DYNAMICS FORMULAS
CH. 12: PARTICLE KINEMATICS (TRANSLATION)
Kinematic Equations
𝑣
𝑡
𝑎 = 𝑑𝑣Τ𝑑𝑡
𝑣 = 𝑣𝑖 + 𝑎𝑡
‫𝑡𝑑)𝑡(𝑎 𝑓 𝑡׬ = 𝑣𝑑 𝑓 𝑣׬‬
𝑣 = 𝑑𝑠Τ𝑑𝑡
𝑖
𝑠𝑓
𝑖
𝑡𝑓
𝑖
𝑖
𝑣
= ‫𝑣 𝑓 𝑣׬‬
𝑖
𝑎𝑑𝑠 = 𝑣𝑑𝑣
𝑑𝑣
𝑣𝑓
Projectile Motion Analysis
𝑅 = 𝑥𝑓 − 𝑥𝑖 = 𝑣𝑥)𝑖 𝑡
𝑣𝑦)𝑓 = 𝑣𝑦)𝑖 − 𝑔𝑡
an = v2/r
𝑤 = න 𝐹𝑑𝑠
2
s = 𝑣𝑖 𝑡 + 2 𝑎𝑡 2
2
− 𝑣𝑖
𝑣 2 𝑦)𝑓 − 𝑣 2 𝑦)𝑖 = −2𝑔(𝑦𝑓 − 𝑦𝑖 )
𝑔 = 32.2
Normal And Tangential Coordinate System
𝑎𝑡 = 𝑑𝑣Τ𝑑𝑡
𝑠2
1
‫𝑡𝑑)𝑡(𝑣 𝑡׬ = 𝑠𝑑 𝑠׬‬
𝑠
‫𝑠𝑑)𝑠(𝑎 𝑓 𝑠׬‬
𝑖
CH. 14: ENERGY METHODS (Particle)
1
1
1
1
2
2
2
𝑚𝑣
+
𝑚𝑔ℎ
+
𝑘𝑠
+
𝑊
=
𝑚𝑣
+
𝑚𝑔ℎ
+
𝑘𝑠𝑓 2
𝑖
𝑖
𝑖
𝑛.𝑐.𝑓.
𝑓
𝑓
2
2
2
2
𝑃𝑜 =
= 2𝑎𝑠
1
𝑓𝑡
𝑦𝑓 − 𝑦𝑖 = 𝑣𝑦)𝑖 𝑡 − 2 𝑔𝑡 2
a = [(at)2 + (an)2]0.5
𝜌=
𝑑𝑦 2
𝑑𝑥
𝑑2 𝑦
𝑑𝑥2
𝜀=
1
𝑚𝑣𝑖 2
2
1 hp = 550 (ft · lb)/s = 746 W
1
1
+ 2 𝐼𝐺 𝜔𝑖 2 + 2 𝑘𝑠𝑖 2 + 𝑚𝑔ℎ𝑖 + 𝑊𝑛.𝑐.𝑓. =
1
𝑚𝑣𝑓 2
2
1
1
+ 2 𝐼𝐺 𝜔𝑓 2 + 2 𝑘𝑠𝑓 2 + 𝑚𝑔ℎ𝑓
𝜃
3ൗ
2
w = ‫ 𝜃׬‬2 𝑀𝑑𝜃
1
Radial And Transverse Coordinate System
ሶ 𝑛
𝑣 = 𝑟𝑒
ሶ 𝑟 + 𝑟𝜃𝑒
𝑎 = 𝑟ሷ − 𝑟𝜃ሶ 2 𝑒𝑟 + 𝑟𝜃ሷ + 2𝑟ሶ 𝜃ሶ 𝑒𝑛
Relative Motion Analysis
𝑣𝐵/𝐴 = 𝑣𝐵 − 𝑣𝐴
𝑎𝐵/𝐴 = 𝑎𝐵 − 𝑎𝐴
CH. 16: RIGID BODY KINEMATICS
𝑣2 = 𝑣1 + 𝜔12 × 𝑟2/1
= 𝐹. 𝑣
𝑠1
𝑃𝑜
𝑃𝑖
CH. 18: ENERGY METHODS (Rigid Body)
ൗ𝑠2 = 9.8 𝑚Τ𝑠2
1+
𝑑𝑊
𝑑𝑡
CH. 15: LINEAR IMPULSE AND MOMENTUM (Particle)
𝑡
Linear Impulse and Momentum: σ 𝑚𝑣𝑖 + σ ‫ = 𝑡𝑑𝐹 𝑓 𝑡׬‬σ 𝑚𝑣𝑓
2
𝑎2 = 𝑎1 + 𝛼12 × 𝑟2/1 − 𝜔12 𝑟2/1
𝑖
𝑣𝐿2 −𝑣𝐿1 𝑓
𝑣𝐿1 −𝑣𝐿2 𝑖
LOI:
𝑒=
POI:
𝑚1 𝑣𝑃1 𝑖 = 𝑚1 𝑣𝑃1
𝑚1 𝑣𝐿1 + 𝑚2 𝑣𝐿2 𝑖 = 𝑚1 𝑣𝐿1 + 𝑚2 𝑣𝐿2
𝑓
Angular Impulse and Momentum:
𝑚2 𝑣𝑃2 𝑖 =
𝒕𝟐
σ ‫ = 𝒕𝒅 𝑴 𝒕׬‬σ 𝒓𝒇
𝟏
𝑚2 𝑣𝑃2
𝑓
× 𝒎𝒗𝒇 − σ 𝒓𝒊 × 𝒎𝒗𝒊
CH. 19: ANGULAR IMPULSE AND MOMENTUM (Rigid Body)
𝑡
Conservation of Momentum: 𝐼𝐺 𝜔𝑖 + σ ‫𝑓𝜔 𝐺𝐼 = 𝑡𝑑𝑀 𝑓 𝑡׬‬
𝐼𝐺 𝜔 + 𝑚𝑣 𝑟 𝑖 =
𝑖
[𝑚1 𝑣1 ]𝑟1 + 𝐼𝐺)1 𝜔1 + [𝑚2 𝑣2 ]𝑟2 + 𝐼𝐺)2 𝜔2
CH. 13: PARTICLE KINETICS
σ 𝐹𝑥 = 𝑚𝑎𝑥 σ 𝐹𝑦 = 𝑚𝑎𝑦
σ 𝐹𝑡 = 𝑚𝑎𝑡
𝑣2
σ 𝐹𝑛 = 𝑚𝑎𝑛 = 𝑚
𝜌
2
ሶ
ሶ
σ 𝐹𝑟 = 𝑚𝑎𝑟 = 𝑚(𝑟ሷ − 𝑟𝜃 )
σ 𝐹𝜃 = 𝑚𝑎𝜃 = 𝑚(𝑟𝜃ሷ + 2𝑟ሶ 𝜃)
𝑟
𝑑𝑦
𝑡𝑎𝑛𝜑 = 𝑑𝑟
𝜃=
ൗ𝑑𝜃
𝑑𝑥
CH. 17: RIGID BODY KINETICS
Parallel axis theorem: 𝐼1 = 𝐼2 + 𝑚 𝑟12 2
𝐼=𝑚 𝑘 2
1
2
𝐼𝐺−𝑟𝑜𝑑 = 𝑚𝑙 2
𝐼𝐺−𝑠𝑝ℎ𝑒𝑟𝑒 = 𝑚𝑟 2
12
5
σ 𝑴𝒐 = 𝑰𝑮 𝜶 + 𝒎𝒂𝑮)𝟏 𝒓1 + 𝒎𝒂𝑮)𝟐 𝒓𝟐
σ 𝐹1 = 𝑚𝑎1
σ 𝐹2 = 𝑚𝑎2
1
𝐼𝐺−𝑑𝑖𝑠𝑘 = 𝑚𝑟 2
2
𝑖
=
𝑓
𝐼𝐺 𝜔 + 𝑚𝑣 𝑟
[𝑚1 𝑣1 ]𝑟1 + 𝐼𝐺)1 𝜔1 + [𝑚2 𝑣2 ]𝑟2 + 𝐼𝐺)2 𝜔2
𝑓
𝑓