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
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
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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