CHEAT SHEET PHYS1205: Integrated Physics

Final Displacement with Avg. Velocity

Vector Multiplication/Division by a Scalar – Only magnitude is multiplied or divided. Direction is reversed for negative scalars.

1 𝑥! = 𝑥! +   (𝑣!" + 𝑣!" )𝑡 2

University of Newcastle

Final Displacement with Velocity and Acceleration

1

1D Motion

Vector – A measurement with both magnitude and direction (e.g. Displacement) Scalar – A measurement with only magnitude (e.g. distance)

1 𝑥! = 𝑥! + 𝑣!" 𝑡 + 𝑎! 𝑡 ! 2

Vector Components •

Length 𝐴 =

Final Velocity without Time ! ! 𝑣!" = 𝑣!" + 2𝑎! 𝑥! − 𝑥!

•

∆𝑥 ∆𝑡

Objects in Freefall – Acceleration is –g (9.8m/s )

Instantaneous Velocity 𝑣!"#$

2

Vectors and 2D Motion

Vector Addition – Tip to Tail 𝑑𝑥 = 𝑑𝑡

𝜃 =   tan!!

∆𝑣 ∆𝑡

Instantaneous Acceleration 𝑎!"#$ =

𝑑𝑥 𝑑𝑡

Final Velocity

𝐴!

𝐴 = 𝐴 x𝚤 + 𝐴 y𝚥 Projectile Motion •

Vector Subtraction – From the negative to the positive, or add the negative (𝐴 − 𝐵 = 𝐴 + (−𝐵))

𝐴!

Unit Vectors:

Position 1 𝑟! = 𝑟! + 𝑣! 𝑡 + 𝑔𝑡 ! 2

Average Acceleration 𝑎!"# =

!

Direction

2

Average Velocity 𝑣!"# =

!

𝐴! + 𝐴!  

•

Initial Horizontal Velocity 𝑣!" = 𝑣! cos 𝜃

•

Initial Vertical Velocity 𝑣!" = 𝑣! sin 𝜃

𝑣!" = 𝑣!" + 𝑎! 𝑡

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CHEAT SHEET Uniform Circular Motion •

Equilibrium

•

Kinetic Energy

Kinetic Friction

𝐾𝐸 = 𝐹 = 𝜇! 𝑁

!

𝑎! + 𝑎!

!

•

𝛴𝑊 = 𝛥𝐾𝐸

𝐹 ≤ 𝜇! 𝑁 2𝜋𝑟 𝑣

𝑣! 𝐹 = 𝑚𝑎! = 𝑚 𝑟

𝑟!" = 𝑟!" + 𝑣!" 𝑡

Force and Motion st

Newton’s 1 Law - In the absence of external forces, when viewed from an inertial reference frame, an object at rest will remain at rest and an object in motion continues in motion with a constant velocity

𝛴𝐹 = 𝑚𝑎

4

𝐹!" =   −𝐹!"

Gravitational 𝑈 = 𝑚𝑔𝛥𝑦 Elastic

Work, Energy and Power

Scalar/Dot Product 𝐴 ∙ 𝐵 = 𝐴𝐵𝑐𝑜𝑠𝜃 Work •

Same Direction as Displacement 𝑊 = 𝐹∆𝑟

•

Different Direction to Displacement

𝑈=

•

Non-conservative Force - Work done dependent on the motion of the object (e.g. Friction) Conservation of Energy •

𝑊!"# =

!! !!

Mechanical Energy 𝐸!"#! = 𝐾𝐸 + 𝑈

•

Work by Varying Force

1 ! 𝑘𝑥 2

Conservative Force - Work done is independent of the path taken by an object (e.g. Gravity)

𝑊 = 𝐹∆𝑟𝑐𝑜𝑠𝜃

rd

Newton’s 3 Law - If two objects interact, the force that object one is exerting on object 2 is equal and opposite to that object two is exerting on object one

•

•

nd

Newton’s 2 Law - Net Force is the product of Mass and Acceleration

Potential Energy

Circular Motion Dynamics

Relative Velocity

1 𝑚𝑣 ! 2

Work-Kinetic Energy Theorem

Static Friction

Period 𝑇=

3

𝐹! = −𝑘𝑥

Friction

𝑣! 𝑟

Overall Acceleration |𝑎| =

•

𝛴𝐹 = 0

Centripetal Acceleration 𝑎! =

•

Hooke’s Law

Total Energy 𝐸!"! = 𝐾𝐸 + 𝑈 + 𝐸!"#

𝐹! 𝑑𝑥

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CHEAT SHEET •

Non-Conservative Force Absent

•

Conservation of KE (Elastic Collisions)

∆𝐸!"#! = 0 •

𝐾𝐸! = 𝐾𝐸!

Non-Conservative Force Present

•

Perfectly Inelastic

∆𝐸!"! = 0

𝑚! 𝑣!! + 𝑚! 𝑣!! = (𝑚! + 𝑚! )𝑣!

Power

• 𝑑𝑊 𝜑= 𝑑𝑡

5

Perfectly Elastic 𝑚! 𝑣!! + 𝑚! 𝑣!! = 𝑚! 𝑣!! + 𝑚! 𝑣!! 1 1 1 1 𝑚 𝑣! + 𝑚 𝑣! = 𝑚 𝑣! + 𝑚 𝑣! 2 ! !! 2 ! !! 2 ! !! 2 ! !!

Momentum 6 𝑝 = 𝑚𝑣

Definition

Final Angular Displacement 𝜃! =   θ! + ωt + αt ! Final Angular Velocity without Time ω!! = 𝜔!! +  2α(θ! − θ! ) 1 θ! = θ! + (ω! + ω! )t 2

Arc Length

Kinetic Energy of Rotation 𝐾! =  

Translational Velocity

For Constant Force For Non-Constant Force 𝐼=

!!

𝑣 = 𝜔𝑟 Translational Acceleration

𝐼 = 𝐹𝑡

𝐹. 𝑑𝑡

!!

•

𝐼 =  

Average Angular Velocity 𝜔!"#

Conservation of Momentum (All Collisions) 𝑝! = 𝑝!

𝜔𝒊𝒏𝒔𝒕

𝑑𝜃 = 𝑑𝑡

Instantaneous Angular Acceleration

𝑚! 𝑟!!

General

𝑎 = 𝛼𝑟 ∆𝜃 = ∆𝑡

𝜔! 2

Moment of Inertia

•

Instantaneous Angular Velocity

Collisions •

𝜔! = 𝜔! +  αt

𝑠 = 𝑟𝜃

𝐼 = ∆𝑝

•

Final Angular Velocity

Rotation

Impulse

•

𝑑𝜔 𝑑𝑡

Final Angular Displacement with Avg. Velocity

Momentum

•

𝛼!"#$ =

•

𝜌𝑟 !  𝑑𝑉

Sphere 𝐼=

2 𝑚𝑟 ! 5

𝐼=

1 𝑚𝑟 ! 2

Cylinder

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CHEAT SHEET •

7

Disk 𝐼 = 𝑚𝑟 !

Parallel Axis Theorem

Waves, Oscillations and SHM

Wave Number

•

𝐸!"#! =

Wave Equation

Torque

•

𝑣 =   ±𝜔 𝐴! − 𝑥 !

𝜏 = 𝑟𝐹𝑠𝑖𝑛𝜙 •

Using Perpendicular Distance

•

𝑣=

Net Torque 𝛴𝜏 = 𝐼𝛼 and 𝐸! = 𝑚𝑐

•

•

•

𝑇 = 2𝜋

General Equation Acceleration

Angular Momentum

𝐼 𝑑𝑚𝑔

𝑇 = 2𝜋 𝑎! = −𝜔 ! 𝑥

•

Angular Frequency

The Conservation of Momentum 𝜔=

𝐿! = 𝐿! •

𝑘 𝑚

Period

Frequency

8

Sound and EM Waves

Bulk Modulus 𝐵=−

𝑇= •

𝐿 𝑔

Physical Pendulum

o

𝑥 𝑡 = 𝐴𝑐𝑜𝑠(𝜔𝑡 + 𝜙) •

𝐿 = 𝐼𝜔

•

𝑇 𝜇

SHM and Circular Motion – Uses SHM formulae for each direction of movement SHM and the Pendulum o Period

Simple Harmonic Motion

!

Angular Momentum

•

Speed of Wave on a String

𝜏 = 𝐹𝑑

1 ! 𝑘𝐴 2

Velocity

𝑦 𝑥, 𝑡 = 𝐴𝑠𝑖𝑛 𝑘𝑥 − 𝜔𝑡 + 𝜙

Using Radius

𝜔 1 = 2𝜋 𝑇

Energy

2𝜋 𝑘= 𝜆

𝐼 = 𝐼!" ×𝑀𝐷 !

•

𝑓=

2𝜋 𝜔

∆𝑃 ∆𝑉/𝑉

Sound Wave Displacement 𝑠 𝑥, 𝑡 = 𝑠!"# cos  (𝑘𝑥 − 𝜔𝑡)

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CHEAT SHEET Sound Wave Pressure •

•

Including Bulk Modulus

𝐼=

∆𝑃 = 𝐵𝑠!"# sin  (𝑘𝑥 − 𝜔𝑡) •

•

Without Bulk Modulus

𝐼≡

Formula

𝑃𝑜𝑤𝑒𝑟!"# 4𝜋𝑟 !

𝑦 = 2𝐴𝑠𝑖𝑛 𝑘𝑥 cos  (𝜔𝑡) •

Amplitude 𝑎𝑚𝑝 = 2𝐴𝑠𝑖𝑛(𝑘𝑥)

•

𝑣 + 𝑣! 𝑓 𝑣 − 𝑣!

•

𝑇! 273

• •

EM Waves

•

Electrical Component 𝐸 = 𝐸! sin  (𝑘𝑥 − 𝜔𝑡) Magnetic Component 𝐵 = 𝐵! sin  (𝑘𝑥 − 𝜔𝑡)

When a pulse hits a fixed boundary, reflection is inverted When a pulse hits a free boundary, reflection is not inverted When a pulse moves from a light to a heavy string the reflected pulse is inverted When a pulse moves from a heavy to a light string, the reflection is not inverted

𝑦 = 2𝐴𝑠𝑖𝑛 𝑘𝑥 − 𝜔𝑡 +

𝜙 𝜙 cos 2 2

𝑛𝜆  (𝑤ℎ𝑒𝑟𝑒  𝑛 = 0, 1, 2 … ) 2

Antinodes 𝑛𝜆  𝑤ℎ𝑒𝑟𝑒  𝑛 = 1, 3, 5 … ) 4

Boundary Conditions on a String 𝑓! =

𝑛 𝑇 2𝐿 𝜇

Standing Waves in an Air Column •

Superposition

Interference

Nodes

𝑥=

Reflection of a Pulse

•

Intensity of a Sound Wave

Formula

𝑥= 𝑓′ =

Dependence on Temperature 𝑣 =  331 1 +

•

•

Doppler Effect 𝐵 𝜈 =   𝜌

•

Standing Waves on a String

𝐼 𝛽 = 10 log 𝐼!

Speed of Sound

•

!

Sound Levels in Decibels

𝑚 𝜌= 𝑉

•

∆𝑃!"# 2𝜌𝜈

In Three Dimensions

∆𝑃!"# = 𝜌𝑣𝜔𝑠!"# Density

𝑝𝑎𝑡ℎ  𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 ×2𝜋 = 𝑝ℎ𝑎𝑠𝑒  𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝜆

Per Unit Area

Closed Pipe 𝑓! =

•

𝑛𝜈  (𝑤ℎ𝑒𝑟𝑒  𝑛 = 1, 3, 5 … ) 4𝐿

Open Pipe 𝑓! =

𝑛𝜈  (𝑤ℎ𝑒𝑟𝑒  𝑛 = 1, 2, 3 … ) 2𝐿

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CHEAT SHEET •

End Effects 𝐿=

•

Equation of Continuity

𝑛𝜆 − 2  ×  𝑒𝑛𝑑  𝑒𝑓𝑓𝑒𝑐𝑡𝑠 2

𝑣! 𝐴! = 𝑣! 𝐴!

Image Formation •

Magnification 𝑚=−

•

9

Fluids

1 1 𝑝! + 𝑝𝑣!! + 𝑝𝑔𝑦! = 𝑝! + 𝑝𝑣!! + 𝑝𝑔𝑦! 2 2

Fluids at Rest •

Density 𝑝=

•

𝑚 𝑉

10

𝐹 𝐴

𝑝! = 𝑝 − 1𝑎𝑡𝑚

Total Internal Reflection

•

𝑛! 𝜃! = sin!! 𝑛!

•

•

Barometers 𝑝!"#$% = 𝑝𝑔ℎ

•

•

Focal Length 1 𝑟 2

•

Archimedes Principle 𝐹! = 𝑝! 𝑉! 𝑔

Fluids in Motion

•

𝑃𝑓 𝑃−𝑓

Thin lens equation in P 𝑖𝑓 𝑖−𝑓

Magnification in terms of P and f 𝑚=

Manometers 𝑝!"#$% = 1𝑎𝑡𝑚 + 𝑝𝑔ℎ

Thin Lens Equation in i

𝑃= 𝑓=

1 1 − 𝑟! 𝑟!

Focal Length (convex lens)

𝑖=

Spherical Mirrors •

Focal Length

1 𝑛−1 = 𝑓 𝑟!

𝑛! 𝑠𝑖𝑛𝜃! = 𝑛! 𝑠𝑖𝑛𝜃!  (𝑆𝑛𝑒𝑙𝑙 ! 𝑠  𝐿𝑎𝑤)

Pressure in Liquids Gauge Pressure

•

•

Refraction

𝑝 = 𝑝! + 𝑝𝑔𝑑 •

Thin Lens Equations 1 = 𝑛−1 𝑓

Ray Optics

Pressure 𝑃=

•

Bernoulli’s Equation

𝑖 𝑝

𝑓 𝑃−𝑓

Image Distance (thin lens equation) 1 1 1 + = 𝑃 𝑖 𝑓

•

Magnification in terms of i and f 𝑚=

𝑖−𝑓 𝑓

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CHEAT SHEET Thin Lens Images

•

Lens

P

Real?

Orientation

m

Convex

F

Yes

↓

↑

= 2F

Yes

↓

-

> 2F

Yes

↓

↓

F

No

↑

↑

= 2F

No

↑

-

> 2F

No

↑

↓

Concave

𝑚!"! = 𝑚! 𝑚!

𝑚!" = −

11

Distances of Dark Fringes

𝑓!" 𝑓!"

𝑦′! = •

∆𝑦 =

Path Difference •

Constructive

•

Angles for Dark Fringes 𝜃! =

Destructive 𝛥𝑟 = 𝑚 +

•

1 𝜆  (𝑤ℎ𝑒𝑟𝑒  𝑚 = 1, 2, 3 … ) 2

Angles for Bright Fringes 𝜃! =

Simple Magnifying Lens 𝑚! =

25 𝑓

Compound Microscope 𝑚=−

𝑠 25 × 𝑓!" 𝑓!"

•

𝑚𝜆 𝑑

Distances of Bright Fringes 𝐿𝑚𝜆 𝑦! = 𝑑

•

𝜆𝐿 𝑑

Interference of Light in Single Slit

𝛥𝑟 = 𝑚𝜆  (𝑤ℎ𝑒𝑟𝑒  𝑚 = 1, 2, 3 … ) •

1 𝐿𝜆 2 𝑑

𝑚+

Distances Between Fringes

Wave Optics

Optical Instruments

•

•

•

𝑦! =

Angles for Dark Fringes 𝜃′! =

𝑚+ 𝑑

1 𝜆 2

•

𝑝𝜆 𝑎

Distances for Dark Fringes

Interference of Light in Double Slit

Two Lens System

•

Refracting Telescope

𝑝𝜆𝐿 𝑎

Width of Central Maximum 𝑤=

2𝜆𝐿 𝑎

Circular Aperture Diffraction 𝑤=

2.44𝜆𝐿 𝐷

Interferometer 𝑑=

𝑁𝜆 2

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CHEAT SHEET 12

Charge

Ohm’s Law

Charge

Coulomb’s Law

𝐼= 𝐾𝑞! 𝑞! 𝐹= 𝑟!

Power and Energy •

Electrical Field •

Power delivered by an emf 𝑃!"# = 𝐼𝛦

Vector Equation 𝐸=

•

𝛥𝑉 𝑅

•

𝐹 𝑞

Electrical Field of a Point Charge 𝐸=

𝐾𝑞 𝑟!

Electrical Potential 𝑈!"!# = 𝑉𝑞

14

-19

e = -1.60 × 10

•

K = 8.99 × 10 Nm /C

•

V = J/C

9

C 2

2

Electrical Circuits

Power Dissipated by a Resistor 𝛥𝑉! 𝑃! = 𝑅

-

•

!

•

A = C/s

•

Ω = V/A

 

Units and Constants

Particle Kinematics in One Dimension •

g = 9.8m/s

2

Particle Dynamics

13

Electrical Circuits Definition 𝐼= •

N = kg.m/s

2

Work and Energy

Current •

•

𝛥𝑞 𝛥𝑡

Conservation of Charge 𝛴𝐼!" = 𝛴𝐼!"#

2

•

J = Nm = kg.m /s

•

W = J/s

•

1Hp = 746W

Fluids •

Pa = N/m

2

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