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PHYS 2310 Engineering Physics I Formula Sheets
Chapters 1-
Chapter 1/Important Numbers Chapter 2
Units for SI Base Quantities
Quantity Unit Name Unit Symbol
Length Meter M
Time Second s
Mass (not weight) Kilogram kg
Common Conversions
1 kg or 1 m 1000 g or m 1 m 1 × 10
6
1 m 100 cm 1 inch 2.54 cm
1 m 1000 mm 1 day 86400 seconds
1 second 1000 milliseconds 1 hour 3600 seconds
1 m 3.281 ft 360 ° 2 𝜋 rad
Important Constants/Measurements
Mass of Earth 5. 98 × 10
24
kg
Radius of Earth 6. 38 × 10
6
m
1 u (Atomic Mass Unit)
- 661 × 10
− 27
kg
Density of water 1 𝑔/𝑐𝑚
3
or 1000 𝑘𝑔/𝑚
3
g (on earth) 9.8 m/s
2
Density
Common geometric Formulas
Circumference 𝐶 = 2 𝜋𝑟 Area circle
2
Surface area
(sphere)
2
Volume (sphere) 𝑉 =
3
Volume (rectangular solid)
Velocity
Average Velocity 𝑉
𝑎𝑣𝑔
Average Speed
𝑎𝑣𝑔
Instantaneous Velocity 𝑣 = lim
∆𝑡→ 0
Acceleration
Average Acceleration 𝑎
𝑎𝑣𝑔
Instantaneous
Acceleration
2
2
Motion of a particle with constant acceleration
0
0
0
2
2
0
2
Chapter 3 Chapter 4
Adding Vectors
Geometrically
Adding Vectors
Geometrically
(Associative Law)
Components of Vectors
𝑥
𝑦
Magnitude of vector | 𝑎
𝑥
2
𝑦
2
Angle between x axis
and vector
𝑦
𝑥
Unit vector notation 𝑎⃗ = 𝑎 𝑥
𝑦
𝑧
Adding vectors in
Component Form
𝑥
𝑥
𝑥
𝑦
𝑦
𝑦
𝑧
𝑧
𝑧
Scalar (dot product) 𝑎⃗ ∙ 𝑏
Scalar (dot product)
𝑎⃗ ∙ 𝑏
⃗⃗
= (𝑎
𝑥
𝑖̂ + 𝑎
𝑦
𝑗̂ + 𝑎
𝑧
𝑘
̂
) ∙ (𝑏
𝑥
𝑖̂ + 𝑏
𝑦
𝑗 ̂+ 𝑏
𝑧
𝑘
̂
)
𝑥
𝑥
𝑦
𝑦
𝑧
𝑧
Projection of 𝑎⃗ 𝑜𝑛 𝑏
or
component of 𝑎⃗ 𝑜𝑛 𝑏
Vector (cross) product
magnitude
Vector (cross product)
𝑎⃗ 𝑥𝑏
⃗⃗
= (𝑎
𝑥
𝑖 ̂+ 𝑎
𝑦
𝑗̂ + 𝑎
𝑧
𝑘
̂
)𝑥(𝑏
𝑥
𝑖̂ + 𝑏
𝑦
𝑗 ̂+ 𝑏
𝑧
𝑘
̂
)
= (𝑎
𝑦
𝑏
𝑧
− 𝑏
𝑦
𝑎
𝑧
)𝑖̂ + (𝑎
𝑧
𝑏
𝑥
− 𝑏
𝑧
𝑎
𝑥
)𝑗̂
𝑥
𝑏
𝑦
− 𝑏
𝑥
𝑎
𝑦
)𝑘
̂
or
𝑥
𝑦
𝑧
𝑥
𝑦
𝑧
Position vector 𝑟⃗ = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧𝑘
displacement ∆𝑟⃗ = ∆𝑥𝑖̂ + ∆𝑦𝑗̂ + ∆𝑧𝑘
Average Velocity 𝑉
𝑎𝑣𝑔
Instantaneous Velocity
𝑣⃗ =
𝑥
𝑦
𝑧
Average Acceleration
𝑎⃗
𝑎𝑣𝑔
Instantaneous
Acceleration
𝑥
𝑦
𝑧
Projectile Motion
𝑦
0
0
0
𝑥
2
or ∆𝑥 = 𝑣
0
𝑐𝑜𝑠𝜃𝑡 if 𝑎
𝑥
0
2
𝑦
2
0
0
2
− 2 𝑔∆y
𝑦
0
0
Trajectory 𝑦 = (𝑡𝑎𝑛𝜃
0
2
0
0
2
Range 𝑅 =
0
2
sin
0
Relative Motion
𝐴𝐶
𝐴𝐵
𝐵𝐶
𝐴𝐵
𝐵𝐴
Uniform Circular
Motion
𝑣
2
𝑟
2 𝜋𝑟
𝑣
Chapter 7 Chapter 8
Kinetic Energy
2
Work done by constant
Force 𝑊 = 𝐹𝑑𝑐𝑜𝑠𝜃 = 𝐹
Work- Kinetic Energy
Theorem
𝑓
0
Work done by gravity
𝑔
Work done by
lifting/lowering object
𝑎
𝑔
𝑎
Spring Force (Hooke’s
law)
𝑠
𝑥
= −𝑘𝑥 (along x-axis)
Work done by spring
𝑠
𝑖
2
𝑓
2
Work done by Variable
Force
𝑥
𝑥
𝑓
𝑥
𝑖
𝑦
𝑦
𝑓
𝑦
𝑖
𝑧
𝑧
𝑓
𝑧
𝑖
Average Power
(rate at which that
force does work on an
object)
𝑎𝑣𝑔
Instantaneous Power 𝑃 =
Potential Energy ∆𝑈 = −𝑊 = − ∫ 𝐹
𝑥𝑓
𝑥𝑖
Gravitational Potential
Energy
Elastic Potential Energy
2
Mechanical Energy
𝑚𝑒𝑐
Principle of
conservation of
mechanical energy
1
1
2
2
𝑚𝑒𝑐
Force acting on particle
Work on System by
external force
With no friction
𝑚𝑒𝑐
Work on System by
external force
With friction
𝑚𝑒𝑐
𝑡ℎ
Change in thermal
energy
𝑡ℎ
𝑘
Conservation of Energy
*if isolated W=
𝑚𝑒𝑐
𝑡ℎ
𝑖𝑛𝑡
Average Power 𝑃
𝑎𝑣𝑔
Instantaneous Power
_In General Physics, Kinetic Energy is abbreviated to KE and Potential Energy is PE_**
Chapter 9
Impulse and Momentum
Impulse
𝑡
𝑓
𝑡
𝑖
𝑛𝑒𝑡
Linear Momentum
Impulse-Momentum
Theorem
𝑓
𝑖
Newton’s 2
nd
law
𝑛𝑒𝑡
System of Particles
𝑛𝑒𝑡
𝑐𝑜𝑚
𝑐𝑜𝑚
𝑛𝑒𝑡
Collision
Final Velocity of 2
objects in a head-on
collision where one
object is initially at rest
1: moving object
2: object at rest
1 𝑓
1
2
1
2
1 𝑖
2 𝑓
1
1
2
1 𝑖
Conservation of Linear
Momentum (in 1D)
𝑖
𝑓
Elastic Collision
1 𝑖
2 𝑖
1 𝑓
2 𝑓
1
𝑖 1
2
12
1
𝑓 1
2
𝑓 2
1 𝑖
2 𝑖
1 𝑓
2 𝑓
Collision continued…
Inelastic Collision
1
01
2
02
1
2
𝑓
Conservation of Linear
Momentum (in 2D)
1 𝑖
2 𝑖
1 𝑓
2 𝑓
Average force
𝑎𝑣𝑔
𝑎𝑣𝑔
Center of Mass
Center of mass location 𝑟⃗
𝑐𝑜𝑚
𝑖
𝑛
𝑖= 1
𝑖
Center of mass velocity 𝑣⃗
𝑐𝑜𝑚
𝑖
𝑛
𝑖= 1
𝑖
Rocket Equations
Thrust (Rv rel
𝑟𝑒𝑙
Change in velocity
𝑟𝑒𝑙
𝑖
𝑓
Moments of Inertia I for various rigid objects of Mass M
Thin walled hollow cylinder or hoop
about central axis
2
Annular cylinder (or ring) about
central axis
1
2
2
2
Solid cylinder or disk about central
axis
2
Solid cylinder or disk about central
diameter
2
2
Solid Sphere, axis through center
2
Solid Sphere, axis tangent to surface
2
Thin Walled spherical shell, axis
through center
2
Thin rod, axis perpendicular to rod
and passing though center
2
Thin rod, axis perpendicular to rod
and passing though end
2
Thin Rectangular sheet (slab), axis
parallel to sheet and passing though
center of the other edge
2
Thin Rectangular sheet (slab_, axis
along one edge
2
Thin rectangular sheet (slab) about
perpendicular axis through center
2
2
Chapter 11
Rolling Bodies (wheel)
Speed of rolling wheel 𝑣
𝑐𝑜𝑚
Kinetic Energy of Rolling
Wheel
𝑐𝑜𝑚
2
𝑐𝑜𝑚
2
Acceleration of rolling
wheel
𝑐𝑜𝑚
Acceleration along x-axis
extending up the ramp
𝑐𝑜𝑚,𝑥
𝑐𝑜𝑚
2
Torque as a vector
Torque
𝜏⃗ = 𝑟⃗ × 𝐹
Magnitude of torque 𝜏 = 𝑟𝐹 ⊥
⊥
Newton’s 2
nd
Law
𝑛𝑒𝑡
Angular Momentum
Angular Momentum
= 𝑟⃗⃗ × 𝑝⃗⃗⃗ = 𝑚(𝑟⃗⃗ × 𝑣⃗⃗⃗)
Magnitude of Angular
Momentum
⊥
⊥
Angular momentum of a
system of particles
𝑖
𝑛
𝑖= 1
𝑛𝑒𝑡
Angular Momentum continued
Angular Momentum of a
rotating rigid body
Conservation of angular
momentum
𝑖
𝑓
Precession of a Gyroscope
Precession rate
Chapter 14 Chapter 15
Density
Pressure
Pressure and depth in
a static Fluid
P1 is higher than P
2
1
1
2
0
Gauge Pressure
Archimedes’ principle 𝐹
𝑏
𝑓
Mass Flow Rate
𝑚
𝑉
Volume flow rate
𝑉
Bernoulli’s Equation
2
Equation of continuity
𝑚
𝑉
Equation of continuity
when
𝑉
Frequency
cycles per time
displacement 𝑥 = 𝑥
𝑚
cos(𝜔𝑡 + 𝜙) 15.
Angular frequency 𝜔 =
Velocity 𝑣 = −𝜔𝑥
𝑚
sin(𝜔𝑡 + 𝜙)
Acceleration 𝑎 = −𝜔
2
𝑚
cos(𝜔𝑡 + 𝜙)
Kinetic and Potential
Energy
2
2
Angular frequency
𝜔 =
Period 𝑇 = 2 𝜋
Torsion pendulum
𝑇 = 2 𝜋
Simple Pendulum 𝑇 = 2 𝜋√
Physical Pendulum 𝑇 = 2 𝜋√
Damping force 𝐹
𝑑
displacement
𝑚
−
𝑏𝑡
2 𝑚
cos(𝜔
′
Angular frequency
′
2
2
Mechanical Energy 𝐸(𝑡) ≈
𝑚
2
−
𝑏𝑡
Chapter 16
Sinusoidal Waves
Mathematical form
(positive direction)
𝑚
sin(𝑘𝑥 − 𝜔𝑡) 16.
Angular wave number 𝑘 =
Angular frequency 𝜔 =
Wave speed 𝑣 =
Average Power
𝑎𝑣𝑔
2
𝑚
2
Traveling Wave Form 𝑦(𝑥, 𝑡) = ℎ(𝑘𝑥 ± 𝜔𝑡) 16.
Wave speed on
stretched string
Resulting wave when 2
waves only differ by
phase constant
𝑦
′
( 𝑥, 𝑡
) = [ 2 𝑦
𝑚
cos (
1
2
𝜙)] sin (𝑘𝑥 − 𝜔𝑡 +
1
2
Standing wave 𝑦
′
(𝑥, 𝑡) = [ 2 𝑦
𝑚
sin(𝑘𝑥)]cos (𝜔𝑡) 16.
Resonant frequency 𝑓 =
𝑣
𝜆
𝑣
2 𝐿
for n=1,2,… 16.
Chapter 1 8
Temperature Scales
Fahrenheit to Celsius
𝐶
𝐹
Celsius to Fahrenheit
𝐹
𝐶
Celsius to Kelvin
𝐶
Thermal Expansion
Linear Thermal Expansion ∆𝐿 = 𝐿𝛼∆𝑇 18.
Volume Thermal Expansion
Heat
Heat and temperature
change
𝑓
𝑖
𝑓
𝑖
Heat and phase change
Power
P=Q/t
Power (Conducted)
𝑐𝑜𝑛𝑑
𝐻
𝐶
Rate objects absorbs
energy
𝑎𝑏𝑠
𝑒𝑛𝑣
4
Power from radiation
𝑟𝑎𝑑
4
𝜎 = 5. 6704 × 10
− 8
2
4
First Law of Thermodynamics
First Law of
Thermodynamics
𝑖𝑛𝑡
𝑖𝑛𝑡,𝑓
𝑖𝑛𝑡,𝑖
𝑖𝑛𝑡
Note:
𝑖𝑛𝑡
Change in Internal Energy
Q (heat) is positive when the system absorbs heat and negative when it
loses heat. W (work) is work done by system. W is positive when expanding
and negative contracts because of an external force
Applications of First Law
Adiabatic
(no heat flow)
Q=
𝑖𝑛𝑡
(constant volume)
W=
𝑖𝑛𝑡
Cyclical process
𝑖𝑛𝑡
Q=W
Free expansions
𝑖𝑛𝑡
Misc.
Work Associated with
Volume Change
𝑉
𝑓
𝑉 𝑖
Revised 7/20/
