PHY 231 Exam 1 Help Sheet | Study notes Physics

CONVERSIONS

km/h m/s:

→ *(1000𝑚/3600𝑠)

Multiplication/Division;

smallest # of sigfigs

Addition/Subtraction;

smallest # of decimals

CALCULATION

RULES

10m* 10n= 10m+n

10m/ 10n= 10m-n

(10m)n= 10m*n

Frictions

𝑓𝑠,𝑚𝑎𝑥= µ𝑠𝑁

𝑓𝑘= µ𝑘𝑚𝑔

FREELY FALLING

OBJECTS:

𝑎=−9.81𝑚/𝑠2=−𝑔

Instantaneous acc. at max

height of thrown object is

↑

-9.81 m/s^2

PIECEWISE

ACCELERATION:

*Area under the graph of v(t)

is the displacement ∆x

*Area under the graph of a(t)

is the velocity change ∆v

AVG/INST. SPEED/VELOCITY

𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒/∆𝑡

𝑎𝑣𝑔 𝑣𝑒𝑙𝑜𝑐 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡/∆𝑡 →𝑣 =(𝑥𝑏−𝑥𝑎)/(𝑡𝑏

𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 →𝑑𝑟𝑎𝑤 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 𝑎𝑛𝑑 𝑡𝑎𝑘𝑒 𝑠

𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛: 𝑎=(𝑣𝑓−𝑣𝑖)/(𝑡𝑓−𝑡𝑖)

𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛: 𝑎=𝑙𝑖𝑚(∆𝑣/∆𝑡)

PROJECTILE MOTION

X and y directions are independent of each other

ax= 0 (x direction: uniform motion)

ay= -g (y direction is free fall)

EQUATIONS OF PROJECTILE MOTION

x) 𝑎𝑥=0 𝑉𝑥𝑓=𝑉𝑥𝑖𝑥𝑓=𝑥𝑖+𝑉𝑥𝑖∆𝑡

y) 𝑎𝑦=−𝑔 𝑉𝑦𝑓=𝑉𝑦𝑖−𝑔∆𝑡

𝑦𝑓=𝑦𝑖+𝑉𝑦𝑖∆𝑡−(1/2)𝑔𝑡2

CONSTANT ACCELERATION (1D)

𝐶𝐴2: 𝑣𝑓=𝑣𝑖+𝑎𝑡 𝐶𝐴3: 𝑥𝑓=𝑥𝑖+𝑣𝑖𝑡+1/2 𝑎𝑡2

𝐶𝐴4:𝑣𝑓2−𝑣𝑖2=2𝑎∆𝑥=2𝑎(𝑥𝑓−𝑥𝑖)

𝐶𝐴1: 𝑎𝑓=𝑎𝑖

VECTORS:“Tip to tail”

𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛: 𝐴−𝐵 ⇒𝐴+(−𝐵)

𝑣𝑒𝑐𝑡𝑜𝑟 𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛: 𝑉=|𝑉|= 𝑉𝑥2+𝑉𝑦2

𝑉𝑥=𝑉𝑐𝑜𝑠(θ)𝑉𝑦=𝑉𝑠𝑖𝑛(θ)𝑡𝑎𝑛(θ)=𝑉𝑦/𝑉𝑥

RELATIVE MOTION

Acceleration down: 𝑎<0, 𝑇<𝑚𝑔, 𝑚. 𝑠𝑐𝑎𝑙𝑒<𝑚

Equilibrium:𝑎=0, 𝑇=𝑚𝑔, 𝑚. 𝑠𝑐𝑎𝑙𝑒=𝑚

Acceleration up: 𝑎>0, 𝑇>𝑚𝑔, 𝑚. 𝑠𝑐𝑎𝑙𝑒>𝑚

Sliding box on an inclined frictionless table

x) x t x

𝐹𝑛𝑒𝑡, 𝑥=𝑚𝑎 𝑚𝑔𝑠𝑖𝑛(θ)=𝑚𝑎

y) y

𝐹𝑛𝑒𝑡, 𝑦=𝑚𝑎 𝑛+(−𝑚𝑔𝑐𝑜𝑠(θ))=𝑚×0→𝑎𝑦=

UNIFORM CIRC MOTION - CONSTANT SPEED

𝑎𝑐=𝑣2/𝑟 =(𝑟ω)2/𝑟=ω2𝑟=𝑐𝑒𝑛𝑡𝑟𝑖𝑝. 𝑎𝑐𝑐

ac: points to center t360°=2π 𝑟𝑎𝑑=1 𝑓𝑢𝑙𝑙 𝑡𝑢𝑟𝑛

∆𝑠=𝑟∆θ=𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑓=1/𝑇=𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦

𝑇=2π𝑟/𝑣=𝑝𝑒𝑟𝑖𝑜𝑑=𝑡𝑖𝑚𝑒 𝑓𝑜𝑟 𝑜𝑛𝑒 𝑡𝑢𝑟𝑛

𝑣=𝑐/𝑇=2π𝑅/𝑇=∆𝑠/∆𝑡=𝑟(∆θ/∆𝑡)=𝑟𝑤

ω=2π𝑓=2π/𝑇=∆θ/∆𝑡 =𝑎𝑛𝑔 𝑣𝑒𝑙𝑐/𝑓𝑟𝑒𝑞

ω: 𝑡ℎ𝑒 𝑠𝑝𝑒𝑒𝑑 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑐ℎ𝑎𝑛𝑔𝑖𝑛𝑔 𝑖𝑛 𝑟𝑎𝑑/

MULTIPLE OBJECTS:

𝑇=((2𝑚1𝑚2)/(𝑚1+𝑚2))𝑔 [𝑁]

𝑎=((𝑚2−𝑚1)/(𝑚1−𝑚2))𝑔

;𝑚1) 𝑇−𝑚1𝑔=𝑚1𝑎 𝑇=𝑚1𝑎+𝑚1𝑔

;𝑚2) 𝑚2𝑔−𝑇=𝑚2𝑎 𝑚2𝑔−(𝑚1𝑎+𝑚1𝑔)=𝑚2𝑎

ANGULAR FREQUENCY - ANGULAR SPEED

Earth orbits sun w/ period of 365dys: T = 3.15*107sec

Frequency: 3.17*10-8 Hz

𝐹𝑔𝑠𝑖𝑛(θ)𝑚𝑎𝑥= µ𝑠*𝑛

𝑚𝑔𝑠𝑖𝑛(θ)𝑚𝑎𝑥= µ𝑠*𝑚𝑔𝑐𝑜𝑠(θ)𝑚𝑎𝑥

𝑡𝑎𝑛(θ)𝑚𝑎𝑥= µ𝑠

θ𝑚𝑎𝑥=𝑎𝑡𝑎𝑛(µ𝑠)

θ𝑚𝑎𝑥=𝑎𝑡𝑎𝑛(0.50)=27°

Elevator with scale

t𝑚𝑎=𝑇+𝐹𝑔 𝑇=𝑚(𝑎+𝑔)

𝑚.𝑠𝑐𝑎𝑙𝑒=𝑇/𝑔=𝑚(1+(𝑎/𝑔))

Partial preview of the text

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CONVERSIONS km/h →m/s: * (1000𝑚/3600𝑠) Multiplication/Division; smallest # of sigfigs Addition/Subtraction; smallest # of decimals

CALCULATION

RULES

10 m^ * 10n^ = 10m+n 10 m^ / 10n^ = 10m-n (10m)n^ = 10m*n Frictions 𝑓𝑠, 𝑚𝑎𝑥 = μ𝑠𝑁 𝑓𝑘 = μ𝑘𝑚𝑔

FREELY FALLING

OBJECTS :

2 =− 𝑔 Instantaneous acc. at max height of ↑thrown object is -9.81 m/s^ PIECEWISE ACCELERATION :

Area under the graph of v(t) is the displacement ∆x
Area under the graph of a(t) is the velocity change ∆v AVG/INST. SPEED/VELOCITY 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒/∆𝑡 𝑎𝑣𝑔 𝑣𝑒𝑙𝑜𝑐 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡/∆𝑡 → 𝑣 = (𝑥𝑏 − 𝑥𝑎)/(𝑡𝑏 𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 → 𝑑𝑟𝑎𝑤 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒 𝑎𝑛𝑑 𝑡𝑎𝑘𝑒 𝑠 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛: 𝑎 = (𝑣𝑓 − 𝑣𝑖)/(𝑡𝑓 − 𝑡𝑖) 𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛: 𝑎 = 𝑙𝑖𝑚(∆𝑣/∆𝑡)

PROJECTILE MOTION

X and y directions are independent of each other ax = 0 (x direction: uniform motion) ay = -g (y direction is free fall) EQUATIONS OF PROJECTILE MOTION x) 𝑎𝑥 = 0 𝑉𝑥 𝑓 = 𝑉𝑥 𝑖 𝑥𝑓 = 𝑥𝑖 + 𝑉𝑥 𝑖 ∆𝑡 y) 𝑎𝑦 =− 𝑔 𝑉𝑦 𝑓 = 𝑉𝑦 𝑖 − 𝑔∆𝑡 𝑦𝑓 = 𝑦𝑖 + 𝑉𝑦 𝑖 ∆𝑡 − (1/2)𝑔𝑡 2

CONSTANT ACCELERATION (1D)

2 𝐶𝐴4: 𝑣𝑓 2 − 𝑣𝑖 2 = 2𝑎∆𝑥 = 2𝑎(𝑥𝑓 − 𝑥𝑖) 𝐶𝐴1: 𝑎𝑓 = 𝑎𝑖 VECTORS :“Tip to tail” 𝑣𝑒𝑐𝑡𝑜𝑟 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛: 𝐴 − 𝐵 ⇒ 𝐴 + (− 𝐵) 𝑣𝑒𝑐𝑡𝑜𝑟 𝑎𝑑𝑑𝑖𝑡𝑖𝑜𝑛: 𝑉 = |𝑉| = 𝑉𝑥 2

𝑉𝑦 2 𝑉𝑥 = 𝑉𝑐𝑜𝑠(θ) 𝑉𝑦 = 𝑉𝑠𝑖𝑛(θ) 𝑡𝑎𝑛(θ) = 𝑉𝑦/𝑉𝑥

RELATIVE MOTION

Acceleration down: 𝑎 < 0, 𝑇 < 𝑚𝑔, 𝑚. 𝑠𝑐𝑎𝑙𝑒 < 𝑚 Equilibrium: 𝑎 = 0, 𝑇 = 𝑚𝑔, 𝑚. 𝑠𝑐𝑎𝑙𝑒 = 𝑚 Acceleration up: 𝑎 > 0, 𝑇 > 𝑚𝑔, 𝑚. 𝑠𝑐𝑎𝑙𝑒 > 𝑚 Sliding box on an inclined frictionless table x) 𝐹𝑛𝑒𝑡, 𝑥 = 𝑚𝑎x t 𝑚𝑔𝑠𝑖𝑛(θ) = 𝑚𝑎x y) 𝐹𝑛𝑒𝑡, 𝑦 = 𝑚𝑎y 𝑛 + (− 𝑚𝑔𝑐𝑜𝑠(θ)) = 𝑚 × 0 → 𝑎𝑦 =

UNIFORM CIRC MOTION - CONSTANT SPEED

2 /𝑟 = (𝑟ω) 2 /𝑟 = ω 2 𝑟 = 𝑐𝑒𝑛𝑡𝑟𝑖𝑝. 𝑎𝑐𝑐 ac: points to center t 360° = 2π 𝑟𝑎𝑑 = 1 𝑓𝑢𝑙𝑙 𝑡𝑢𝑟𝑛 ∆𝑠 = 𝑟∆θ = 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ t𝑓 = 1/𝑇 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑇 = 2π𝑟/𝑣 = 𝑝𝑒𝑟𝑖𝑜𝑑 = 𝑡𝑖𝑚𝑒 𝑓𝑜𝑟 𝑜𝑛𝑒 𝑡𝑢𝑟𝑛 𝑣 = 𝑐/𝑇 = 2π𝑅/𝑇 = ∆𝑠/∆𝑡 = 𝑟(∆θ/∆𝑡) = 𝑟𝑤 ω = 2π𝑓 = 2π/𝑇 = ∆θ/∆𝑡 = 𝑎𝑛𝑔 𝑣𝑒𝑙𝑐/𝑓𝑟𝑒𝑞 ω: 𝑡ℎ𝑒 𝑠𝑝𝑒𝑒𝑑 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑖𝑠 𝑐ℎ𝑎𝑛𝑔𝑖𝑛𝑔 𝑖𝑛 𝑟𝑎𝑑/

MULTIPLE OBJECTS:

𝑇 = ((2𝑚1𝑚2)/(𝑚1 + 𝑚2))𝑔 [𝑁]

ANGULAR FREQUENCY - ANGULAR SPEED

Earth orbits sun w/ period of 365dys: T = 3.1510^7 sec Frequency: 3.1710-8^ Hz 𝐹𝑔𝑠𝑖𝑛(θ)𝑚𝑎𝑥 = μ𝑠 * 𝑛 𝑚𝑔𝑠𝑖𝑛(θ)𝑚𝑎𝑥 = μ𝑠 * 𝑚𝑔𝑐𝑜𝑠(θ)𝑚𝑎𝑥 𝑡𝑎𝑛(θ)𝑚𝑎𝑥 = μ𝑠 θ𝑚𝑎𝑥 = 𝑎𝑡𝑎𝑛(μ𝑠) θ𝑚𝑎𝑥 = 𝑎𝑡𝑎𝑛(0. 50) = 27° Elevator with scale 𝑚𝑎 = 𝑇 + 𝐹𝑔 t 𝑇 = 𝑚(𝑎 + 𝑔) 𝑚. 𝑠𝑐𝑎𝑙𝑒 = 𝑇/𝑔 = 𝑚(1 + (𝑎/𝑔))

NEWTON’S LAWS :

1. If F=0, a=0 and v=constant At rest: F=0, which means a=0 and v=constant Fnet=F1+F2+F3…+Fn 2. F=ma: 𝑎 = 𝐹/𝑚; 𝐹 = 𝑚𝑎 t𝑤 = 𝑚𝑔 3. F 12 = -F 12 Box at rest:𝑚𝑎 = 𝐹𝑔 + 𝑛 = 0 Moving box, no friction: 𝑚𝑎 = 𝐹𝑔 + 𝑛 + 𝐹𝑝 = 𝐹𝑝 𝑦) 𝐹𝑦 t (^) x 𝑛𝑒𝑡 = 𝑛 − 𝑚𝑔 = 0 𝑥) 𝐹𝑥 𝑛𝑒𝑡 =+ 𝐹𝑝 = 𝑚𝑎 x) direction of motion t y) perpendicular to x If F=ma, x) 𝑇 = 𝑚𝑎x t y) 𝑛 − 𝑚𝑔 = 0 = 𝑚𝑎y STATIC FRICTION If: 𝐹𝑝≤μ𝑠𝑁, 𝑎 = 0 and 𝑓𝑠 =− 𝐹𝑝(stat fric cancels app force) If: 𝐹𝑝 > μ𝑠𝑁, kinetic friction starts and object starts to move Kinetic friction y) 𝑛 − 𝑚𝑔 = 𝑚 * 0 → 𝑛 = 𝑚𝑔 x) -𝑓𝑘 = 𝑚𝑎𝑥 → − μ𝑘𝑁 = 𝑚𝑎𝑥 → − μ𝑘(𝑚𝑔) = 𝑚𝑎𝑥 → − μ𝑘𝑔 = 𝑎𝑥 Inclined surface-kinetic friction 𝑥) 𝑚𝑔𝑠𝑖𝑛(θ) − μ𝑘𝑁 = 𝑚𝑎𝑥 𝑦) 𝑛 − 𝑚𝑔𝑐𝑜𝑠(θ) = 0 → 𝑛 = 𝑚𝑔𝑐𝑜𝑠(θ)

PHY 231 Exam 1 Help Sheet, Study notes of Physics

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Partial preview of the text

Download PHY 231 Exam 1 Help Sheet and more Study notes Physics in PDF only on Docsity!

CALCULATION

RULES

FREELY FALLING

OBJECTS :

PROJECTILE MOTION

CONSTANT ACCELERATION (1D)

RELATIVE MOTION

UNIFORM CIRC MOTION - CONSTANT SPEED

MULTIPLE OBJECTS:

𝑇 = ((2𝑚1𝑚2)/(𝑚1 + 𝑚2))𝑔 [𝑁]

ANGULAR FREQUENCY - ANGULAR SPEED

NEWTON’S LAWS :