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Formula Sheet Cascarano Scalar Product Math Stuff ⃗⃗ = |𝐴⃗||𝐵 ⃗⃗| cos 𝜃 𝐴⃗ ∙ 𝐵 Vector Product ⃗⃗ = |𝐴⃗||𝐵 ⃗⃗| sin 𝜃 𝐴⃗ × 𝐵 Quadratic formula (ax2+bx+c=0) 𝑥= Kinematics 𝑣𝑥 = 𝑣𝑜𝑥 + 𝑎𝑥 𝑡 1 𝑥 = 𝑥𝑜 + 𝑣𝑜𝑥 𝑡 + 2𝑎𝑥 𝑡 2 2 𝑣𝑥2 = 𝑣𝑜𝑥 + 2𝑎𝑥 (𝑥 − 𝑥𝑜 ) 𝑣𝑜𝑥 + 𝑣𝑥 𝑥 − 𝑥𝑜 = ( )𝑡 2 Velocity Addition 𝑣⃗𝐴𝐵 = 𝑣⃗𝐴𝐶 + 𝑣⃗𝐶𝐵 −𝑏 ± √𝑏 2 − 4𝑎𝑐 2𝑎 Acceleration of gravity Constants g = 9.80 m/s2 G = 6.67x10-11 Nm2/kg2 Universal gravitation 𝑀𝐸 = 5.97𝑥1024 𝑘𝑔 𝑅𝐸 = 6.38𝑥106 𝑚 Mass of Earth Radius of Earth Trig Identities Law of cosines: Newton’s Laws / Friction / Circular motion ∆𝑝⃗ 2 Law ∑ 𝐹⃗ = 𝑚𝑎⃗ = 𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝛼 Gravitation Law of sines: Static friction Kinetic friction nd ∆𝑡 𝑎 𝑏 𝑐 = = sin 𝛼 sin 𝛽 sin 𝛾 a b 𝐺𝑚1 𝑚2 𝐹= 𝑟2 𝑓𝑠 ≤ 𝜇𝑠 𝑁 𝑓𝑘 = 𝜇𝑘 𝑁 Centripetal acceleration 𝑎𝑐𝑝 = 1 Work done by a spring 𝑊𝑠𝑝 = 2𝑘𝑥 2 Conservative forces 𝑊𝑐 = −∆𝑈 Non-conservative forces Power 𝑊𝑛𝑐 = ∆𝐸 𝑊𝑜𝑟𝑘 𝑃= 𝑡𝑖𝑚𝑒 Impulse and Momentum 𝑝⃗ = 𝑚𝑣⃗ Energy 1 𝐾 = 2𝑚𝑣 2 Kinetic energy Gravitational potential energy Spring potential energy Impulse momentum 𝑈𝑔 = 𝑚𝑔𝑦 = − 𝐺𝑚𝑀𝐸 𝑟 1 𝑈𝑠𝑝 = 2𝑘𝑥 2 𝐹 = 𝑘𝑥 𝑊𝑡𝑜𝑡𝑎𝑙 = ∆𝐾 Hooke’s Law (spring force) Work-Energy Theorem Center of Mass Center of mass (point objects) 𝑥𝐶𝑀 = 𝐼⃗ = 𝐹⃗𝑎𝑣𝑒 ∆𝑡 𝐼⃗ = ∆𝑝⃗ Impulse 𝑣2 𝑟 c Work & Power 𝑊 = 𝐹𝑑𝑐𝑜𝑠𝜃 Work by constant force Linear momentum Physics 2A ∑𝑛 𝑚𝑛 𝑥𝑛 ∑𝑛 𝑚 𝑛 𝑝⃗𝑡𝑜𝑡𝑎𝑙 = 𝑀𝑡𝑜𝑡𝑎𝑙 𝑣⃗𝐶𝑀 Total momentum ∑ 𝐹⃗𝑒𝑥𝑡 = 𝑀𝑡𝑜𝑡𝑎𝑙 𝑎⃗𝐶𝑀 Rotational Kinematics 𝜔 = 𝜔𝑜 + 𝛼𝑡 1 Linear / Angular 𝑠 = 𝑟𝜃 (𝜃 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠) 𝜃 = 𝜃𝑜 + 𝜔𝑜 𝑡 + 2𝛼𝑡 2 𝑣𝑡 = 𝑟𝜔 𝜔2 = 𝜔𝑜2 + 2𝛼(𝜃 − 𝜃𝑜 ) 𝜔𝑜 + 𝜔 𝜃 − 𝜃𝑜 = ( )𝑡 2 𝑎𝑡 = 𝑟𝛼 Torque 𝑎𝑐𝑝 = 𝑣2 = 𝑟𝜔2 𝑟 Torque and Angular Momentum 𝜏⃗ = 𝑟⃗ × 𝐹⃗ = 𝑟𝐹𝑠𝑖𝑛∅ = 𝑟⊥ 𝐹 Angular momentum 𝐿⃗⃗ = 𝑟⃗ × 𝑝⃗ = 𝑚𝑣𝑟⊥ = 𝐼𝜔 ⃗⃗ Rotational Inertia and Energy 𝐼 = ∑𝑖 𝑚𝑖 𝑟𝑖 2 Rotational inertia (point objects) Parallel axis theorem 𝐼𝑃 = 𝐼𝐶𝑀 + 𝑀𝑑 2 1 𝐾 = 2𝐼𝜔2 Rotational kinetic energy Rotational Motion Newton’s 2nd law ∑ 𝜏⃗ = 𝐼𝛼⃗ = ⃗⃗ 𝑑𝐿 𝑑𝑡 Work 𝑊 = 𝜏∆𝜃 = ∆𝐾𝑟𝑜𝑡 Power 𝑃 = 𝜏𝜔 Cascarano Displacement Formula Sheet Simple Harmonic Motion 𝑥(𝑡) = 𝐴𝑐𝑜𝑠(𝜔𝑡 + ∅) Velocity Period Angular Frequency Mass on a spring Simple pendulum 𝑣(𝑡) = −𝜔𝐴𝑠𝑖𝑛(𝜔𝑡 + ∅) 2𝜋 𝑇= 𝜔 Frequency (j) Thin hoop rotating on axis through any diameter of the hoop: 1 𝐼 = 𝑀𝑅 2 2 Physics 2A 𝑘 𝜔=√ 𝑚 𝑔 𝜔=√ 𝑙 1 𝜔 𝑓= = 𝑇 2𝜋
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