PHYS 2310 Engineering Physics I Formula Sheets
Chapters 1-18
Chapter 1/Important Numbers
Chapter 2
Velocity
Quantity
Units for SI Base Quantities
Unit Name
Unit Symbol
Length
Meter
M
Time
Second
s
Mass (not weight)
Kilogram
kg
1 kg or 1 m
1m
1m
1 second
1m
Common Conversions
1000 g or m
1m
100 cm
1 inch
1000 mm
1 day
1000 milliseconds 1 hour
3.281 ft
360°
Average Velocity
Average Speed
1 × 106 𝜇𝑚
2.54 cm
86400 seconds
3600 seconds
2𝜋 rad
Important Constants/Measurements
Mass of Earth
5.98 × 1024 kg
Radius of Earth
6.38 × 106 m
1 u (Atomic Mass Unit)
1.661 × 10−27 kg
Density of water
1 𝑔/𝑐𝑚3 or 1000 𝑘𝑔/𝑚3
g (on earth)
9.8 m/s 2
Circumference
Surface area
(sphere)
Density
Common geometric Formulas
Area circle
𝐶 = 2𝜋𝑟
𝑆𝐴 = 4𝜋𝑟 2
Volume (rectangular solid)
Instantaneous Velocity
𝑉𝑎𝑣𝑔 =
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 ∆𝑥
=
𝑡𝑖𝑚𝑒
∆𝑡
𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑖𝑚𝑒
̅ 𝑑𝑥
∆𝑥
𝑣 = lim
=
∆𝑡→0 ∆𝑡
𝑑𝑡
𝑠𝑎𝑣𝑔 =
Average Acceleration
∆𝑣
∆𝑡
𝑑𝑣 𝑑2 𝑥
𝑎=
=
𝑑𝑡 𝑑𝑡 2
𝑎𝑎𝑣𝑔 =
Motion of a particle with constant acceleration
𝐴 = 𝜋𝑟
4
Volume (sphere)
𝑉 = 𝜋𝑟 3
3
𝑉 =𝑙∙𝑤∙ℎ
𝑉 = 𝑎𝑟𝑒𝑎 ∙ 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠
2.3
2.4
Acceleration
Instantaneous
Acceleration
2
2.2
𝑣 = 𝑣0 + 𝑎𝑡
1
∆𝑥 = (𝑣0 + 𝑣)𝑡
2
1
∆𝑥 = 𝑣0 𝑡 + 𝑎𝑡 2
2
𝑣 2 = 𝑣02 + 2𝑎∆𝑥
2.11
2.17
2.15
2.16
2.7
2.8
2.9
Chapter 3
Adding Vectors
Geometrically
Adding Vectors
Geometrically
(Associative Law)
Chapter 4
𝑎⃗ + 𝑏⃗⃗ = 𝑏⃗⃗ + 𝑎⃗
3.2
(𝑎⃗ + 𝑏⃗⃗) + 𝑐⃗ = 𝑎⃗ + (𝑏⃗⃗ + 𝑐⃗)
3.3
3.5
Magnitude of vector
|𝑎| = 𝑎 = √𝑎𝑥2 + 𝑎𝑦2
3.6
Angle between x axis
and vector
𝑎𝑦
𝑡𝑎𝑛𝜃 =
𝑎𝑥
3.6
Unit vector notation
𝑎⃗ = 𝑎𝑥 𝑖̂ + 𝑎𝑦 𝑗̂ + 𝑎𝑧 𝑘̂
3.7
𝑟𝑥 = 𝑎𝑥 + 𝑏𝑥
𝑟𝑦 = 𝑎𝑦 + 𝑏𝑦
𝑟𝑧 = 𝑎𝑧 + 𝑏𝑧
3.10
3.11
3.12
𝑎⃗ ∙ 𝑏⃗⃗ = 𝑎𝑏𝑐𝑜𝑠𝜃
3.20
Adding vectors in
Component Form
Scalar (dot product)
Scalar (dot product)
Projection of 𝑎⃗ 𝑜𝑛 𝑏⃗⃗ or
component of 𝑎⃗ 𝑜𝑛 𝑏⃗⃗
Vector (cross) product
magnitude
𝑎⃗ ∙ 𝑏⃗⃗ = (𝑎𝑥 𝑖̂ + 𝑎𝑦 𝑗̂ + 𝑎𝑧 𝑘̂) ∙ (𝑏𝑥 𝑖̂ + 𝑏𝑦 𝑗̂ + 𝑏𝑧 𝑘̂ )
𝑎⃗ ∙ 𝑏⃗⃗ = 𝑎𝑥 𝑏𝑥 + 𝑎𝑦 𝑏𝑦 + 𝑎𝑧 𝑏𝑧
displacement
Average Acceleration
Instantaneous
Acceleration
4.4
∆𝑥
∆𝑡
4.8
𝑑𝑟⃗
= 𝑣𝑥 𝑖̂ + 𝑣𝑦 𝑗̂ + 𝑣𝑧 𝑘̂
𝑑𝑡
∆𝑣⃗
𝑎⃗𝑎𝑣𝑔 =
∆𝑡
𝑑𝑣⃗
𝑎⃗ =
𝑑𝑡
𝑎⃗ = 𝑎𝑥 𝑖̂ + 𝑎𝑦 𝑗̂ + 𝑎𝑧 𝑘̂
𝑣⃗ =
Projectile Motion
𝑣𝑦 = 𝑣0 𝑠𝑖𝑛𝜃0 − 𝑔𝑡
3.24
1
∆𝑦 = 𝑣0 𝑠𝑖𝑛𝜃𝑡 − 𝑔𝑡 2
2
2
2
𝑣𝑦 = (𝑣0 𝑠𝑖𝑛𝜃0 ) − 2𝑔∆y
𝑘̂
𝑎𝑧 |
𝑏𝑧
4.16
4.17
4.22
4.24
4.23
2
Trajectory
Range
3.26
𝑗
𝑎𝑦
𝑏𝑦
4.15
4.21
𝑣𝑦 = 𝑣0 𝑠𝑖𝑛𝜃0 − 𝑔𝑡
𝑐 = 𝑎𝑏𝑠𝑖𝑛𝜙
4.10
4.11
4.23
1
∆𝑥 = 𝑣0 𝑐𝑜𝑠𝜃𝑡 + 𝑎𝑥 𝑡 2
2
or ∆𝑥 = 𝑣0 𝑐𝑜𝑠𝜃𝑡 if 𝑎𝑥 =0
𝑎⃗ ∙ 𝑏⃗⃗
|𝑏|
or
𝑖̂
⃗⃗
𝑎⃗𝑥𝑏 = 𝑑𝑒𝑡 |𝑎𝑥
𝑏𝑥
∆𝑟⃗ = ∆𝑥𝑖̂ + ∆𝑦𝑗̂ + ∆𝑧𝑘̂
⃗⃗𝑎𝑣𝑔 =
𝑉
Instantaneous Velocity
3.22
𝑎⃗𝑥𝑏⃗⃗ = (𝑎𝑥 𝑖̂ + 𝑎𝑦 𝑗̂ + 𝑎𝑧 𝑘̂)𝑥(𝑏𝑥 𝑖̂ + 𝑏𝑦 𝑗̂ + 𝑏𝑧 𝑘̂ )
= (𝑎𝑦 𝑏𝑧 − 𝑏𝑦 𝑎𝑧 )𝑖̂ + (𝑎𝑧 𝑏𝑥 − 𝑏𝑧 𝑎𝑥 )𝑗̂
+ (𝑎𝑥 𝑏𝑦 − 𝑏𝑥 𝑎𝑦 )𝑘̂
Vector (cross product)
4.4
Average Velocity
𝑎𝑥 = 𝑎𝑐𝑜𝑠𝜃
𝑎𝑦 = 𝑎𝑠𝑖𝑛𝜃
Components of Vectors
𝑟⃗ = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧𝑘̂
Position vector
Relative Motion
Uniform Circular
Motion
𝑦 = (𝑡𝑎𝑛𝜃0 )𝑥 −
𝑅=
𝑔𝑥
2(𝑣0 𝑐𝑜𝑠𝜃0 )2
𝑣02
sin(2𝜃0 )
𝑔
⃗⃗⃗⃗⃗⃗⃗
𝑣𝐴𝐶 = ⃗⃗⃗⃗⃗⃗⃗
𝑣𝐴𝐵 + ⃗⃗⃗⃗⃗⃗⃗
𝑣𝐵𝐶
𝑎𝐴𝐵 = ⃗⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗
𝑎𝐵𝐴
𝑎=
𝑣2
𝑟
𝑇=
2𝜋𝑟
𝑣
4.25
4.26
4.44
4.45
4.34
4.35
General
Component form
Chapter 5
Chapter 6
Newton’s Second Law
Friction
𝐹⃗𝑛𝑒𝑡 = 𝑚𝑎⃗
𝐹𝑛𝑒𝑡,𝑥 = 𝑚𝑎𝑥
𝐹𝑛𝑒𝑡,𝑦 = 𝑚𝑎𝑦
𝐹𝑛𝑒𝑡,𝑧 = 𝑚𝑎𝑦
5.1
Kinetic Frictional
Weight
𝐹𝑔 = 𝑚𝑔
𝑊 = 𝑚𝑔
𝑓⃗𝑠,𝑚𝑎𝑥 = 𝜇𝑠 𝐹𝑁
6.1
𝑓⃗𝑘 = 𝜇𝑘 𝐹𝑁
6.2
5.2
Drag Force
Gravitational Force
Gravitational Force
Static Friction
(maximum)
Terminal velocity
1
𝐷 = 𝐶𝜌𝐴𝑣 2
2
2𝐹𝑔
𝑣𝑡 = √
𝐶𝜌𝐴
6.14
6.16
5.8
5.12
Centripetal
acceleration
Centripetal Force
𝑣2
𝑎=
𝑅
𝐹=
𝑚𝑣 2
𝑅
6.17
6.18
Chapter 7
Work- Kinetic Energy
Theorem
7.1
𝑊 = 𝐹𝑑𝑐𝑜𝑠𝜃 = 𝐹⃗ ∙ 𝑑⃗
7.7
7.8
Spring Force (Hooke’s
law)
Work done by spring
Work done by Variable
Force
Average Power
(rate at which that
force does work on an
object)
Instantaneous Power
𝑥𝑓
𝑦𝑓
𝑥𝑖
𝑦𝑖
𝑃𝑎𝑣𝑔 =
Mechanical Energy
𝐸𝑚𝑒𝑐 = 𝐾 + 𝑈
8.12
7.15
Principle of
conservation of
mechanical energy
𝐾1 + 𝑈1 = 𝐾2 + 𝑈2
𝐸𝑚𝑒𝑐 = ∆𝐾 + ∆𝑈 = 0
8.18
8.17
7.20
7.21
Force acting on particle
𝑧𝑓
𝑃=
7.36
𝑧𝑖
𝑊
∆𝑡
𝑑𝑊
= 𝐹𝑉𝑐𝑜𝑠𝜃 = 𝐹⃗ ∙ 𝑣⃗
𝑑𝑡
8.7
8.11
7.25
𝑊 = ∫ 𝐹𝑥 𝑑𝑥 + ∫ 𝐹𝑦 𝑑𝑦 + ∫ 𝐹𝑧 𝑑𝑧
∆𝑈 = 𝑚𝑔∆𝑦
1 2
𝑘𝑥
2
7.12
∆𝐾 = 𝑊𝑎 + 𝑊𝑔
𝑊𝑎 = 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝐹𝑜𝑟𝑐𝑒
𝐹⃗𝑠 = −𝑘𝑑⃗
𝐹𝑥 = −𝑘𝑥 (along x-axis)
Gravitational Potential
Energy
8.1
8.6
𝑈(𝑥) =
𝑊𝑔 = 𝑚𝑔𝑑𝑐𝑜𝑠𝜙
1 2 1 2
𝑘𝑥 − 𝑘𝑥
2 𝑖 2 𝑓
∆𝑈 = −𝑊 = − ∫ 𝐹(𝑥)𝑑𝑥
Elastic Potential Energy
7.10
𝑊𝑠 =
Potential Energy
𝑥𝑖
∆𝐾 = 𝐾𝑓 − 𝐾0 = 𝑊
Work done by gravity
Work done by
lifting/lowering object
𝑥𝑓
1
𝐾 = 𝑚𝑣 2
2
Kinetic Energy
Work done by constant
Force
Chapter 8
7.42
7.43
7.47
Work on System by
external force
With no friction
Work on System by
external force
With friction
Change in thermal
energy
Conservation of Energy
*if isolated W=0
𝐹(𝑥) = −
𝑑𝑈(𝑥)
𝑑𝑥
8.22
𝑊 = ∆𝐸𝑚𝑒𝑐 = ∆𝐾 + ∆𝑈
8.25
8.26
𝑊 = ∆𝐸𝑚𝑒𝑐 + ∆𝐸𝑡ℎ
8.33
∆𝐸𝑡ℎ = 𝑓𝑘 𝑑𝑐𝑜𝑠𝜃
8.31
𝑊 = ∆𝐸 = ∆𝐸𝑚𝑒𝑐 + ∆𝐸𝑡ℎ + ∆𝐸𝑖𝑛𝑡
8.35
Average Power
Instantaneous Power
**In General Physics, Kinetic Energy is abbreviated to KE and Potential Energy is PE
𝑃𝑎𝑣𝑔 =
𝑃=
∆𝐸
∆𝑡
𝑑𝐸
𝑑𝑡
8.40
8.41
Chapter 9
Impulse and Momentum
𝑡𝑓
Impulse
𝐽⃗ = ∫ 𝐹⃗ (𝑡)𝑑𝑡
𝑡𝑖
9.35
𝑝⃗ = 𝑚𝑣⃗
9.22
𝐽⃗ = Δ𝑝⃗ = 𝑝⃗𝑓 − 𝑝⃗𝑖
𝑑𝑝⃗
𝑑𝑡
𝐹⃗𝑛𝑒𝑡 = 𝑚⃗𝑎⃗𝑐𝑜𝑚
𝑃⃗⃗ = 𝑀𝑣⃗𝑐𝑜𝑚
𝑑⃗⃗⃗
𝑃
𝐹⃗𝑛𝑒𝑡 =
Newton’s 2nd law
System of Particles
9.30
𝐽 = 𝐹𝑛𝑒𝑡 ∆𝑡
Linear Momentum
Impulse-Momentum
Theorem
Collision continued…
𝐹⃗𝑛𝑒𝑡 =
9.31
9.32
Inelastic Collision
Conservation of Linear
Momentum (in 2D)
Average force
Elastic Collision
𝑛
𝑛
∆𝑝 = − 𝑚∆𝑣
∆𝑡
∆𝑡
∆𝑚
𝐹𝑎𝑣𝑔 = −
∆𝑣
∆𝑡
2𝑚1
=(
)𝑣
𝑚1 + 𝑚2 1𝑖
𝑛
9.14
9.25
9.27
Center of mass location
𝑟⃗𝑐𝑜𝑚
1
= ∑ 𝑚𝑖 𝑟⃗𝑖
𝑀
9.67
𝑖=1
𝑛
𝑣⃗𝑐𝑜𝑚 =
1
∑ 𝑚𝑖 𝑣⃗𝑖
𝑀
Rocket Equations
Thrust (Rvrel)
𝑅𝑣𝑟𝑒𝑙 = 𝑀𝑎
9.68
Change in velocity
9.42
9.43
𝑝⃗1𝑖 + 𝑝⃗2𝑖 = 𝑝⃗1𝑓 + 𝑝⃗2𝑓
9.50
9.51
9.78
𝐾1𝑖 + 𝐾2𝑖 = 𝐾1𝑓 + 𝐾2𝑓
9.8
𝑖=1
𝑃⃗⃗ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑃⃗⃗𝑖 = 𝑃⃗⃗𝑓
𝑚1 𝑣𝑖1 + 𝑚2 𝑣12 = 𝑚1 𝑣𝑓1 + 𝑚2 𝑣𝑓2
9.37
9.40
Center of Mass
𝑑𝑡
𝑚1 − 𝑚2
𝑣1𝑓 = (
)𝑣
𝑚1 + 𝑚2 1𝑖
9.77
9.22
Collision
𝑣2𝑓
𝑃⃗⃗1𝑖 + 𝑃⃗⃗2𝑖 = 𝑃⃗⃗1𝑓 + 𝑃⃗⃗2𝑓
𝐹𝑎𝑣𝑔 = −
Center of mass velocity
Final Velocity of 2
objects in a head-on
collision where one
object is initially at rest
1: moving object
2: object at rest
Conservation of Linear
Momentum (in 1D)
𝑚1 𝑣01 + 𝑚2 𝑣02 = (𝑚1 + 𝑚2 )𝑣𝑓
Δ𝑣 = 𝑣𝑟𝑒𝑙 𝑙𝑛
𝑀𝑖
𝑀𝑓
9.88
9.88
Chapter 10
Angular displacement
(in radians
Average angular
velocity
Instantaneous Velocity
Average angular
acceleration
Instantaneous angular
acceleration
𝑠
𝑟
Δ𝜃 = 𝜃2 − 𝜃1
∆𝜃
𝜔𝑎𝑣𝑔 =
∆𝑡
𝑑𝜃
𝜔=
𝑑𝑡
∆𝜔
𝛼𝑎𝑣𝑔 =
∆𝑡
𝑑𝜔
𝛼=
𝑑𝑡
10.1
10.4
𝜃=
Rotational Kinematics
𝜔 = 𝜔0 + 𝛼𝑡
1
Δ𝜃 = 𝜔0 𝑡 + 𝛼𝑡 2
2
2
2
𝜔 = 𝜔0 + 2𝛼Δ𝜃
1
Δ𝜃 = (𝜔 + 𝜔0 )𝑡
2
1
Δ𝜃 = 𝜔𝑡 − 𝛼𝑡 2
2
10.5
10.6
10.7
10.8
10.13
10.14
10.15
𝐼 = 𝐼𝑐𝑜𝑚 + 𝑀ℎ2
10.36
𝜏 = 𝑟𝐹𝑡 = 𝑟⊥ 𝐹 = 𝑟𝐹𝑠𝑖𝑛𝜃
10.3910.41
Newton’s Second Law
𝜏𝑛𝑒𝑡 = 𝐼𝛼
10.45
Rotational work done
by a toque
𝑊 = ∫ 𝜏𝑑𝜃
Torque
Rotational Kinetic
Energy
Work-kinetic energy
theorem
𝑣 = 𝜔𝑟
10.18
Tangential Acceleration
𝑎𝑡 = 𝛼𝑟
10.19
2
𝑣
= 𝜔2 𝑟
𝑟
10.23
2𝜋𝑟 2𝜋
=
𝑣
𝜔
10.19
10.20
𝑇=
10.35
Power in rotational
motion
Velocity
Period
𝐼 = ∫ 𝑟 2 𝑑𝑚
Rotation inertia
(discrete particle
system)
Parallel Axis Theorem
h=perpendicular
distance between two
axes
𝜃𝑓
Relationship Between Angular and Linear Variables
𝑎𝑟 =
10.34
10.12
10.16
Radical component of 𝑎⃗
𝐼 = ∑ 𝑚𝑖 𝑟𝑖2
Rotation inertia
𝜃𝑖
𝑊 = 𝜏∆𝜃 (𝜏 constant)
𝑑𝑊
𝑃=
= 𝜏𝜔
𝑑𝑡
1
𝐾 = 𝐼𝜔2
2
1
1
∆𝐾 = 𝐾𝑓 − 𝐾𝑖 = 𝐼𝜔𝑓2 − 𝐼𝜔𝑖2 = 𝑊
2
2
10.53
10.54
10.55
10.34
10.52
Moments of Inertia I for various rigid objects of Mass M
Thin walled hollow cylinder or hoop
about central axis
Annular cylinder (or ring) about
central axis
𝐼 = 𝑀𝑅 2
1
𝐼 = 𝑀(𝑅12 + 𝑅22 )
2
Solid Sphere, axis through center
2
𝐼 = 𝑀𝑅 2
5
Thin rod, axis perpendicular to rod
and passing though end
Solid Sphere, axis tangent to surface
7
𝐼 = 𝑀𝑅 2
5
Thin Rectangular sheet (slab), axis
parallel to sheet and passing though
center of the other edge
Solid cylinder or disk about central
axis
1
𝐼 = 𝑀𝑅 2
2
Thin Walled spherical shell, axis
through center
Solid cylinder or disk about central
diameter
1
1
𝐼 = 𝑀𝑅 2 + 𝑀𝐿2
4
12
Thin rod, axis perpendicular to rod
and passing though center
2
𝐼 = 𝑀𝑅 2
3
Thin Rectangular sheet (slab_, axis
along one edge
𝐼=
1
𝑀𝐿2
12
Thin rectangular sheet (slab) about
perpendicular axis through center
1
𝐼 = 𝑀𝐿2
3
𝐼=
1
𝑀𝐿2
12
1
𝐼 = 𝑀𝐿2
3
𝐼=
1
𝑀(𝑎2 + 𝑏 2 )
12
Chapter 11
Rolling Bodies (wheel)
Speed of rolling wheel
Kinetic Energy of Rolling
Wheel
Acceleration of rolling
wheel
Acceleration along x-axis
extending up the ramp
𝑣𝑐𝑜𝑚 = 𝜔𝑅
11.2
Angular Momentum
1
1
2
𝐾 = 𝐼𝑐𝑜𝑚 𝜔2 + 𝑀𝑣𝑐𝑜𝑚
2
2
11.5
Magnitude of Angular
Momentum
𝑎𝑐𝑜𝑚 = 𝛼𝑅
11.6
𝑎𝑐𝑜𝑚,𝑥
𝑔𝑠𝑖𝑛𝜃
=−
𝐼
1 + 𝑐𝑜𝑚2
𝑀𝑅
11.10
Magnitude of torque
Newton’s 2nd Law
𝜏⃗ = 𝑟⃗ × 𝐹⃗
11.14
𝜏 = 𝑟𝐹⊥ = 𝑟⊥ 𝐹 = 𝑟𝐹𝑠𝑖𝑛𝜙
11.1511.17
𝜏⃗𝑛𝑒𝑡 =
⃗⃗
𝑑ℓ
𝑑𝑡
ℓ = 𝑟𝑚𝑣𝑠𝑖𝑛𝜙
ℓ = 𝑟𝑝⊥ = 𝑟𝑚𝑣⊥
11.18
11.1911.21
𝑛
Angular momentum of a
system of particles
⃗⃗ = ∑ ⃗⃗
𝐿
ℓ𝑖
𝑖=1
𝜏⃗𝑛𝑒𝑡 =
Torque as a vector
Torque
Angular Momentum
⃗⃗ × ⃗𝑣⃗)
𝑣 ⃗ℓ⃗ = ⃗𝑟⃗ × ⃗𝑝⃗ = 𝑚(𝑟
⃗⃗
𝑑𝐿
𝑑𝑡
Angular Momentum continued
Angular Momentum of a
𝐿 = 𝐼𝜔
rotating rigid body
Conservation of angular
𝐿⃗⃗ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
momentum
𝐿⃗⃗𝑖 = 𝐿⃗⃗𝑓
11.23
11.26
11.29
11.31
11.32
11.33
Precession of a Gyroscope
Precession rate
Ω=
𝑀𝑔𝑟
𝐼𝜔
11.31
Chapter 12
Chapter 13
Static Equilibrium
12.3
Gravitational Force
(Newton’s law of
gravitation)
𝜏⃗𝑛𝑒𝑡 = 0
12.5
Principle of
Superposition
𝐹⃗𝑛𝑒𝑡,𝑥 = 0, 𝐹⃗𝑛𝑒𝑡,𝑦 = 0
12.7
12.8
𝐹⃗𝑛𝑒𝑡 = 0
If forces lie on the
xy-plane
Stress (force per unit
area)
Strain (fractional
change in length)
Stress (pressure)
Tension/Compression
E: Young’s modulus
Shearing Stress
G: Shear modulus
Hydraulic Stress
B: Bulk modulus
𝜏⃗𝑛𝑒𝑡,𝑧 = 0
𝑠𝑡𝑟𝑒𝑠𝑠 = 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 × 𝑠𝑡𝑟𝑎𝑖𝑛
𝐹
𝑃=
𝐴
𝐹
∆𝐿
=𝐸
𝐴
𝐿
𝐹
∆𝑥
=𝐺
𝐴
𝐿
∆𝑉
𝑝=𝐵
𝑉
12.9
12.22
𝐹=𝐺
𝐹⃗1,𝑛𝑒𝑡 = ∑ 𝐹⃗1𝑖
𝐹⃗1 = ∫ 𝑑𝐹⃗
Escape Speed
Kepler’s 3rd Law
(law of periods)
Energy for bject in
circular orbit
13.6
𝐺𝑀
𝑟2
𝐺𝑚𝑀
𝐹=
𝑟
𝑅3
𝐺𝑀𝑚
𝑈=−
𝑟
13.11
𝑎𝑔 =
Gravitational Potential
Energy
12.24
𝑈 = −(
13.19
13.21
𝐺𝑚1 𝑚2 𝐺𝑚1 𝑚3 𝐺𝑚2 𝑚3
+
+
)
𝑟12
𝑟13
𝑟23
2𝐺𝑀
𝑣=√
𝑅
𝑇2 = (
𝑈=−
4𝜋 2 3
)𝑟
𝐺𝑀
𝐺𝑀𝑚
𝑟
𝐾=
𝐸=−
13.22
13.28
𝐺𝑀𝑚
2𝑟
𝐺𝑀𝑚
2𝑟
𝐺𝑀𝑚
Mechanical Energy
𝐸
=
−
(elliptical orbit)
2𝑎
−11
*Note: 𝐺 = 6.6704 × 10
𝑁 ∙ 𝑚2 /𝑘𝑔2
Mechanical Energy
(circular orbit)
13.5
𝑖=2
Gravitation within a
spherical Shell
12.23
13.1
𝑛
Gravitational Force
acting on a particle
from an extended
body
Gravitational
acceleration
Potential energy on a
system (3 particles)
𝑚1 𝑚2
𝑟2
13.34
13.21
13.38
13.40
13.42
Chapter 14
Density
∆𝑚
∆𝑉
𝑚
𝜌=
𝑉
𝜌=
Pressure and depth in
a static Fluid
P1 is higher than P2
Gauge Pressure
Archimedes’ principle
Mass Flow Rate
Volume flow rate
Bernoulli’s Equation
Equation of continuity
Equation of continuity
when
14.1
14.2
∆𝐹
∆𝐴
𝐹
𝑝=
𝐴
14.3
14.4
𝑝2 = 𝑝1 + 𝜌𝑔(𝑦1 − 𝑦2 )
𝑝 = 𝑝0 + 𝜌𝑔ℎ
14.7
14.8
𝑝=
Pressure
Chapter 15
Angular frequency
Acceleration
Kinetic and Potential
Energy
𝐹𝑏 = 𝑚𝑓 𝑔
14.16
𝑅𝑚 = 𝜌𝑅𝑉 = 𝜌𝐴𝑣
14.25
𝑅𝑉 = 𝐴𝑣
14.24
𝑅𝑉 = 𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
displacement
Velocity
𝜌𝑔ℎ
1
𝑝 + 𝜌𝑣 2 + 𝜌𝑔𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
2
𝑅𝑚 = 𝜌𝑅𝑉 = 𝜌𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Frequency
cycles per time
14.29
Angular frequency
𝑓=
1
𝑇
15.2
𝑥 = 𝑥𝑚 cos(𝜔𝑡 + 𝜙)
𝜔=
2𝜋
= 2𝜋𝑓
𝑇
15.3
15.5
𝑣 = −𝜔𝑥𝑚 sin(𝜔𝑡 + 𝜙)
15.6
𝑎 = −𝜔2 𝑥𝑚 cos(𝜔𝑡 + 𝜙)
15.7
1
1
𝐾 = 𝑚𝑣 2 𝑈 = 𝑘𝑥 2
2
2
𝑘
𝜔=√
𝑚
15.12
Period
𝑇 = 2𝜋√
𝑚
𝑘
15.13
Torsion pendulum
𝐼
𝑇 = 2𝜋√
𝑘
15.23
Simple Pendulum
𝐿
𝑇 = 2𝜋√
𝑔
15.28
Physical Pendulum
𝐼
𝑇 = 2𝜋√
𝑚𝑔𝐿
15.29
14.25
14.24
Damping force
𝐹⃗𝑑 = −𝑏𝑣⃗
displacement
𝑥(𝑡) = 𝑥𝑚 𝑒 −2𝑚 cos(𝜔′ 𝑡 + 𝜙)
15.42
Angular frequency
𝑘
𝑏2
𝜔′ = √ −
𝑚 4𝑚2
15.43
Mechanical Energy
1 2 −𝑏𝑡
𝐸(𝑡) ≈ 𝑘𝑥𝑚
𝑒 𝑚
2
15.44
𝑏𝑡
Chapter 16
Sinusoidal Waves
Mathematical form
(positive direction)
Angular wave number
Angular frequency
Wave speed
Average Power
𝑦(𝑥, 𝑡) = 𝑦𝑚 sin(𝑘𝑥 − 𝜔𝑡)
2𝜋
𝑘=
𝜆
2𝜋
𝜔=
= 2𝜋𝑓
𝑇
𝑣=
𝜔 𝜆
= = 𝜆𝑓
𝑘 𝑇
1
2
𝑃𝑎𝑣𝑔 = 𝜇𝑣𝜔2 𝑦𝑚
2
Traveling Wave Form
16.2
16.5
𝑦(𝑥, 𝑡) = ℎ(𝑘𝑥 ± 𝜔𝑡)
16.17
𝜏
𝑣=√
𝜇
16.26
1
1
𝑦 ′ (𝑥, 𝑡) = [2𝑦𝑚 cos ( 𝜙)] sin (𝑘𝑥 − 𝜔𝑡 + 𝜙)
2
2
16.51
𝑦 ′ (𝑥, 𝑡) = [2𝑦𝑚 sin(𝑘𝑥)]cos(𝜔𝑡)
16.60
Wave speed on
stretched string
16.9
Resulting wave when 2
waves only differ by
phase constant
16.13
Standing wave
16.33
Resonant frequency
𝑣
𝑣
𝑓 = 𝜆 = 𝑛 2𝐿 for n=1,2,…
16.66
Chapter 17
Sound Waves
Standing Waves Patterns in Pipes
𝐵
𝑣=√
𝜌
17.3
𝑠 = 𝑠𝑚 cos(𝑘𝑥 − 𝜔𝑡)
17.12
Change in pressure
Δ𝑝 = Δ𝑝𝑚 sin(𝑘𝑥 − 𝜔𝑡)
17.13
Standing wave
frequency (open at
both ends)
Standing wave
frequency (open at
one end)
Pressure amplitude
Δ𝑝𝑚 = (𝑣𝜌𝜔)𝑠𝑚
17.14
beats
Speed of sound wave
displacement
𝑣
𝑓=𝜆=
𝑣
𝑓=𝜆=
𝑛𝑣
2𝐿
𝑛𝑣
4𝐿
for n=1,2,3
17.39
for n=1,3,5
17.41
𝑓𝑏𝑒𝑎𝑡 = 𝑓1 − 𝑓2
17.46
Interference
Phase difference
Fully Constructive
Interference
Full Destructive
interference
Mechanical Energy
Δ𝐿
2𝜋
𝜆
𝜙 = 𝑚(2𝜋) for m=0,1,2…
Δ𝐿
= 0,1,2
𝜆
𝜙 = (2𝑚 + 1)𝜋 for m=0,12
Δ𝐿
= .5,1.5,2.5 …
𝜆
𝜙=
1 2 −𝑏𝑡
𝐸(𝑡) ≈ 𝑘𝑥𝑚
𝑒 𝑚
2
17.21
17.22
17.23
17.24
17.25
15.44
Doppler Effect
Source Moving toward
stationary observer
Source Moving away
from stationary
observer
Observer moving
toward stationary
source
Observer moving away
from stationary source
Sound Intensity
𝐼=
Intensity
Intensity -uniform in
all directions
𝑃
𝐴
1
2
𝐼 = 𝜌𝑣𝜔2 𝑠𝑚
2
𝐼=
𝑃𝑠
4𝜋𝑟 2
𝑓′ = 𝑓
𝑣
𝑣 − 𝑣𝑠
17.53
𝑓′ = 𝑓
𝑣
𝑣 + 𝑣𝑠
17.54
𝑣 + 𝑣𝐷
𝑣
𝑣 − 𝑣𝐷
𝑓′ = 𝑓
𝑣
𝑓′ = 𝑓
Shockwave
17.26
17.27
17.29
Intensity level in
decibels
𝐼
𝛽 = (10𝑑𝐵) log ( )
𝐼𝑜
17.29
Mechanical Energy
1 2 −𝑏𝑡
𝐸(𝑡) ≈ 𝑘𝑥𝑚
𝑒 𝑚
2
15.44
Half-angle 𝜃 of Mach
cone
𝑠𝑖𝑛𝜃 =
𝑣
𝑣𝑠
17.49
17.51
17.57
Chapter 18
Temperature Scales
Fahrenheit to Celsius
Celsius to Fahrenheit
5
𝑇𝐶 = (𝑇𝐹 − 32)
9
9
𝑇𝐹 = 𝑇𝐶 + 32
5
𝑇 = 𝑇𝐶 + 273.15
Celsius to Kelvin
First Law of Thermodynamics
18.8
18.8
18.7
∆𝐸𝑖𝑛𝑡 = 𝐸𝑖𝑛𝑡,𝑓 − 𝐸𝑖𝑛𝑡,𝑖 = 𝑄 − 𝑊
First Law of
18.26
Thermodynamics
18.27
𝑑𝐸𝑖𝑛𝑡 = 𝑑𝑄 − 𝑑𝑊
Note:
∆𝐸𝑖𝑛𝑡 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
Thermal Expansion
Applications of First Law
Linear Thermal Expansion
∆𝐿 = 𝐿𝛼∆𝑇
18.9
Volume Thermal Expansion
∆𝑉 = 𝑉𝛽∆𝑇
18.10
Q=0
∆𝐸𝑖𝑛𝑡 = −𝑊
Adiabatic
(no heat flow)
W=0
∆𝐸𝑖𝑛𝑡 = 𝑄
∆𝐸𝑖𝑛𝑡 = 0
Q=W
𝑄 = 𝑊 = ∆𝐸𝑖𝑛𝑡 = 0
(constant volume)
Heat
Heat and temperature
change
𝑄 = 𝐶(𝑇𝑓 − 𝑇𝑖 )
𝑄 = 𝑐𝑚(𝑇𝑓 − 𝑇𝑖 )
18.13
18.14
Heat and phase change
𝑄 = 𝐿𝑚
18.16
Power (Conducted)
𝑃𝑐𝑜𝑛𝑑
𝑄
𝑇𝐻 − 𝑇𝐶
= = 𝑘𝐴
𝑡
𝐿
Free expansions
Misc.
𝑉𝑓
P=Q/t
Power
Cyclical process
18.32
Rate objects absorbs
energy
4
𝑃𝑎𝑏𝑠 = 𝜎𝜖𝐴𝑇𝑒𝑛𝑣
18.39
Power from radiation
𝑃𝑟𝑎𝑑 = 𝜎𝜖𝐴𝑇 4
18.38
Work Associated with
Volume Change
𝑊 = ∫ 𝑑𝑊 = ∫ 𝑝𝑑𝑉
𝑉𝑖
18.25
𝑊 = 𝑝∆𝑣
𝜎 = 5.6704 × 10−8 𝑊/𝑚2 ∙ 𝐾 4
Revised 7/20/17