Summary Mechanical Engineering 1st year FSK116 formula sheet

FSK CHEAT SHEET
10−24 yocto Ay 101 deka Ada
10−21 zepto Az 102 hector Ah
10−18 atto Aa 103 kilo Ak
10−15 femto Af 106 mega AM
10−12 pico Ap 109 giga AG
10−9 nano An 1012 tera AT
10−6 micro Aµ 1015 peta AP
10−3 milli Am 1018 exa AE
10−2 centi Ac 1021 zetta AZ
−1 24
10 deci Ad 10 yotta AY

Motion in 1D: Laws of motion: 𝐹𝑠 = −𝑘𝑥
∆𝑥 𝑥𝑓 − 𝑥𝑖 𝐹𝑛𝑒𝑡 = 𝑚𝑎 1
𝑣𝑥,𝑎𝑣𝑔 = = 𝐾 = 𝑚𝑣 2
∆𝑡 𝑡𝑓 − 𝑡𝑖 𝑚𝑀 2
⃗⃗⃗⃗
𝐹𝑔 = 𝐺 2 = 𝑚𝑎⃗
𝑑 𝑟 𝑊𝑒𝑥𝑡 = 𝛥𝐾
𝑣𝑎𝑣𝑔 = 𝑀 𝑈 = 𝑚𝑔ℎ
∆𝑡 𝑔=𝐺 2
∆𝑥 𝑑𝑥 𝑟 𝑊𝑒𝑥𝑡 = 𝛥𝑈
𝑣𝑥,𝑖𝑛𝑠𝑡 = lim = 𝐺 = 6.674 × 10−11 1
∆𝑡→0 ∆𝑡 𝑑𝑡
⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗ 𝑈𝑠 = 𝑘𝑥 2
𝛥𝑣𝑥 𝐹12 = −𝐹 21 2
𝑎𝑥,𝑖𝑛𝑠𝑡 = lim 𝐸𝑚𝑒𝑐ℎ = 𝐾 + 𝑈
𝛥𝑡→0 𝛥𝑡 𝐹𝑠 = 𝜇𝑠 𝐹𝑁
𝛥𝑣𝑥 𝐹𝑘 = 𝜇𝑘 𝐹𝑁 Conservation of energy: Non-
𝑎𝑥,𝑎𝑣𝑔 = isolated
𝛥𝑡 Circular motion: 𝛥𝐸𝑠𝑦𝑠𝑡𝑒𝑚 = 𝛥𝐾 + 𝛥𝑈 + 𝛥𝐸𝑖𝑛𝑡
2 2 system
𝑣𝑥,𝑓 = 𝑣𝑥,𝑖 + 2𝑎𝛥𝑥 𝑣2 𝐸𝑚𝑒𝑐ℎ = 𝛥𝐾 + 𝛥𝑈𝑔 = 0
Vectors: 𝑎𝑐 = Isolated
𝑟 𝐸𝑚𝑒𝑔,𝑓 = 𝐸𝑚𝑒𝑔,𝑖
𝑥 = 𝑟𝑐𝑜𝑠(𝜃) 𝑣2 system
−𝑓𝑘 𝑑 = 𝛥𝐾
𝑦 = 𝑟𝑠𝑖𝑛(𝜃) 𝐹𝑛𝑒𝑡 = 𝑚
𝑟 𝑊
𝑦 𝑇𝑐𝑜𝑠(𝜃) = 𝑚𝑔 𝑃𝑎𝑣𝑔 =
𝜃 = tan−1( ) 𝛥𝑡
𝑥 𝑚𝑣 2 𝑊 Power
𝑟 = √𝑥 2 + 𝑦 2 𝑇𝑠𝑖𝑛(𝜃) = 𝑃𝑖𝑛𝑠𝑡 = lim
𝑟 𝛥𝑡→0 𝛥𝑡
𝑎⃗ • 𝑏⃗⃗ = 𝑎𝑏𝑐𝑜𝑠(∅) 𝑣2 Conical 𝑃 = 𝐹⃗ 𝑣⃗
tan(𝜃) = pendulum
Motion in 2D: 𝑟𝑔 Linear momentum and collisions:
𝛥𝑟⃗ 𝛥𝑥 𝛥𝑦 𝛥𝑧 𝑣 = √𝑟𝑔𝑡𝑎𝑛(𝜃) 𝑝 = 𝑚𝑣
𝑣𝑎𝑣𝑔 =
⃗⃗⃗⃗⃗⃗⃗⃗⃗ = 𝑖̂ + 𝑗̂ + 𝑘̂ 𝑝𝑡𝑜𝑡𝑎𝑙,𝑓 = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑝𝑡𝑜𝑡𝑎𝑙,𝑖
𝛥𝑡 𝛥𝑡 𝛥𝑡 𝛥𝑡 𝑣 = √𝐿𝑔𝑠𝑖𝑛(𝜃) tan(𝜃)
𝛥𝑣⃗ 𝐼⃗ = 𝛥𝑝 ⃗⃗⃗⃗⃗⃗ = 𝐹𝑛𝑒𝑡 𝛥𝑡
𝑎𝑎𝑣𝑔 = 𝑇𝑚𝑎𝑥 𝑟
𝛥𝑡 o Collisions in 1D
𝑑𝑣𝑥 𝑑𝑣𝑦 𝑑𝑣𝑧 𝑣𝑚𝑎𝑥 = √
𝑎𝑖𝑛𝑠𝑡 = 𝑖̂ + 𝑗̂ + 𝑘̂ 𝑚 Elastic coll: ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑝𝑡𝑜𝑡𝑎𝑙,𝑓 = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑝𝑡𝑜𝑡𝑎𝑙,𝑖
𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑚𝑣 2 𝐾𝑖 = 𝐾𝑓
𝑣𝑓 = ⃗𝑣⃗⃗𝑖 + 𝑎⃗𝑡
⃗⃗⃗⃗⃗ 𝐹𝑠,𝑚𝑎𝑥 =
1 ⃗𝑡 2 𝑟 Inelastic coll: ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑝𝑡𝑜𝑡𝑎𝑙,𝑓 = ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑝𝑡𝑜𝑡𝑎𝑙,𝑖
𝛥𝑥 = ⃗𝑣⃗⃗𝑡 𝑖 + ⁄2 𝑎 𝑣𝑚𝑎𝑥 = √𝜇𝑠 𝑔𝑟 𝐾𝑖 ≠ 𝐾𝑓
𝑣𝑖 sin 𝜃𝑖 𝑣2
𝑡max ℎ = (𝑚1 − 𝑚2 ) (2𝑚2 )
𝑔 𝜃 = tan−1 𝑣1,𝑓 = [
⃗⃗⃗⃗⃗⃗⃗ ] ⃗⃗⃗⃗⃗⃗
𝑣1,𝑖 + [ ] ⃗⃗⃗⃗⃗⃗⃗
𝑣
𝑟𝑔 (𝑚1 + 𝑚2 ) (𝑚1 + 𝑚2 ) 2,𝑖
𝑣𝑖 sin 𝜃𝑖
𝑡𝑡𝑜𝑡𝑎𝑙 = 𝑅⃗⃗ = −𝑏𝑣⃗ (2𝑚1 ) (𝑚2 − 𝑚1 )
𝑔 Projectile 𝑣2,𝑓 = [
⃗⃗⃗⃗⃗⃗⃗⃗ ] ⃗⃗⃗⃗⃗⃗
𝑣 +[ ] ⃗⃗⃗⃗⃗⃗⃗
𝑣
𝑅 = 1⁄2 𝐷𝑟𝐴𝑣 2 (𝑚1 + 𝑚2 ) 1,𝑖 (𝑚1 + 𝑚2 ) 2,𝑖
𝑣𝑖2 sin2 𝜃𝑖 motion
ℎ= 𝐷𝜌𝐴 2 o Collisions in 2D
2𝑔 𝑎 = 𝑔−( )𝑣 Resistive
2𝑚 𝑚1 ⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑣1,𝑖𝑥 + 𝑚2 ⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑣2,𝑖𝑥 = 𝑚1 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑣1,𝑓𝑥 + 𝑚2 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑣2,𝑓𝑥
𝑣𝑖2 sin 2𝜃𝑖 force
𝑅= 𝑚1 ⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑣1,𝑖𝑦 + 𝑚2 ⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑣2,𝑖𝑦 = 𝑚1 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑣1,𝑓𝑦 + 𝑚2 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑣2,𝑓𝑦
𝑔 2𝑚𝑔
𝑉𝑇 = √ o Centre of mass
2𝜋𝑟 𝐷𝜌𝐴 𝑚1 𝑥1 + 𝑚2 𝑥2
𝑇=
𝑣 Energy of a system: 𝑥𝐶𝑀 =
2𝜋 𝑚1 + 𝑚2
𝜔= Uniform 𝑊 = 𝐹𝛥𝑥𝑐𝑜𝑠(𝜃) 1
𝑇 circular 𝑥𝑓 𝑣𝐶𝑀 = ∑ 𝑚𝑖 ⃗𝑣⃗⃗𝑖
⃗⃗⃗⃗⃗⃗⃗⃗
𝑣 = 𝑟𝜔 𝑀
motion 𝑊 = ∫ 𝐹𝑥 𝑑𝑥 𝑖
𝑣 2 (𝑟𝜔)2 𝑥𝑖 1
𝑎𝑐 = = = 𝑟𝜔 2 𝑥𝑓
1 1 𝑎𝐶𝑀 = ∑ 𝑚𝑖 ⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗ 𝑎𝑖
𝑟 𝑟 𝑊𝑠 = ∫ −𝑘𝑥 𝑑𝑥 = 𝑘𝑥𝑖2 − 𝑘𝑥𝑓2 𝑀
𝑖
𝑥𝑖 2 2
⃗⃗⃗⃗⃗⃗⃗⃗
𝑝 𝑒𝑥𝑡 = 𝑀𝑣 ⃗⃗⃗⃗⃗⃗⃗⃗
𝐶𝑀
⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝐹 1 2
𝑎𝑝𝑝 = − ⁄2 𝑘𝑥𝑚𝑎𝑥 ⃗⃗⃗⃗⃗⃗⃗⃗
𝑥𝑓
∑𝐹 𝑒𝑥𝑡 = 𝑀𝑎 ⃗⃗⃗⃗⃗⃗⃗⃗
𝐶𝑀
1 1
𝑊𝑎𝑝𝑝 = ∫ 𝑘𝑥 𝑑𝑥 = 𝑘𝑥𝑓2 − 𝑘𝑥𝑖2 𝐼⃗ = 𝛥𝑝
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝑥𝑖 2 2 𝑡𝑜𝑡𝑎𝑙