Applied Mathematics Formula Booklet

Applied Mathematics Formula Booklet 1. Linear vs Rotational Motion Feature Linear Motion Rotational Motion Displacement 𝑠 𝜃 Velocity 𝑣 𝜔 Acceleration 𝑎 𝛼 𝑣 = 𝑢 + 𝑎𝑡 𝜔1 = 𝜔0 + 𝛼𝑡 1 2 𝑠 = 𝑢𝑡 + 𝑎𝑡 2 1 2 𝜃 = 𝜔0 𝑡 + 𝛼𝑡 2 𝑣 2 = 𝑢2 + 2𝑎𝑠 𝜔1 2 = 𝜔0 2 + 2𝛼𝜃 𝑢+𝑣 𝑠= 𝑡 2 𝜔0 + 𝜔1 𝜃= 𝑡 2 Inertia Mass 𝑚 Moment of Inertia 𝐼 Kinetic Energy 1 𝑚𝑣 2 2 1 2 𝐼𝜔 2 Force/Torque 𝐹 = 𝑚𝑎 𝑇 = 𝐼𝛼 Momentum 𝑚𝑣 𝐼𝜔 Work 𝐹𝑠 𝑇𝜃 Power 𝐹𝑣 𝑇𝜔 Constant Acceleration Equations Applied Mathematics Formula Booklet 2. Common Derivatives 𝑦 𝑑𝑦 𝑑𝑥 𝑥𝑛 𝑛𝑥 𝑛−1 sin 𝑎𝑥 𝑎 cos 𝑎𝑥 𝑐𝑜𝑠 𝑎𝑥 − 𝑎𝑠𝑖𝑛 𝑎𝑥 𝑒 𝑎𝑥 𝑎𝑒 𝑎𝑥 ln 𝑎𝑥 1 𝑥 3. Common Integrals 𝑦 𝑥𝑛 𝑐𝑜𝑠 𝑎𝑥 𝑠𝑖𝑛 𝑎𝑥 𝑒 𝑎𝑥 1 𝑥 𝑓′(𝑥) ∫ 𝑑𝑥 𝑓(𝑥) ∫ 𝑦 𝑑𝑥 𝑥 𝑛+1 + 𝑐 𝑛 ≠ −1 𝑛+1 1 𝑠𝑖𝑛 𝑎𝑥 + 𝑐 𝑎 1 − 𝑐𝑜𝑠 𝑎𝑥 + 𝑐 𝑎 1 𝑎𝑥 𝑒 +𝑐 𝑎 ln 𝑥 + 𝑐 ln|𝑓(𝑥)| + 𝑐 Applied Mathematics Formula Booklet 4. Laws of Logarithms ↔ log 𝑎 𝑦 = 𝑥 𝑎𝑥 = 𝑦 ln 𝑦 = 𝑥 𝑒𝑥 = 𝑦 log 𝐴 + log 𝐵 log(𝐴𝐵) log 𝐴 − log 𝐵 log(𝐴/𝐵) log 𝐴𝑛 𝑛 log 𝐴 log 𝐴 = log 𝐵 𝐴=𝐵 5. Momentum and Impulse 𝑡1 𝑚1 𝑣1 + ∫ 𝐹𝑁𝑒𝑡 𝑑𝑡 = 𝑚2 𝑣2 𝑡0 6. Rotational Mechanics Feature Formula Linear/Angular Displacement 𝑠 = 𝑟𝜃 Linear/Angular Velocity 𝑣 = 𝑟𝜔 Linear/Angular Acceleration 𝑎 = 𝑟𝛼 Centripetal Acceleration 𝑣2 𝑎= = 𝜔2 𝑟 𝑟 Torque 𝑇 = 𝐹𝑟 Applied Mathematics Formula Booklet 7. Elasticity Component Elastic Restoration Force 𝐹 = 𝑘𝑥 Spring and String 𝜆 𝐹= 𝑥 𝑙 8. Centre of Gravity Modelling Method Formula System of Particles 𝑥 𝑥 𝑥 𝑥̅ (𝑚1 + 𝑚2 + ⋯ 𝑚𝑛 ) ( ) = 𝑚1 (𝑦1 ) + 𝑚2 (𝑦2 ) + ⋯ 𝑚𝑛 (𝑦𝑛 ) 𝑦̅ 1 2 𝑛 Lamina 𝑥 𝑥 𝑥 𝑥̅ (𝐴1 + 𝐴2 + ⋯ 𝐴𝑛 ) ( ) = 𝐴1 (𝑦1 ) + 𝐴2 (𝑦2 ) + ⋯ 𝐴𝑛 (𝑦𝑛 ) 𝑦̅ 1 2 𝑛 Volume 𝑥1 𝑥2 𝑥𝑛 𝑥̅ (𝑉1 + 𝑉2 + ⋯ 𝑉𝑛 ) (𝑦̅) = 𝑉1 (𝑦1 ) + 𝑉2 (𝑦2 ) + ⋯ 𝑉𝑛 (𝑦𝑛 ) 𝑧1 𝑧2 𝑧𝑛 𝑧̅ 𝑏 Lamina 𝑥̅ = ∫𝑎 𝑥𝑦 𝑑𝑥 𝑦̅ = 𝑏 ∫𝑎 𝑦 𝑑𝑥 𝑏 Volume of Revolution 𝑥̅ = ∫𝑎 𝜋𝑥𝑦 2 𝑑𝑥 𝑏 ∫𝑎 𝜋𝑦 2 𝑑𝑥 𝑏1 ∫𝑎 2 𝑦 2 𝑑𝑥 𝑏 ∫𝑎 𝑦 𝑑𝑥 Applied Mathematics Formula Booklet 9. Simple Harmonic Motion Feature Formula Characteristic Equation 𝑥̈ + 𝜔𝑛2 𝑥 = 0 Undamped Natural Frequency 𝜔𝑛 = √𝑘/𝑚 Displacement (t) 𝑥 = 𝐴 𝑠𝑖𝑛 𝜔𝑛 𝑡 + 𝐵 𝑐𝑜𝑠 𝜔𝑛 𝑡 Velocity 𝑣 2 = 𝜔𝑛2 (𝑎2 − 𝑥 2 ) Maximum Velocity 𝑣𝑚𝑎𝑥 = 𝜔𝑛 𝑎 10. 2𝜋 𝜔𝑛 Time Period 𝜏= Frequency 1 𝑓= 𝜏 Principle of Energy and Work 𝑥1 1 𝑚(𝑣 2 − 𝑢2 ) = ∫ 𝐹𝑁𝑒𝑡 𝑑𝑥 2 𝑥0 Applied Mathematics Formula Booklet 11. Moment of Inertia Object Location of Axis Moment of Inertia Thin Hoop Through Centre 𝑀𝑅02 Thin Hoop Through Central Diameter 1 1 𝑀𝑅02 + 𝑀𝑤 2 2 12 Solid Cylinder Through Centre 1 𝑀𝑅02 2 Hollow Cylinder Through Centre 1 𝑀(𝑅12 + 𝑅22 ) 2 Uniform Sphere Through Centre 2 𝑀𝑅02 5 Long Uniform Rod Through Centre 1 𝑀𝑙 2 12 Long Through End Uniform Rod Rectangular Thin Plate Through Centre 1 2 𝑀𝑙 3 1 𝑀(𝑙 2 + 𝑤 2 ) 12 Applied Mathematics Formula Booklet 12. Centre of Gravity (CoG) Shape Area CoG Location 𝑏 Square/ 𝑏×ℎ Rectangle Triangles 𝑏×ℎ 2 𝑏/2 𝑎 𝑎/2 ℎ ℎ 𝑏/3 1 ℎ 3 ℎ/3 𝑏/2 𝑏 SemiCircle 𝜋𝑟 2 2 QuaterCircle 𝜋𝑟 2 4 2 ℎ 3 𝑏 r 4𝑟 3𝜋 4𝑟 3𝜋 4𝑟 3𝜋

Applied Mathematics Formula Booklet

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