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PHYS1205 Integrated Physics Cheatsheet

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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
π‘₯! = π‘₯! + (𝑣!" + 𝑣!" )𝑑
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
π‘₯! = π‘₯! + 𝑣!" 𝑑 + π‘Ž! 𝑑 !
2
Vector Components
•
Length
𝐴 =
Final Velocity without Time
!
!
𝑣!"
= 𝑣!"
+ 2π‘Ž! π‘₯! − π‘₯!
•
βˆ†π‘₯
βˆ†π‘‘
Objects in Freefall – Acceleration is –g (9.8m/s )
Instantaneous Velocity
𝑣!"#$
2
Vectors and 2D Motion
Vector Addition – Tip to Tail
𝑑π‘₯
=
𝑑𝑑
πœƒ = tan!!
βˆ†π‘£
βˆ†π‘‘
Instantaneous Acceleration
π‘Ž!"#$ =
𝑑π‘₯
𝑑𝑑
Final Velocity
𝐴!
𝐴 = 𝐴 x𝚀 + 𝐴 yπš₯
Projectile Motion
•
Vector Subtraction – From the negative to the
positive, or add the negative (𝐴 − 𝐡 = 𝐴 + (−𝐡))
𝐴!
Unit Vectors:
Position
1
π‘Ÿ! = π‘Ÿ! + 𝑣! 𝑑 + 𝑔𝑑 !
2
Average Acceleration
π‘Ž!"# =
!
Direction
2
Average Velocity
𝑣!"# =
!
𝐴! + 𝐴! •
Initial Horizontal Velocity
𝑣!" = 𝑣! cos πœƒ
•
Initial Vertical Velocity
𝑣!" = 𝑣! sin πœƒ
𝑣!" = 𝑣!" + π‘Ž! 𝑑
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CHEAT SHEET
Uniform Circular Motion
•
Equilibrium
•
Kinetic Energy
Kinetic Friction
𝐾𝐸 =
𝐹 = πœ‡! 𝑁
!
π‘Ž! + π‘Ž!
!
•
π›΄π‘Š = π›₯𝐾𝐸
𝐹 ≤ πœ‡! 𝑁
2πœ‹π‘Ÿ
𝑣
𝑣!
𝐹 = π‘šπ‘Ž! = π‘š
π‘Ÿ
π‘Ÿ!" = π‘Ÿ!" + 𝑣!" 𝑑
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 = π‘šπ‘Ž
4
𝐹!" = −𝐹!"
Gravitational
π‘ˆ = π‘šπ‘”π›₯𝑦
Elastic
Work, Energy and Power
Scalar/Dot Product
𝐴 βˆ™ 𝐡 = π΄π΅π‘π‘œπ‘ πœƒ
Work
•
Same Direction as Displacement
π‘Š = πΉβˆ†π‘Ÿ
•
Different Direction to Displacement
π‘ˆ=
•
Non-conservative Force - Work done dependent on
the motion of the object (e.g. Friction)
Conservation of Energy
•
π‘Š!"# =
!!
!!
Mechanical Energy
𝐸!"#! = 𝐾𝐸 + π‘ˆ
•
Work by Varying Force
1 !
π‘˜π‘₯
2
Conservative Force - Work done is independent of
the path taken by an object (e.g. Gravity)
π‘Š = πΉβˆ†π‘Ÿπ‘π‘œπ‘ πœƒ
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
π‘šπ‘£ !
2
Work-Kinetic Energy Theorem
Static Friction
Period
𝑇=
3
𝐹! = −π‘˜π‘₯
Friction
𝑣!
π‘Ÿ
Overall Acceleration
|π‘Ž| =
•
𝛴𝐹 = 0
Centripetal Acceleration
π‘Ž! =
•
Hooke’s Law
Total Energy
𝐸!"! = 𝐾𝐸 + π‘ˆ + 𝐸!"#
𝐹! 𝑑π‘₯
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CHEAT SHEET
•
Non-Conservative Force Absent
•
Conservation of KE (Elastic Collisions)
βˆ†πΈ!"#! = 0
•
𝐾𝐸! = 𝐾𝐸!
Non-Conservative Force Present
•
Perfectly Inelastic
βˆ†πΈ!"! = 0
π‘š! 𝑣!! + π‘š! 𝑣!! = (π‘š! + π‘š! )𝑣!
Power
•
π‘‘π‘Š
πœ‘=
𝑑𝑑
5
Perfectly Elastic
π‘š! 𝑣!! + π‘š! 𝑣!! = π‘š! 𝑣!! + π‘š! 𝑣!!
1
1
1
1
π‘š 𝑣! + π‘š 𝑣! = π‘š 𝑣! + π‘š 𝑣!
2 ! !! 2 ! !! 2 ! !! 2 ! !!
Momentum
6
𝑝 = π‘šπ‘£
Definition
Final Angular Displacement
πœƒ! = θ! + ωt + αt !
Final Angular Velocity without Time
ω!! = πœ”!! + 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 = πœ”π‘Ÿ
Translational Acceleration
𝐼 = 𝐹𝑑
𝐹. 𝑑𝑑
!!
•
𝐼 = Average Angular Velocity
πœ”!"#
Conservation of Momentum (All Collisions)
𝑝! = 𝑝!
πœ”π’Šπ’π’”π’•
π‘‘πœƒ
=
𝑑𝑑
Instantaneous Angular Acceleration
π‘š! π‘Ÿ!!
General
π‘Ž = π›Όπ‘Ÿ
βˆ†πœƒ
=
βˆ†π‘‘
πœ”!
2
Moment of Inertia
•
Instantaneous Angular Velocity
Collisions
•
πœ”! = πœ”! + αt
𝑠 = π‘Ÿπœƒ
𝐼 = βˆ†π‘
•
Final Angular Velocity
Rotation
Impulse
•
π‘‘πœ”
𝑑𝑑
Final Angular Displacement with Avg. Velocity
Momentum
•
𝛼!"#$ =
•
πœŒπ‘Ÿ ! 𝑑𝑉
Sphere
𝐼=
2
π‘šπ‘Ÿ !
5
𝐼=
1
π‘šπ‘Ÿ !
2
Cylinder
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CHEAT SHEET
•
7
Disk
𝐼 = π‘šπ‘Ÿ !
Parallel Axis Theorem
Waves, Oscillations and SHM
Wave Number
•
𝐸!"#! =
Wave Equation
Torque
•
𝑣 = ±πœ” 𝐴! − π‘₯ !
𝜏 = π‘ŸπΉπ‘ π‘–π‘›πœ™
•
Using Perpendicular Distance
•
𝑣=
Net Torque
π›΄πœ = 𝐼𝛼
and
𝐸! = π‘šπ‘
•
•
•
𝑇 = 2πœ‹
General Equation
Acceleration
Angular Momentum
𝐼
π‘‘π‘šπ‘”
𝑇 = 2πœ‹
π‘Ž! = −πœ” ! π‘₯
•
Angular Frequency
The Conservation of Momentum
πœ”=
𝐿! = 𝐿!
•
π‘˜
π‘š
Period
Frequency
8
Sound and EM Waves
Bulk Modulus
𝐡=−
𝑇=
•
𝐿
𝑔
Physical Pendulum
o
π‘₯ 𝑑 = π΄π‘π‘œπ‘ (πœ”π‘‘ + πœ™)
•
𝐿 = πΌπœ”
•
𝑇
πœ‡
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 !
π‘˜π΄
2
Velocity
𝑦 π‘₯, 𝑑 = 𝐴𝑠𝑖𝑛 π‘˜π‘₯ − πœ”π‘‘ + πœ™
Using Radius
πœ”
1
=
2πœ‹ 𝑇
Energy
2πœ‹
π‘˜=
πœ†
𝐼 = 𝐼!" ×𝑀𝐷 !
•
𝑓=
2πœ‹
πœ”
βˆ†π‘ƒ
βˆ†π‘‰/𝑉
Sound Wave Displacement
𝑠 π‘₯, 𝑑 = 𝑠!"# cos (π‘˜π‘₯ − πœ”π‘‘)
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CHEAT SHEET
Sound Wave Pressure
•
•
Including Bulk Modulus
𝐼=
βˆ†π‘ƒ = 𝐡𝑠!"# sin (π‘˜π‘₯ − πœ”π‘‘)
•
•
Without Bulk Modulus
𝐼≡
Formula
π‘ƒπ‘œπ‘€π‘’π‘Ÿ!"#
4πœ‹π‘Ÿ !
𝑦 = 2𝐴𝑠𝑖𝑛 π‘˜π‘₯ cos (πœ”π‘‘)
•
Amplitude
π‘Žπ‘šπ‘ = 2𝐴𝑠𝑖𝑛(π‘˜π‘₯)
•
𝑣 + 𝑣!
𝑓
𝑣 − 𝑣!
•
𝑇!
273
•
•
EM Waves
•
Electrical Component
𝐸 = 𝐸! sin (π‘˜π‘₯ − πœ”π‘‘)
Magnetic Component
𝐡 = 𝐡! sin (π‘˜π‘₯ − πœ”π‘‘)
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 π‘˜π‘₯ − πœ”π‘‘ +
πœ™
πœ™
cos
2
2
π‘›πœ†
(π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑛 = 0, 1, 2 … )
2
Antinodes
π‘›πœ†
π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑛 = 1, 3, 5 … )
4
Boundary Conditions on a String
𝑓! =
𝑛 𝑇
2𝐿 πœ‡
Standing Waves in an Air Column
•
Superposition
Interference
Nodes
π‘₯=
Reflection of a Pulse
•
Intensity of a Sound Wave
Formula
π‘₯=
𝑓′ =
Dependence on Temperature
𝑣 = 331 1 +
•
•
Doppler Effect
𝐡
𝜈 = 𝜌
•
Standing Waves on a String
𝐼
𝛽 = 10 log
𝐼!
Speed of Sound
•
!
Sound Levels in Decibels
π‘š
𝜌=
𝑉
•
βˆ†π‘ƒ!"#
2𝜌𝜈
In Three Dimensions
βˆ†π‘ƒ!"# = πœŒπ‘£πœ”π‘ !"#
Density
π‘π‘Žπ‘‘β„Ž π‘‘π‘–π‘“π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’
×2πœ‹ = π‘β„Žπ‘Žπ‘ π‘’ π‘‘π‘–π‘“π‘“π‘’π‘Ÿπ‘’π‘›π‘π‘’
πœ†
Per Unit Area
Closed Pipe
𝑓! =
•
π‘›πœˆ
(π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑛 = 1, 3, 5 … )
4𝐿
Open Pipe
𝑓! =
π‘›πœˆ
(π‘€β„Žπ‘’π‘Ÿπ‘’ 𝑛 = 1, 2, 3 … )
2𝐿
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CHEAT SHEET
•
End Effects
𝐿=
•
Equation of Continuity
π‘›πœ†
− 2 × π‘’π‘›π‘‘ 𝑒𝑓𝑓𝑒𝑐𝑑𝑠
2
𝑣! 𝐴! = 𝑣! 𝐴!
Image Formation
•
Magnification
π‘š=−
•
9
Fluids
1
1
𝑝! + 𝑝𝑣!! + 𝑝𝑔𝑦! = 𝑝! + 𝑝𝑣!! + 𝑝𝑔𝑦!
2
2
Fluids at Rest
•
Density
𝑝=
•
π‘š
𝑉
10
𝐹
𝐴
𝑝! = 𝑝 − 1π‘Žπ‘‘π‘š
Total Internal Reflection
•
𝑛!
πœƒ! = sin!!
𝑛!
•
•
Barometers
𝑝!"#$% = π‘π‘”β„Ž
•
•
Focal Length
1
π‘Ÿ
2
•
Archimedes Principle
𝐹! = 𝑝! 𝑉! 𝑔
Fluids in Motion
•
𝑃𝑓
𝑃−𝑓
Thin lens equation in P
𝑖𝑓
𝑖−𝑓
Magnification in terms of P and f
π‘š=
Manometers
𝑝!"#$% = 1π‘Žπ‘‘π‘š + π‘π‘”β„Ž
Thin Lens Equation in i
𝑃=
𝑓=
1 1
−
π‘Ÿ! π‘Ÿ!
Focal Length (convex lens)
𝑖=
Spherical Mirrors
•
Focal Length
1 𝑛−1
=
𝑓
π‘Ÿ!
𝑛! π‘ π‘–π‘›πœƒ! = 𝑛! π‘ π‘–π‘›πœƒ! (𝑆𝑛𝑒𝑙𝑙 ! 𝑠 πΏπ‘Žπ‘€)
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
π‘š=
𝑖−𝑓
𝑓
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CHEAT SHEET
Thin Lens Images
•
Lens
P
Real?
Orientation
m
Convex
<F
No
↑
↑
=F
N/A
N/A
N/A
>F
Yes
↓
↑
= 2F
Yes
↓
-
> 2F
Yes
↓
↓
<F
No
↑
↑
=F
N/A
N/A
N/A
>F
No
↑
↑
= 2F
No
↑
-
> 2F
No
↑
↓
Concave
π‘š!"! = π‘š! π‘š!
π‘š!" = −
11
Distances of Dark Fringes
𝑓!"
𝑓!"
𝑦′! =
•
βˆ†π‘¦ =
Path Difference
•
Constructive
•
Angles for Dark Fringes
πœƒ! =
Destructive
π›₯π‘Ÿ = π‘š +
•
1
πœ† (π‘€β„Žπ‘’π‘Ÿπ‘’ π‘š = 1, 2, 3 … )
2
Angles for Bright Fringes
πœƒ! =
Simple Magnifying Lens
π‘š! =
25
𝑓
Compound Microscope
π‘š=−
𝑠
25
×
𝑓!" 𝑓!"
•
π‘šπœ†
𝑑
Distances of Bright Fringes
πΏπ‘šπœ†
𝑦! =
𝑑
•
πœ†πΏ
𝑑
Interference of Light in Single Slit
π›₯π‘Ÿ = π‘šπœ† (π‘€β„Žπ‘’π‘Ÿπ‘’ π‘š = 1, 2, 3 … )
•
1
πΏπœ†
2
𝑑
π‘š+
Distances Between Fringes
Wave Optics
Optical Instruments
•
•
•
𝑦! =
Angles for Dark Fringes
πœƒ′! =
π‘š+
𝑑
1
πœ†
2
•
π‘πœ†
π‘Ž
Distances for Dark Fringes
Interference of Light in Double Slit
Two Lens System
•
Refracting Telescope
π‘πœ†πΏ
π‘Ž
Width of Central Maximum
𝑀=
2πœ†πΏ
π‘Ž
Circular Aperture Diffraction
𝑀=
2.44πœ†πΏ
𝐷
Interferometer
𝑑=
π‘πœ†
2
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CHEAT SHEET
12
Charge
Ohm’s Law
Charge
Coulomb’s Law
𝐼=
πΎπ‘ž! π‘ž!
𝐹=
π‘Ÿ!
Power and Energy
•
Electrical Field
•
Power delivered by an emf
𝑃!"# = 𝐼𝛦
Vector Equation
𝐸=
•
π›₯𝑉
𝑅
•
𝐹
π‘ž
Electrical Field of a Point Charge
𝐸=
πΎπ‘ž
π‘Ÿ!
Electrical Potential
π‘ˆ!"!# = π‘‰π‘ž
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
π›₯𝑉!
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
•
•
π›₯π‘ž
π›₯𝑑
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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