗ or 𝑎⃗ ∙ 𝑏⃗⃗ 𝑣𝑦 = (𝑣0𝑠𝑖𝑛𝜃0) − 2𝑔∆y 4.24 ⃗⃗ component of 𝑎⃗ 𝑜𝑛 𝑏 |𝑏| 𝑣𝑦 = 𝑣0𝑠𝑖𝑛𝜃0 − 𝑔𝑡 4.23 Vector (cross) product 𝑔𝑥2 𝑐 = 𝑎𝑏𝑠𝑖𝑛𝜙 3.24 Trajectory 4.25 magnitude 𝑦 = (𝑡𝑎𝑛𝜃0)𝑥 − 2 2(𝑣0𝑐𝑜𝑠𝜃0) 𝑎⃗𝑥𝑏⃗⃗ = (𝑎 𝑖̂ + 𝑎 𝑗̂ + 𝑎 𝑘̂)𝑥(𝑏 𝑖̂ + 𝑏 𝑗̂ + 𝑏 𝑘̂) 2 𝑥 𝑦 𝑧 𝑥 𝑦 𝑧 𝑣0 = (𝑎 𝑏 − 𝑏 𝑎 )𝑖̂ + (𝑎 𝑏 − 𝑏 𝑎 )𝑗̂ Range 𝑅 = sin(2𝜃 ) 4.26 𝑦 𝑧 𝑦 𝑧 𝑧 𝑥 𝑧 𝑥 𝑔 0 ̂ + (𝑎𝑥𝑏𝑦 − 𝑏𝑥𝑎𝑦)𝑘 Vector (cross product) or 3.26 ⃗𝑣⃗⃗⃗⃗⃗⃗ = ⃗𝑣⃗⃗⃗⃗⃗⃗ + 𝑣⃗⃗⃗⃗⃗⃗⃗ 𝑖̂ 𝑗 𝑘̂ Relative Motion 𝐴𝐶 𝐴𝐵 𝐵𝐶 4.44 𝑎⃗⃗⃗𝐴𝐵⃗⃗⃗⃗ = 𝑎⃗⃗⃗⃗𝐵𝐴⃗⃗⃗⃗ 𝑎⃗𝑥𝑏⃗⃗ = 𝑑𝑒𝑡 |𝑎𝑥 𝑎𝑦 𝑎𝑧| 4.45 2 𝑏𝑥 𝑏𝑦 𝑏𝑧 Uniform Circular 𝑣 2𝜋𝑟 4.34 𝑎 = 𝑇 = Motion 𝑟 𝑣 4.35 Chapter 5 Chapter 6 Newton’s Second Law Friction 𝐹⃗𝑛𝑒𝑡 = 𝑚𝑎⃗ Static Friction General 5.1 𝑓⃗𝑠,𝑚𝑎𝑥 = 𝜇𝑠𝐹𝑁 6.1 (maximum) 𝐹𝑛𝑒𝑡,𝑥 = 𝑚𝑎𝑥 Kinetic Frictional 𝑓⃗𝑘 = 𝜇𝑘𝐹𝑁 6.2 Component form 𝐹𝑛𝑒𝑡,𝑦 = 𝑚𝑎𝑦 5.2 𝐹𝑛𝑒𝑡,𝑧 = 𝑚𝑎𝑦 1 2 Drag Force 𝐷 = 𝐶𝜌𝐴𝑣 6.14 2 Gravitational Force 2𝐹𝑔 Terminal velocity 𝑣 = √ 6.16 𝑡 𝐶𝜌𝐴 Gravitational Force 5.8 𝐹 = 𝑚𝑔 𝑔 Centripetal 𝑣2 Weight 𝑎 = 6.17 𝑊 = 𝑚𝑔 5.12 acceleration 𝑅 2 Centripetal Force 𝑚𝑣 𝐹 = 6.18 𝑅 Chapter 7 Chapter 8 𝑥𝑓 Kinetic Energy 1 2 8.1 𝐾 = 𝑚𝑣 7.1 Potential Energy ∆𝑈 = −𝑊 = − ∫ 𝐹(𝑥)𝑑𝑥 2 𝑥𝑖 8.6 Work done by constant 7.7 Gravitational Potential Force ⃗ ⃗ ∆𝑈 = 𝑚𝑔∆𝑦 8.7 𝑊 = 𝐹𝑑𝑐𝑜𝑠𝜃 = 𝐹 ∙ 𝑑 7.8 Energy 1 Work- Kinetic Energy Elastic Potential Energy 𝑈(𝑥) = 𝑘𝑥2 8.11 ∆𝐾 = 𝐾 − 𝐾 = 𝑊 7.10 2 Theorem 𝑓 0 Work done by gravity Mechanical Energy 𝐸𝑚𝑒𝑐 = 𝐾 + 𝑈 8.12 𝑊 = 𝑚𝑔𝑑𝑐𝑜𝑠𝜙 7.12 𝑔 Principle of Work done by ∆𝐾 = 𝑊 + 𝑊 𝐾1 + 𝑈1 = 𝐾2 + 𝑈2 8.18 𝑎 𝑔 7.15 conservation of 𝐸𝑚𝑒𝑐 = ∆𝐾 + ∆𝑈 = 0 8.17 lifting/lowering object 𝑊𝑎 = 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝐹𝑜𝑟𝑐𝑒 mechanical energy Spring Force (Hooke’s 𝐹⃗𝑠 = −𝑘𝑑⃗ 7.20 𝑑𝑈(𝑥) Force acting on particle 𝐹(𝑥) = − 8.22 law) 𝐹𝑥 = −𝑘𝑥 (along x-axis) 7.21 𝑑𝑥 1 1 Work on System by Work done by spring 2 2 7.25 8.25 𝑊𝑠 = 𝑘𝑥𝑖 − 𝑘𝑥𝑓 external force 𝑊 = ∆𝐸 = ∆𝐾 + ∆𝑈 2 2 𝑚𝑒𝑐 8.26 𝑥 𝑦 𝑧 With no friction Work done by Variable 𝑓 𝑓 𝑓 𝑊 = ∫ 𝐹𝑥𝑑𝑥 + ∫ 𝐹𝑦𝑑𝑦 + ∫ 𝐹𝑧𝑑𝑧 7.36 Work on System by Force 𝑥𝑖 𝑦𝑖 𝑧𝑖 external force 𝑊 = ∆𝐸𝑚𝑒𝑐 + ∆𝐸𝑡ℎ 8.33 Average Power With friction (rate at which that 𝑊 𝑃 = 7.42 Change in thermal force does work on an 𝑎𝑣𝑔 ∆𝑡 ∆𝐸 = 𝑓 𝑑𝑐𝑜𝑠𝜃 8.31 energy 𝑡ℎ 𝑘 object) Conservation of Energy 𝑑𝑊 7.43 𝑊 = ∆𝐸 = ∆𝐸 + ∆𝐸 + ∆𝐸 8.35 Instantaneous Power 𝑃 = = 𝐹𝑉𝑐𝑜𝑠𝜃 = 𝐹⃗ ∙ 𝑣⃗ *if isolated W=0 𝑚𝑒𝑐 𝑡ℎ 𝑖𝑛𝑡 𝑑𝑡 7.47 ∆𝐸 Average Power 𝑃 = 8.40 𝑎𝑣𝑔 ∆𝑡 𝑑𝐸 Instantaneous Power 𝑃 = 8.41 𝑑𝑡 **In General Physics, Kinetic Energy is abbreviated to KE and Potential Energy is PE Chapter 9 Impulse and Momentum Collision continued… 𝑡 Inelastic Collision 𝑚 𝑣 + 𝑚 𝑣 = (𝑚 + 𝑚 )𝑣 𝑓 9.30 1 01 2 02 1 2 𝑓 𝐽⃗ = ∫ 𝐹⃗(𝑡)𝑑𝑡 Impulse 𝑡𝑖 9.35 Conservation of Linear 𝑃⃗⃗ + 𝑃⃗⃗ = 𝑃⃗⃗ + 𝑃⃗⃗ 𝐽 = 𝐹𝑛𝑒𝑡∆𝑡 1𝑖 2𝑖 1𝑓 2𝑓 9.77 Momentum (in 2D) Linear Momentum 𝑝⃗ = 𝑚𝑣⃗ 9.22 𝑛 𝑛 𝐹 = − ∆𝑝 = − 𝑚∆𝑣 𝑎𝑣𝑔 9.37 Impulse-Momentum 9.31 Average force ∆𝑡 ∆𝑡 𝐽⃗ = Δ𝑝⃗ = 𝑝⃗ − 𝑝⃗ ∆𝑚 9.40 Theorem 𝑓 𝑖 9.32 𝐹 = − ∆𝑣 𝑎𝑣𝑔 ∆𝑡 𝑑𝑝⃗ nd Newton’s 2 law 𝐹⃗𝑛𝑒𝑡 = 9.22 𝑑𝑡 Center of Mass 𝐹⃗ = 𝑚𝑎⃗⃗⃗ 𝑛 𝑛𝑒𝑡 𝑐𝑜𝑚 9.14 1 ⃗⃗ Center of mass location 𝑟⃗ = ∑ 𝑚 𝑟⃗ 9.8 System of Particles 𝑃 = 𝑀𝑣⃗𝑐𝑜𝑚 9.25 𝑐𝑜𝑚 𝑖 𝑖 ⃗⃗⃗ 𝑀 𝑑𝑃 9.27 𝑖=1 𝐹⃗𝑛𝑒𝑡 = 𝑛 𝑑𝑡 1 Center of mass velocity 𝑣⃗ = ∑ 𝑚 𝑣⃗ 𝑐𝑜𝑚 𝑀 𝑖 𝑖 Collision 𝑖=1 Final Velocity of 2 𝑚1 − 𝑚2 objects in a head-on 𝑣 = ( ) 𝑣 Rocket Equations 1𝑓 𝑚 + 𝑚 1𝑖 9.67 collision where one 1 2 Thrust (Rvrel) 𝑅𝑣𝑟𝑒𝑙 = 𝑀𝑎 object is initially at rest 9.88 2𝑚1 9.68 1: moving object 𝑣2𝑓 = ( ) 𝑣1𝑖 𝑚1 + 𝑚2 Change in velocity 𝑀𝑖 2: object at rest Δ𝑣 = 𝑣𝑟𝑒𝑙𝑙𝑛 9.88 𝑀𝑓 Conservation of Linear 𝑃⃗⃗ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 9.42 Momentum (in 1D) 𝑃⃗⃗ = 𝑃⃗⃗ 9.43 𝑖 𝑓 𝑝⃗1𝑖 + 𝑝⃗2𝑖 = 𝑝⃗1𝑓 + 𝑝⃗2𝑓 9.50 Elastic Collision 𝑚1𝑣𝑖1 + 𝑚2𝑣12 = 𝑚1𝑣𝑓1 + 𝑚2𝑣𝑓2 9.51 𝐾1𝑖 + 𝐾2𝑖 = 𝐾1𝑓 + 𝐾2𝑓 9.78 Chapter 10 𝑠 Angular displacement 𝜃 = 10.1 𝑟 Rotation inertia 𝐼 = ∑ 𝑚 𝑟2 10.34 (in radians 10.4 𝑖 𝑖 Δ𝜃 = 𝜃2 − 𝜃1 Average angular ∆𝜃 Rotation inertia 𝜔 = 10.5 2 velocity 𝑎𝑣𝑔 ∆𝑡 (discrete particle 𝐼 = ∫ 𝑟 𝑑𝑚 10.35 system) 𝑑𝜃 Instantaneous Velocity 𝜔 = 10.6 Parallel Axis Theorem 𝑑𝑡 h=perpendicular 𝐼 = 𝐼 + 𝑀ℎ2 10.36 Average angular ∆𝜔 distance between two 𝑐𝑜𝑚 𝛼 = 10.7 acceleration 𝑎𝑣𝑔 ∆𝑡 axes Instantaneous angular 𝑑𝜔 10.39- 𝛼 = 10.8 Torque 𝜏 = 𝑟𝐹𝑡 = 𝑟⊥𝐹 = 𝑟𝐹𝑠𝑖𝑛𝜃 acceleration 𝑑𝑡 10.41 Rotational Kinematics Newton’s Second Law 𝜏𝑛𝑒𝑡 = 𝐼𝛼 10.45 𝜔 = 𝜔0 + 𝛼𝑡 10.12 𝜃𝑓 Rotational work done 𝑊 = ∫ 𝜏𝑑𝜃 10.53 1 2 Δ𝜃 = 𝜔0𝑡 + 𝛼𝑡 10.13 by a toque 𝜃𝑖 10.54 2 𝑊 = 𝜏∆𝜃 (𝜏 constant) 2 2 𝜔 = 𝜔0 + 2𝛼Δ𝜃 10.14 Power in rotational 𝑑𝑊 𝑃 = = 𝜏𝜔 10.55 1 motion 𝑑𝑡 Δ𝜃 = (𝜔 + 𝜔0)𝑡 10.15 2 Rotational Kinetic 1 𝐾 = 𝐼𝜔2 10.34 1 Energy Δ𝜃 = 𝜔𝑡 − 𝛼𝑡2 10.16 2 2 Work-kinetic energy 1 1 ∆𝐾 = 𝐾 − 𝐾 = 𝐼𝜔2 − 𝐼𝜔2 = 𝑊 10.52 theorem 𝑓 𝑖 2 𝑓 2 𝑖 Relationship Between Angular and Linear Variables Velocity 𝑣 = 𝜔𝑟 10.18 Tangential Acceleration 𝑎𝑡 = 𝛼𝑟 10.19 𝑣2 Radical component of 𝑎⃗ 𝑎 = = 𝜔2𝑟 10.23 𝑟 𝑟 2𝜋𝑟 2𝜋 10.19 Period 𝑇 = = 𝑣 𝜔 10.20 Moments of Inertia I for various rigid objects of Mass M Thin walled hollow cylinder or hoop Annular cylinder (or ring) about Solid cylinder or disk about central Solid cylinder or disk about central about central axis central axis axis diameter 2 1 𝐼 = 𝑀𝑅 𝐼 = 𝑀(𝑅2 + 𝑅2) 1 2 1 2 𝐼 = 𝑀𝑅2 2 1 1 𝐼 = 𝑀𝑅2 + 𝑀𝐿2 4 12 Solid Sphere, axis through center Solid Sphere, axis tangent to surface Thin Walled spherical shell, axis Thin rod, axis perpendicular to rod through center and passing though center 2 𝐼 = 𝑀𝑅2 7 5 𝐼 = 𝑀𝑅2 2 1 5 𝐼 = 𝑀𝑅2 𝐼 = 𝑀𝐿2 3 12 Thin rod, axis perpendicular to rod Thin Rectangular sheet (slab), axis Thin Rectangular sheet (slab_, axis Thin rectangular sheet (slab) about and passing though end parallel to sheet and passing though along one edge perpendicular axis through center center of the other edge 1 𝐼 = 𝑀𝐿2 3 1 1 𝐼 = 𝑀𝐿2 𝐼 = 𝑀(𝑎2 + 𝑏2) 1 3 12 𝐼 = 𝑀𝐿2 12 Chapter 11 Rolling Bodies (wheel) Angular Momentum Speed of rolling wheel 𝑣𝑐𝑜𝑚 = 𝜔𝑅 11.2 Angular Momentum 𝑣ℓ⃗⃗ = 𝑟⃗⃗ × 𝑝⃗⃗⃗ = 𝑚(𝑟⃗⃗ × 𝑣⃗⃗⃗) 11.18 Kinetic Energy of Rolling 1 1 2 2 11.5 Magnitude of Angular ℓ = 𝑟𝑚𝑣𝑠𝑖𝑛𝜙 11.19- Wheel 𝐾 = 𝐼𝑐𝑜𝑚𝜔 + 𝑀𝑣𝑐𝑜𝑚 2 2 Momentum ℓ = 𝑟𝑝⊥ = 𝑟𝑚𝑣⊥ 11.21 Acceleration of rolling 𝑛 𝑎𝑐𝑜𝑚 = 𝛼𝑅 11.6 wheel ⃗⃗ 𝐿⃗⃗ = ∑ ℓ𝑖 11.26 𝑔𝑠𝑖𝑛𝜃 Angular momentum of a Acceleration along x-axis 𝑖=1 11.29 𝑎𝑐𝑜𝑚,𝑥 = − 11.10 system of particles extending up the ramp 𝐼𝑐𝑜𝑚 𝑑𝐿⃗⃗ 1 + 2 𝜏⃗ = 𝑀𝑅 𝑛𝑒𝑡 𝑑𝑡 Torque as a vector Angular Momentum continued Angular Momentum of a Torque 𝜏⃗ = 𝑟⃗ × 𝐹⃗ 11.14 𝐿 = 𝐼𝜔 11.31 rotating rigid body 11.15- ⃗⃗ Magnitude of torque 𝜏 = 𝑟𝐹 = 𝑟 𝐹 = 𝑟𝐹𝑠𝑖𝑛𝜙 Conservation of angular 𝐿 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 11.32 ⊥ ⊥ 11.17 momentum 𝐿⃗⃗𝑖 = 𝐿⃗⃗𝑓 11.33 nd 𝑑ℓ⃗⃗ Newton’s 2 Law 𝜏⃗ = 11.23 𝑛𝑒𝑡 𝑑𝑡 Precession of a Gyroscope 𝑀𝑔𝑟 Precession rate Ω = 11.31 𝐼𝜔 Chapter 12 Chapter 13 Static Equilibrium Gravitational Force 𝑚1𝑚2 (Newton’s law of 𝐹 = 𝐺 13.1 𝑟2 𝐹⃗𝑛𝑒𝑡 = 0 12.3 gravitation) 𝑛 Principle of 𝐹⃗ = ∑ 𝐹⃗ 13.5 𝜏⃗𝑛𝑒𝑡 = 0 12.5 Superposition 1,𝑛𝑒𝑡 1𝑖 𝑖=2 12.7 Gravitational Force 𝐹⃗ = 0, 𝐹⃗ = 0 If forces lie on the 𝑛𝑒𝑡,𝑥 𝑛𝑒𝑡,𝑦 12.8 acting on a particle 𝐹⃗1 = ∫ 𝑑𝐹⃗ 13.6 xy-plane from an extended 𝜏⃗𝑛𝑒𝑡,𝑧 = 0 12.9 body Gravitational 𝐺𝑀 13.11 𝑎𝑔 = 2 Stress (force per unit acceleration 𝑟 area) Gravitation within a 𝐺𝑚𝑀 𝑠𝑡𝑟𝑒𝑠𝑠 = 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 × 𝑠𝑡𝑟𝑎𝑖𝑛 12.22 𝐹 = 𝑟 13.19 Strain (fractional spherical Shell 𝑅3 change in length) Gravitational Potential 𝐺𝑀𝑚 𝑈 = − 13.21 𝐹 Energy Stress (pressure) 𝑃 = 𝑟 𝐴 Potential energy on a 𝐺𝑚1𝑚2 𝐺𝑚1𝑚3 𝐺𝑚2𝑚3 Tension/Compression 𝐹 ∆𝐿 𝑈 = − ( + + ) 13.22 = 𝐸 12.23 system (3 particles) 𝑟12 𝑟13 𝑟23 E: Young’s modulus 𝐴 𝐿 Shearing Stress 𝐹 ∆𝑥 2𝐺𝑀 = 𝐺 12.24 Escape Speed 𝑣 = √ 13.28 G: Shear modulus 𝐴 𝐿 𝑅 Hydraulic Stress ∆𝑉 2 rd 𝑝 = 𝐵 Kepler’s 3 Law 2 4𝜋 3 B: Bulk modulus 𝑉 𝑇 = ( ) 𝑟 13.34 (law of periods) 𝐺𝑀 Energy for bject in 𝐺𝑀𝑚 𝐺𝑀𝑚 13.21 𝑈 = − 𝐾 = circular orbit 𝑟 2𝑟 13.38 Mechanical Energy 𝐺𝑀𝑚 𝐸 = − 13.40 (circular orbit) 2𝑟 Mechanical Energy 𝐺𝑀𝑚 𝐸 = − 13.42 (elliptical orbit) 2𝑎 *Note: 𝐺 = 6.6704 × 10−11 𝑁 ∙ 𝑚2/𝑘𝑔2 Chapter 14 Chapter 15 ∆𝑚 Frequency 1 𝜌 = 14.1 𝑓 = 15.2 Density ∆𝑉 cycles per time 𝑇 𝑚 14.2 𝜌 = 𝑉 displacement 𝑥 = 𝑥𝑚cos (𝜔𝑡 + 𝜙) 15.3 ∆𝐹 𝑝 = 2𝜋 14.3 Angular frequency 𝜔 = = 2𝜋𝑓 15.5 Pressure ∆𝐴 𝐹 14.4 𝑇 𝑝 = 𝐴 Velocity 𝑣 = −𝜔𝑥𝑚sin(𝜔𝑡 + 𝜙) 15.6 Pressure and depth in 𝑝 = 𝑝 + 𝜌𝑔(𝑦 − 𝑦 ) 14.7 a static Fluid 2 1 1 2 Acceleration 𝑎 = −𝜔2𝑥 cos (𝜔𝑡 + 𝜙) 15.7 𝑝 = 𝑝 + 𝜌𝑔ℎ 14.8 𝑚 P1 is higher than P2 0 Kinetic and Potential 1 1 𝐾 = 𝑚𝑣2 𝑈 = 𝑘𝑥2 Gauge Pressure 𝜌𝑔ℎ Energy 2 2 Archimedes’ principle 𝐹𝑏 = 𝑚𝑓𝑔 14.16 𝑘 Angular frequency 𝜔 = √ 15.12 𝑚 Mass Flow Rate 𝑅𝑚 = 𝜌𝑅𝑉 = 𝜌𝐴𝑣 14.25 𝑚 Volume flow rate 𝑅 = 𝐴𝑣 14.24 Period 𝑇 = 2𝜋√ 15.13 𝑉 𝑘 1 Bernoulli’s Equation 𝑝 + 𝜌𝑣2 + 𝜌𝑔𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 14.29 𝐼 2 Torsion pendulum 𝑇 = 2𝜋√ 15.23 𝑘 Equation of continuity 𝑅𝑚 = 𝜌𝑅𝑉 = 𝜌𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 14.25 𝐿 Simple Pendulum 𝑇 = 2𝜋√ 15.28 Equation of continuity 𝑔 𝑅 = 𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 14.24 when 𝑉 𝐼 Physical Pendulum 𝑇 = 2𝜋√ 15.29 𝑚𝑔𝐿 Damping force 𝐹⃗𝑑 = −𝑏𝑣⃗ 𝑏𝑡 displacement − ′ 15.42 𝑥(𝑡) = 𝑥𝑚𝑒 2𝑚cos (𝜔 𝑡 + 𝜙) 𝑘 𝑏2 Angular frequency 𝜔′ = √ − 15.43 𝑚 4𝑚2 𝑏𝑡 1 − Mechanical Energy 𝐸(𝑡) ≈ 𝑘𝑥2 𝑒 𝑚 15.44 2 𝑚 Chapter 16 Sinusoidal Waves Traveling Wave Form 𝑦(𝑥, 𝑡) = ℎ(𝑘𝑥 ± 𝜔𝑡) 16.17 Mathematical form 𝑦(𝑥, 𝑡) = 𝑦 sin (𝑘𝑥 − 𝜔𝑡) 16.2 (positive direction) 𝑚 Wave speed on 𝜏 𝑣 = √ 16.26 2𝜋 stretched string 𝜇 Angular wave number 𝑘 = 16.5 𝜆 Resulting wave when 2 1 1 2𝜋 waves only differ by 𝑦′(𝑥, 𝑡) = [2𝑦 cos ( 𝜙)] sin (𝑘𝑥 − 𝜔𝑡 + 𝜙) 16.51 Angular frequency 𝜔 = = 2𝜋𝑓 16.9 𝑚 2 2 𝑇 phase constant 𝜔 𝜆 ′ Wave speed 𝑣 = = = 𝜆𝑓 16.13 Standing wave 𝑦 (𝑥, 𝑡) = [2𝑦𝑚 sin(𝑘𝑥)]cos (𝜔𝑡) 16.60 𝑘 𝑇 1 𝑣 𝑣 2 2 Resonant frequency 𝑓 = = 𝑛 for n=1,2,… 16.66 Average Power 𝑃𝑎𝑣𝑔 = 𝜇𝑣𝜔 𝑦𝑚 16.33 𝜆 2𝐿 2 Chapter 17 Sound Waves Standing Waves Patterns in Pipes Standing wave 𝐵 𝑣 𝑛𝑣 frequency (open at 𝑓 = = for n=1,2,3 17.39 Speed of sound wave 𝑣 = √ 17.3 𝜆 2𝐿 𝜌 both ends) Standing wave displacement 𝑠 = 𝑠 cos (𝑘𝑥 − 𝜔𝑡) 17.12 𝑣 𝑛𝑣 𝑚 frequency (open at 𝑓 = = for n=1,3,5 17.41 𝜆 4𝐿 one end) Change in pressure Δ𝑝 = Δ𝑝𝑚 sin(𝑘𝑥 − 𝜔𝑡) 17.13 Pressure amplitude Δ𝑝 = (𝑣𝜌𝜔)𝑠 17.14 𝑚 𝑚 beats 𝑓𝑏𝑒𝑎𝑡 = 𝑓1 − 𝑓2 17.46 Interference Δ𝐿 Doppler Effect Phase difference 𝜙 = 2𝜋 17.21 Source Moving toward 𝑣 𝜆 𝑓′ = 𝑓 17.53 𝜙 = 𝑚(2𝜋) for m=0,1,2… stationary observer 𝑣 − 𝑣𝑠 Fully Constructive 17.22 Δ𝐿 Source Moving away 𝑣 Interference = 0,1,2 17.23 ′ 𝜆 from stationary 𝑓 = 𝑓 17.54 𝑣 + 𝑣𝑠 𝜙 = (2𝑚 + 1)𝜋 for m=0,12 observer Full Destructive 17.24 Δ𝐿 Observer moving interference 17.25 𝑣 + 𝑣𝐷 = .5,1.5,2.5 … toward stationary 𝑓′ = 𝑓 17.49 𝜆 𝑣 𝑏𝑡 source 1 2 − Mechanical Energy 𝐸(𝑡) ≈ 𝑘𝑥𝑚𝑒 𝑚 15.44 Observer moving away 𝑣 − 𝑣𝐷 2 𝑓′ = 𝑓 17.51 from stationary source 𝑣 Sound Intensity 𝑃 Shockwave 𝐼 = Half-angle 𝜃 of Mach 𝑣 𝐴 17.26 𝑠𝑖𝑛𝜃 = 17.57 Intensity cone 𝑣 1 17.27 𝑠 𝐼 = 𝜌𝑣𝜔2𝑠2 2 𝑚 Intensity -uniform in 𝑃𝑠 𝐼 = 17.29 all directions 4𝜋𝑟2 Intensity level in 𝐼 𝛽 = (10𝑑𝐵) log ( ) 17.29 decibels 𝐼𝑜 𝑏𝑡 1 − Mechanical Energy 𝐸(𝑡) ≈ 𝑘𝑥2 𝑒 𝑚 15.44 2 𝑚 Chapter 18 Temperature Scales First Law of Thermodynamics 5 First Law of ∆𝐸𝑖𝑛𝑡 = 𝐸𝑖𝑛𝑡,𝑓 − 𝐸𝑖𝑛𝑡,𝑖 = 𝑄 − 𝑊 18.26 Fahrenheit to Celsius 𝑇𝐶 = (𝑇𝐹 − 32) 18.8 9 Thermodynamics 𝑑𝐸𝑖𝑛𝑡 = 𝑑𝑄 − 𝑑𝑊 18.27 9 Note: Celsius to Fahrenheit 𝑇 = 𝑇 + 32 18.8 ∆𝐸 Change in Internal Energy 𝐹 5 𝐶 𝑖𝑛𝑡 Q (heat) is positive when the system absorbs heat and negative when it loses heat. W (work) is work done by system. W is positive when expanding Celsius to Kelvin 𝑇 = 𝑇𝐶 + 273.15 18.7 and negative contracts because of an external force Thermal Expansion Applications of First Law Q=0 Linear Thermal Expansion ∆𝐿 = 𝐿𝛼∆𝑇 18.9 Adiabatic (no heat flow) ∆𝐸𝑖𝑛𝑡 = −𝑊 Volume Thermal Expansion ∆𝑉 = 𝑉𝛽∆𝑇 18.10 W=0 (constant volume) ∆𝐸 = 𝑄 Heat 𝑖𝑛𝑡 ∆𝐸 = 0 Cyclical process 𝑖𝑛𝑡 Heat and temperature 𝑄 = 𝐶(𝑇𝑓 − 𝑇𝑖) 18.13 Q=W change 𝑄 = 𝑐𝑚(𝑇 − 𝑇 ) 18.14 𝑓 𝑖 Free expansions 𝑄 = 𝑊 = ∆𝐸𝑖𝑛𝑡 = 0 Heat and phase change 𝑄 = 𝐿𝑚 18.16 Misc. 𝑉 Power 𝑓 P=Q/t Work Associated with 𝑊 = ∫ 𝑑𝑊 = ∫ 𝑝𝑑𝑉 𝑉 18.25 𝑄 𝑇𝐻 − 𝑇𝐶 Volume Change 𝑖 Power (Conducted) 𝑃 = = 𝑘𝐴 18.32 𝑊 = 𝑝∆𝑣 𝑐𝑜𝑛𝑑 𝑡 𝐿 Rate objects absorbs 𝑃 = 𝜎𝜖𝐴𝑇4 18.39 energy 𝑎𝑏𝑠 𝑒𝑛𝑣 4 Power from radiation 𝑃𝑟𝑎𝑑 = 𝜎𝜖𝐴𝑇 18.38 −8 2 4 𝜎 = 5.6704 × 10 𝑊/𝑚 ∙ 𝐾 Revised 7/20/17