F2019 Formula sheet PHYS 211 221 PDF

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Physics 211/221 Formula sheet, Fall 2019

Page 1 of 2

FORMULA SHEET KINEMATIC EQUATIONS FOR CONSTANT ACCELERATION:

𝑥𝑓 = 𝑥𝑖 + 𝑣𝑖𝑥 ∆𝑡 +

1

2

𝑎𝑥 ∆𝑡 2

;

𝑣𝑓𝑥 = 𝑣𝑖𝑥 + 𝑎𝑥 ∆𝑡

2 2 𝑣𝑓𝑥 + 2𝑎𝑥 ∆𝑥 = 𝑣𝑖𝑥

KINEMATIC EQUATIONS FOR CONSTANT ANGULAR ACCELERATION: 1

𝜃𝑓 = 𝜃𝑖 + 𝜔𝑖 ∆𝑡 + 𝛼∆𝑡 2 2

∆𝑡 = 𝑡𝑓 − 𝑡𝑖

𝜔𝑓 = 𝜔𝑖 + 𝛼∆𝑡

𝜔𝑓2 = 𝜔𝑖2 + 2𝛼∆𝜃

∆𝑥 = 𝑥𝑓 − 𝑥𝑖

;

SOLUTIONS TO A QUADRATIC EQUATION: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0

 𝑥=

;

∆𝜃 = 𝜃𝑓 − 𝜃𝑖

INTEGRALS: 𝑡𝑓

−𝑏±√𝑏2−4𝑎𝑐

∫ 𝑡 𝑛 𝑑𝑡 =

2𝑎

Derivatives:

∆𝑡 = 𝑡𝑓 − 𝑡𝑖

;

𝑡𝑖 𝑡𝑓

𝑑 𝑛 (𝑡 ) = 𝑛𝑡 𝑛−1 𝑑𝑡 𝑑𝑓 𝑑𝑔 𝑑 (𝑓 + 𝑔) = + 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑 𝑑𝑓 (𝐶𝑓) = 𝐶 (𝐶 = constant) 𝑑𝑡 𝑑𝑡

𝑡

𝑡 𝑛+1 𝑓 𝑡𝑓 𝑛+1 𝑡𝑖 𝑛+1 ] = − 𝑛+1 𝑡 𝑛+1 𝑛+1 𝑖

(𝑛 ≠ −1)

𝑡𝑓

𝑡𝑓

∫ (𝑓 + 𝑔)𝑑𝑡 = ∫ 𝑓 𝑑𝑡 + ∫ 𝑔 𝑑𝑡 𝑡𝑖 𝑡𝑓

𝑡𝑖 𝑡𝑓

∫ (𝐶𝑓)𝑑𝑡 = 𝐶 ∫ 𝑓 𝑑𝑡 𝑡𝑖

𝑡𝑖

𝑡𝑖

(𝐶 = constant)

TRIGONOMETRY: Law of Cosines: 𝐶 2 = 𝐴2 + 𝐵 2 − 2𝐴𝐵 cos 𝑐 Law of Sines:

sin 𝑎 𝐴

=

sin 𝑏 𝐵

=

sin 𝑐 𝐶

Pythagorean Theorem: 𝐶 2 = 𝐴2 + 𝐵 2 ; sin 𝜃 =

opp

hyp

; cos 𝜃 =

adj

; tan 𝜃 =

hyp

opp adj

VECTOR ALGEBRA IN 2D For 𝑅󰇍 = 𝑅𝑥 𝑖 + 𝑅𝑦 𝑗 󰇍 ⋅𝐵 󰇍 = 𝐴𝐵 cos(𝜃) A

|𝑅| = √𝑅x2 + 𝑅𝑦2 , tan 𝜃 =

For 𝐴 = 𝐴𝑥 𝑖 + 𝐴𝑦 𝑗, 𝐵󰇍 = 𝐵𝑥 𝑖 + 𝐵𝑦 𝑗

𝑅𝑦

𝑅𝑥

𝐴 ⋅ 𝐵󰇍 = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦

𝐴 × 𝐵󰇍 = (𝐴𝑥 𝐵𝑦 − 𝐴𝑦 𝐵𝑥 )𝑘

|𝐴× 𝐵󰇍 | = 𝐴𝐵 sin(𝜃) GRAVITY 𝐹𝑔 =

Near the Earth’s surface: 𝑈𝑔 = 𝑚𝑔𝑦

𝐺𝑀𝑚 𝑟2

Outside a spherically symmetric mass distribution: 𝑈𝑔 = − ELASTICITY

𝐹𝑥 = −𝑘∆𝑥

𝑈𝑠𝑝 = FORCES

𝑓𝑘 = 𝜇𝑘 𝐹𝑛 𝐹𝑔 = 𝑚𝑔

1 𝑘(∆𝑥)2 2

𝑓𝑠 ≤ 𝜇𝑠 𝐹𝑛 𝐹𝑑𝑟𝑎𝑔 =

1 𝐶𝜌𝐴𝑣 2 2

𝐺𝑀𝑚 𝑟

Physics 211/221 Formula sheet, Fall 2019

Page 2 of 2 TORQUE

𝜏= 𝑟× 𝐹

𝜏 = 𝑟𝐹 sin(𝜃)

𝜏 = 𝑟⊥ 𝐹

𝜏 = 𝑟𝐹⊥

IMPULSE AND MOMENTUM (𝑣𝑓𝑥 )1 =

𝐽𝑥 = ∆𝑝𝑥 𝑡𝑓

(𝑣𝑓𝑥 )2 =

𝐽𝑥 = ∫ 𝐹𝑥 𝑑𝑡 𝑡𝑖

𝑚1 (𝑣𝑖𝑥 )1 + 𝑚2 [2(𝑣𝑖𝑥 )2 − (𝑣𝑖𝑥 )1 ] 𝑚1 + 𝑚2

𝑚2 (𝑣𝑖𝑥 )2 + 𝑚1 [2(𝑣𝑖𝑥 )1 − (𝑣𝑖𝑥 )2 ] 𝑚1 + 𝑚2

𝑝 = 𝑚𝑣 WORK AND ENERGY 𝑊 = Δ𝐾

𝑊 = ∫ 𝐹 ⋅ 𝑑𝑟

𝑊𝑛𝑐 = Δ𝐸 = Δ𝐾 + Δ𝑈

𝐹 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡: 𝑊 = 𝐹 ⋅ Δ𝑟

𝑊𝑐 = −Δ𝑈

𝐾=

1 𝑚𝑣 2 2

POWER 𝑃=

𝑑𝑊 𝑑𝑡

𝐹 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡: 𝑃 = 𝐹 ⋅ 𝑣 CIRCULAR MOTION

𝑎𝑡 = 𝛼𝑅

𝑣 = 𝜔𝑅

𝑣=

𝜔=

2𝜋𝑅 𝑇

𝑣2 = 𝜔2 𝑅 𝑅 1 𝑓= 𝑇

𝑎 = √𝑎𝑡2 + 𝑎𝑐2

𝑎𝑐 =

2𝜋 𝑇

𝜔=

LAWS OF MOTION 𝐹𝑛𝑒𝑡 =

𝑑𝑝 𝑑𝑡

𝑚 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡: 𝐹𝑛𝑒𝑡 = 𝑚𝑎

CONVERSION FACTORS 1 foot = 0.3048 m

CONSTANTS

1 inch = 0.0254 m

1 mile = 1.609 × 103 m 1 lb = 0.454 kg 𝑇[°𝐶] =

5

9

(𝑇[°𝐹] − 32o )

g = 9.81

m s2

G = 6.67 × 10−11 1 US gallon = 3.78 litre

1 imp. gallon = 4.55 litre 𝑇[°𝐹] =

9 𝑇[°𝐶] + 32o 5

N m2 kg 2

𝑅𝐸𝑎𝑟𝑡ℎ = 6.37 × 106 m

∆𝜃 ∆𝑡...