PHYS 2310 Engineering Physics I Formula Sheets Chapters 1-18 PDF

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PHYS 2310 Engineering Physics I Formula Sheets Chapters 1-18

Chapter 1/Important Numbers

Chapter 2

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°

Velocity

1 × 106 𝜇𝑚 2.54 cm 86400 seconds 3600 seconds 2𝜋 rad

Circumference Surface area (sphere)

𝑆𝐴 =

Volume (rectangular solid)

4𝜋𝑟 2

𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 ∆𝑥 = 𝑡𝑖𝑚𝑒 ∆𝑡

𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑖𝑚𝑒 ∆𝑥 𝑑𝑥 𝑣 = lim = ∆𝑡→0 ∆𝑡 𝑑𝑡

𝑠𝑎𝑣𝑔 =

Average Speed

Important Constants/Measurements Mass of Earth 5.98 × 1024 kg 6.38 × 106 m Radius of Earth 1.661 × 10−27 kg 1 u (Atomic Mass Unit) 1 𝑔/𝑐𝑚3 or 1000 𝑘𝑔/𝑚3 Density of water 9.8 m/s2 g (on earth) Density Common geometric Formulas Area circle 𝐶 = 2𝜋𝑟

𝑉𝑎𝑣𝑔 =

Average Velocity

Instantaneous Velocity

∆𝑣 ∆𝑡 𝑑𝑣 𝑑 2 𝑥 = 𝑎= 𝑑𝑡 𝑑𝑡 2 𝑎𝑎𝑣𝑔 =

Average Acceleration

Motion of a particle with constant acceleration

𝐴= 4 Volume (sphere) 𝑉 = 𝜋𝑟 3 3 𝑉 =𝑙∙𝑤∙ℎ 𝑉 = 𝑎𝑟𝑒𝑎 ∙ 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠

2.3 2.4

Acceleration

Instantaneous Acceleration

𝜋𝑟 2

2.2

𝑣 = 𝑣0 + 𝑎𝑡 1 ∆𝑥 = (𝑣0 + 𝑣)𝑡 2 1 ∆𝑥 = 𝑣0 𝑡 + 𝑎𝑡 2 2 𝑣2

=

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) Components of Vectors Magnitude of vector Angle between x axis and vector Unit vector notation Adding vectors in Component Form Scalar (dot product) Scalar (dot product) Projection of 𝑎 𝑜𝑛 𝑏󰇍 or component of 𝑎 𝑜𝑛 𝑏󰇍 Vector (cross) product magnitude

Vector (cross product)

Chapter 4

󰇍 = 󰇍𝑏 + 𝑎 𝑎 + 𝑏

3.2

(𝑎 + 𝑏󰇍) + 𝑐 = 𝑎 + (𝑏󰇍 + 𝑐) 𝑎𝑥 = 𝑎𝑐𝑜𝑠𝜃 𝑎𝑦 = 𝑎𝑠𝑖𝑛𝜃 𝑎𝑦 𝑡𝑎𝑛𝜃 = 𝑎𝑥

𝑎 = 𝑎𝑥 𝑖 + 𝑎𝑦 𝑗 + 𝑎𝑧 𝑘 𝑟𝑥 = 𝑎𝑥 + 𝑏𝑥 𝑟𝑦 = 𝑎𝑦 + 𝑏𝑦 𝑟𝑧 = 𝑎𝑧 + 𝑏𝑧

𝑎 ∙ 󰇍𝑏 = 𝑎𝑏𝑐𝑜𝑠𝜃

𝑎 ∙ 𝑏󰇍 = 𝑎𝑥 𝑏𝑥 + 𝑎𝑦 𝑏𝑦 + 𝑎𝑧 𝑏𝑧 𝑎 ∙ 󰇍𝑏 |𝑏|

𝑐 = 𝑎𝑏𝑠𝑖𝑛𝜙

𝑎𝑥𝑏󰇍 = (𝑎𝑥 𝑖 + 𝑎𝑦 𝑗 + 𝑎𝑧 𝑘 )𝑥(𝑏𝑥 𝑖 + 𝑏𝑦 𝑗 + 𝑏𝑧 𝑘 ) = (𝑎𝑦 𝑏𝑧 − 𝑏𝑦 𝑎𝑧 )𝑖 + (𝑎𝑧 𝑏𝑥 − 𝑏𝑧 𝑎𝑥 )𝑗 + (𝑎𝑥 𝑏𝑦 − 𝑏𝑥 𝑎𝑦 )𝑘

𝑗 𝑎𝑦 𝑏𝑦

3.6 3.6 3.7

Instantaneous Velocity Average Acceleration Instantaneous Acceleration

3.10 3.11 3.12

𝑘 𝑎𝑧 | 𝑏𝑧

Trajectory Range

3.26

∆𝑥 ∆𝑡

𝑑𝑟 = 𝑣𝑥 𝑖 + 𝑣𝑦 𝑗 + 𝑣𝑧 𝑘 𝑑𝑡 ∆𝑣 𝑎𝑎𝑣𝑔 = ∆𝑡 𝑑𝑣 𝑎 = 𝑑𝑡 𝑎 = 𝑎𝑥 𝑖 + 𝑎𝑦 𝑗 + 𝑎𝑧 𝑘

𝑣 =

Relative Motion Uniform Circular Motion

4.4 4.4 4.8 4.10 4.11 4.15 4.16 4.17

1 ∆𝑥 = 𝑣0 𝑐𝑜𝑠𝜃𝑡 + 𝑎𝑥 𝑡 2 2 or ∆𝑥 = 𝑣0 𝑐𝑜𝑠𝜃𝑡 if 𝑎𝑥 =0

4.23

𝑣𝑦 = 𝑣0 𝑠𝑖𝑛𝜃0 − 𝑔𝑡

4.24

4.21

1 ∆𝑦 = 𝑣0 𝑠𝑖𝑛𝜃𝑡 − 𝑔𝑡 2 2 𝑣𝑦2 = (𝑣0 𝑠𝑖𝑛𝜃0 )2 − 2𝑔∆y

3.22

3.24

𝑉󰇍𝑎𝑣𝑔 =

Projectile Motion 𝑣𝑦 = 𝑣0 𝑠𝑖𝑛𝜃0 − 𝑔𝑡

3.20

𝑎 ∙ 𝑏󰇍 = (𝑎𝑥 𝑖 + 𝑎𝑦 𝑗 + 𝑎𝑧 𝑘 ) ∙ (𝑏𝑥 𝑖 + 𝑏𝑦 𝑗 + 𝑏𝑧 𝑘)

∆𝑟 = ∆𝑥𝑖 + ∆𝑦𝑗 + ∆𝑧𝑘

displacement Average Velocity

3.5

|𝑎| = 𝑎 = √𝑎𝑥2 + 𝑎2𝑦

or 𝑖 󰇍 = 𝑑𝑒𝑡 |𝑎𝑥 𝑎𝑥𝑏 𝑏𝑥

3.3

𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘

Position vector

4.22

𝑔𝑥 2 𝑦 = (𝑡𝑎𝑛𝜃0 )𝑥 − 2(𝑣0 𝑐𝑜𝑠𝜃0 )2 2 𝑣0 sin(2𝜃0 ) 𝑅= 𝑔

𝑣𝐴𝐶 = 𝑣󰇍󰇍󰇍󰇍󰇍 󰇍󰇍󰇍󰇍󰇍󰇍 󰇍󰇍󰇍󰇍 𝐴𝐵 + 󰇍𝑣 𝐵𝐶 󰇍󰇍󰇍󰇍󰇍󰇍 󰇍󰇍󰇍󰇍󰇍󰇍  𝑎 𝐵𝐴 𝐴𝐵 = 𝑎 𝑎=

𝑣2 𝑟

𝑇=

2𝜋𝑟 𝑣

4.23 4.25 4.26

4.44 4.45 4.34 4.35

Chapter 5

Chapter 6 Friction

Newton’s Second Law General

Component form

𝐹𝑛𝑒𝑡 = 𝑚𝑎

𝐹𝑛𝑒𝑡,𝑥 = 𝑚𝑎𝑥 𝐹𝑛𝑒𝑡,𝑦 = 𝑚𝑎𝑦 𝐹𝑛𝑒𝑡,𝑧 = 𝑚𝑎𝑦

5.1

Kinetic Frictional 5.2 Drag Force

Gravitational Force Gravitational Force Weight

𝐹𝑔 = 𝑚𝑔

𝑊 = 𝑚𝑔

Static Friction (maximum)

Terminal velocity 5.8

5.12

Centripetal acceleration Centripetal Force

𝑓𝑠,𝑚𝑎𝑥 = 𝜇𝑠 𝐹𝑁 𝑓𝑘 = 𝜇𝑘 𝐹𝑁

𝐷=

1

2

𝐶𝜌𝐴𝑣 2

𝑣𝑡 = √

𝑎=

𝐹=

2𝐹𝑔 𝐶𝜌𝐴

𝑣2 𝑅

𝑚𝑣 2 𝑅

6.1 6.2

6.14 6.16

6.17 6.18

Chapter 7 1 2 𝐾 = 2 𝑚𝑣

Kinetic Energy

𝑊 = 𝐹𝑑𝑐𝑜𝑠𝜃 = 𝐹 ∙ 𝑑

Work done by constant Force

∆𝐾 = 𝐾𝑓 − 𝐾0 = 𝑊

Work- Kinetic Energy Theorem

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

1 2 1 2 𝑘𝑥 − 𝑘𝑥 2 𝑖 2 𝑓 𝑦𝑓

𝑧𝑓

𝑊 = ∫ 𝐹𝑥 𝑑𝑥 + ∫ 𝐹𝑦 𝑑𝑦 + ∫ 𝐹𝑧 𝑑𝑧 𝑥𝑖

𝑃=

𝑦𝑖

𝑃𝑎𝑣𝑔 =

𝑊 ∆𝑡

Potential Energy

7.7 7.8

Gravitational Potential Energy

7.12

∆𝐾 = 𝑊𝑎 + 𝑊𝑔 𝑊𝑎 = 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝐹𝑜𝑟𝑐𝑒 𝐹𝑠 = −𝑘𝑑 𝐹𝑥 = −𝑘𝑥 (along x-axis)

𝑊𝑠 =

7.1

7.10

𝑊𝑔 = 𝑚𝑔𝑑𝑐𝑜𝑠𝜙

Work done by gravity Work done by lifting/lowering object

Chapter 8

𝑧𝑖

𝑑𝑊 = 𝐹𝑉𝑐𝑜𝑠𝜃 = 𝐹 ∙ 𝑣 𝑑𝑡

Mechanical Energy

7.15 7.20 7.21

Force acting on particle

7.36

7.42 7.43 7.47

𝑥𝑖

∆𝑈 = 𝑚𝑔∆𝑦

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

1 2 𝑘𝑥 2

𝑈(𝑥) =

Elastic Potential Energy

Principle of conservation of mechanical energy

7.25

𝑥𝑓

∆𝑈 = −𝑊 = − ∫ 𝐹(𝑥)𝑑𝑥

𝐸𝑚𝑒𝑐 = 𝐾 + 𝑈

𝐾1 + 𝑈1 = 𝐾2 + 𝑈2 𝐸𝑚𝑒𝑐 = ∆𝐾 + ∆𝑈 = 0 𝐹(𝑥) = −

𝑑𝑈(𝑥) 𝑑𝑥

𝑊 = ∆𝐸𝑚𝑒𝑐 = ∆𝐾 + ∆𝑈 𝑊 = ∆𝐸𝑚𝑒𝑐 + ∆𝐸𝑡ℎ ∆𝐸𝑡ℎ = 𝑓𝑘 𝑑𝑐𝑜𝑠𝜃

𝑊 = ∆𝐸 = ∆𝐸𝑚𝑒𝑐 + ∆𝐸𝑡ℎ + ∆𝐸𝑖𝑛𝑡

Average Power Instantaneous Power

**In General Physics, Kinetic Energy is abbreviated to KE and Potential Energy is PE

𝑃𝑎𝑣𝑔 =

𝑃=

∆𝐸 ∆𝑡

𝑑𝐸 𝑑𝑡

8.1 8.6 8.7 8.11 8.12 8.18 8.17 8.22 8.25 8.26 8.33 8.31 8.35 8.40 8.41

Chapter 9 Impulse and Momentum

Impulse

𝑡𝑓

𝐽 = ∫ 𝐹 (𝑡)𝑑𝑡 𝑡𝑖

𝐽 = 𝐹𝑛𝑒𝑡 ∆𝑡

𝑝 = 𝑚𝑣

Linear Momentum Impulse-Momentum Theorem

𝐽 = Δ𝑝 = 𝑝𝑓 − 𝑝𝑖 𝑑𝑝 𝑑𝑡  𝐹𝑛𝑒𝑡 = 𝑚𝑎󰇍𝑐𝑜𝑚 󰇍 = 𝑀𝑣𝑐𝑜𝑚 𝑃 󰇍 𝑑𝑃 𝐹𝑛𝑒𝑡 =

Newton’s 2nd law

System of Particles

𝐹𝑛𝑒𝑡 =

𝑑𝑡

Collision continued… 9.30 9.35

Inelastic Collision Conservation of Linear Momentum (in 2D)

9.22 9.31 9.32

Average force

9.22

Elastic Collision

𝑣1𝑓 𝑣2𝑓

𝑚1 − 𝑚2 )𝑣 =( 𝑚1 + 𝑚2 1𝑖

2𝑚1 )𝑣 =( 𝑚1 + 𝑚2 1𝑖

󰇍 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑃 𝑃󰇍𝑖 = 𝑃󰇍𝑓

𝑝1𝑖 + 𝑝2𝑖 = 𝑝1𝑓 + 𝑝2𝑓

𝑚1 𝑣𝑖1 + 𝑚2 𝑣12 = 𝑚1 𝑣𝑓1 + 𝑚2 𝑣𝑓2

𝐾1𝑖 + 𝐾2𝑖 = 𝐾1𝑓 + 𝐾2𝑓

𝑃󰇍1𝑖 + 𝑃󰇍2𝑖 = 𝑃󰇍1𝑓 + 𝑃󰇍2𝑓

𝑛 𝑛 ∆𝑝 = − 𝑚∆𝑣 ∆𝑡 ∆𝑡 ∆𝑚 𝐹𝑎𝑣𝑔 = − ∆𝑣 ∆𝑡

𝐹𝑎𝑣𝑔 = −

Center of Mass 9.14 9.25 9.27

9.67

𝑟𝑐𝑜𝑚

Center of mass velocity

𝑣𝑐𝑜𝑚 =

Change in velocity

9.50 9.51 9.78

𝑖=1 𝑛

9.77

9.37 9.40

9.8

1 ∑ 𝑚𝑖 𝑣𝑖 𝑀 𝑖=1

𝑅𝑣𝑟𝑒𝑙 = 𝑀𝑎

Rocket Equations Thrust (Rvrel)

9.68

9.42 9.43

𝑛

1 ∑ 𝑚𝑖 𝑟𝑖 = 𝑀

Center of mass location

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 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 𝜃=𝑟 Δ𝜃 = 𝜃2 − 𝜃1 ∆𝜃 𝜔𝑎𝑣𝑔 = ∆𝑡 𝑑𝜃 𝜔= 𝑑𝑡 ∆𝜔 𝛼𝑎𝑣𝑔 = ∆𝑡 𝑑𝜔 𝛼= 𝑑𝑡 Rotational Kinematics 𝜔 = 𝜔0 + 𝛼𝑡 1 Δ𝜃 = 𝜔0 𝑡 + 𝛼𝑡 2 2 𝜔2 = 𝜔20 + 2𝛼Δ𝜃

10.1 10.4 10.5 10.6 10.7 10.8

Rotational work done by a toque

10.14

Power in rotational motion

10.15

Rotational Kinetic Energy

10.16

Work-kinetic energy theorem

Relationship Between Angular and Linear Variables

Radical component of 𝑎 Period

𝑣 = 𝜔𝑟

𝑎𝑡 = 𝛼𝑟

𝑎𝑟 =

𝑇=

𝑣2 𝑟

Torque Newton’s Second Law

10.13

1 Δ𝜃 = 𝜔𝑡 − 𝛼𝑡 2 2

Tangential Acceleration

Rotation inertia (discrete particle system) Parallel Axis Theorem h=perpendicular distance between two axes

10.12

1 Δ𝜃 = (𝜔 + 𝜔0 )𝑡 2

Velocity

Rotation inertia

= 𝜔2 𝑟

2𝜋𝑟 2𝜋 = 𝑣 𝜔

10.18 10.19 10.23 10.19 10.20

𝐼 = ∑ 𝑚𝑖 𝑟𝑖2

10.34

𝐼 = ∫ 𝑟2 𝑑𝑚

10.35

𝐼 = 𝐼𝑐𝑜𝑚 + 𝑀ℎ2

10.36

𝜏 = 𝑟𝐹𝑡 = 𝑟⊥ 𝐹 = 𝑟𝐹𝑠𝑖𝑛𝜃

10.3910.41

𝜏𝑛𝑒𝑡 = 𝐼𝛼 𝜃𝑓

𝑊 = ∫ 𝜏𝑑𝜃 𝜃𝑖

𝑊 = 𝜏∆𝜃 (𝜏 constant) 𝑑𝑊 = 𝜏𝜔 𝑃= 𝑑𝑡 1 𝐾 = 𝐼𝜔2 2 1 2 1 2 ∆𝐾 = 𝐾𝑓 − 𝐾𝑖 = 𝐼𝜔𝑓 − 𝐼𝜔𝑖 = 𝑊 2 2

10.45 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

Solid cylinder or disk about central axis

𝐼 = 𝑀𝑅 2

1 𝐼 = 𝑀(𝑅21 + 𝑅22 ) 2

1 𝐼 = 𝑀𝑅2 2

Solid Sphere, axis through center

Solid Sphere, axis tangent to surface

Thin Walled spherical shell, axis through center

2 𝐼 = 𝑀𝑅 2 5

Thin rod, axis perpendicular to rod and passing though end

𝐼=

1 𝑀𝐿2 3

7 𝐼 = 𝑀𝑅 2 5

Thin Rectangular sheet (slab), axis parallel to sheet and passing though center of the other edge

1 𝑀𝐿2 𝐼= 12

Solid cylinder or disk about central diameter

1 1 𝑀𝐿2 𝐼 = 𝑀𝑅 2 + 12 4

Thin rod, axis perpendicular to rod and passing though center

2 𝐼 = 𝑀𝑅2 3

Thin Rectangular sheet (slab_, axis along one edge

𝐼=

1

3

𝑀𝐿2

𝐼=

1 𝑀𝐿2 12

Thin rectangular sheet (slab) about perpendicular axis through center

𝐼=

1 𝑀(𝑎 2 + 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

𝐾=

1

2

Angular Momentum

𝑣𝑐𝑜𝑚 = 𝜔𝑅

11.2

Angular Momentum

11.5

𝑎𝑐𝑜𝑚 = 𝛼𝑅

Magnitude of Angular Momentum

11.6

2 1 𝑀𝑣𝑐𝑜𝑚 𝐼𝑐𝑜𝑚 𝜔2 + 2

𝑎𝑐𝑜𝑚,𝑥 = −

𝑔𝑠𝑖𝑛𝜃 𝐼 1 + 𝑐𝑜𝑚2 𝑀𝑅

11.10

Torque as a vector Torque Magnitude of torque Newton’s 2nd Law

𝜏 = 𝑟 × 𝐹

11.14

𝑑ℓ󰇍 𝑑𝑡

11.23

𝜏 = 𝑟𝐹⊥ = 𝑟⊥ 𝐹 = 𝑟𝐹𝑠𝑖𝑛𝜙 𝜏𝑛𝑒𝑡 =

11.1511.17

󰇍󰇍 = 𝑚(𝑟󰇍 × 𝑣󰇍 ) 𝑣󰇍ℓ = 󰇍 𝑟× 𝑝

ℓ = 𝑟𝑚𝑣𝑠𝑖𝑛𝜙 ℓ = 𝑟𝑝⊥ = 𝑟𝑚𝑣⊥ 𝑛

Angular momentum of a system of particles

󰇍 𝐿 = ∑ ℓ󰇍 𝑖 𝑖=1

𝜏𝑛𝑒𝑡 =

𝑑𝐿󰇍 𝑑𝑡

11.18 11.1911.21 11.26 11.29

Angular Momentum continued Angular Momentum of a 𝐿 = 𝐼𝜔 rotating rigid body

11.31

Conservation of angular momentum

11.32 11.33

󰇍 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐿 󰇍𝐿 𝑖 = 𝐿󰇍𝑓

𝑀𝑔𝑟 𝐼𝜔

Precession of a Gyroscope Precession rate

Ω=

11.31

Chapter 12

Chapter 13

Static Equilibrium

𝐹𝑛𝑒𝑡 = 0 𝜏𝑛𝑒𝑡 = 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

𝐹𝑛𝑒𝑡,𝑥 = 0, 𝐹𝑛𝑒𝑡,𝑦 = 0 𝜏𝑛𝑒𝑡,𝑧 = 0

𝑠𝑡𝑟𝑒𝑠𝑠 = 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 × 𝑠𝑡𝑟𝑎𝑖𝑛 𝑃=

𝐹 𝐴

𝐹 ∆𝐿 =𝐸 𝐿 𝐴 ∆𝑥 𝐹 =𝐺 𝐿 𝐴 ∆𝑉 𝑝=𝐵 𝑉

12.3

Gravitational Force (Newton’s law of gravitation)

12.5

Principle of Superposition

12.7 12.8 12.9

12.22

Gravitational Force acting on a particle from an extended body Gravitational acceleration Gravitation within a spherical Shell Gravitational Potential Energy

12.23

Potential energy on a system (3 particles)

12.24

Escape Speed Kepler’s 3rd Law (law of periods) Energy for bject in circular orbit Mechanical Energy (circular orbit)

𝐹=𝐺

𝑚1 𝑚 𝑟2 2 𝑛

13.1

𝐹1,𝑛𝑒𝑡 = ∑ 𝐹1𝑖

13.5

𝐹1 = ∫ 𝑑𝐹

13.6

𝑖=2

𝐺𝑀 𝑟2 𝐺𝑚𝑀 𝑟 𝐹= 𝑅3 𝐺𝑀𝑚 𝑈=− 𝑟 𝐺𝑚1 𝑚2 𝐺𝑚1 𝑚3 𝐺𝑚2 𝑚3 𝑈 = −( + + ) 𝑟12 𝑟13 𝑟23 𝑎𝑔 =

2𝐺𝑀 𝑣 =√ 𝑅

𝑇2 = (

𝑈=−

𝐸=−

𝐾=

𝐺𝑀𝑚

13.19 13.21 13.22

13.28

4𝜋 2 3 )𝑟 𝐺𝑀

𝐺𝑀𝑚 𝑟

13.11

𝐺𝑀𝑚 2𝑟

2𝑟 𝐺𝑀𝑚 Mechanical Energy 𝐸=− (elliptical orbit) 2𝑎 *Note: 𝐺 = 6.6704 × 10−11 𝑁 ∙ 𝑚2 /𝑘𝑔2

13.34 13.21 13.38 13.40 13.42

Chapter 14 ∆𝑚 𝜌 = ∆𝑉 𝑚 𝜌= 𝑉

Density

∆𝐹 𝑝= ∆𝐴 𝐹 𝑝= 𝐴

Pressure Pressure and depth in a static Fluid P1 is higher than P2

14.16

𝑅𝑉 = 𝐴𝑣

14.24

𝑅𝑚 = 𝜌𝑅𝑉 = 𝜌𝐴𝑣

Equation of continuity when

1 2 𝜌𝑣 + 𝜌𝑔𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 2 𝑅𝑚 = 𝜌𝑅𝑉 = 𝜌𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑝+

𝑅𝑉 = 𝐴𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Frequency cycles per time displacement Angular frequency Velocity

𝐹𝑏 = 𝑚𝑓 𝑔

𝜌𝑔ℎ

Mass Flow Rate

Equation of continuity

14.3 14.4 14.7 14.8

Archimedes’ principle

Bernoulli’s Equation

14.1 14.2

𝑝2 = 𝑝1 + 𝜌𝑔(𝑦1 − 𝑦2 ) 𝑝 = 𝑝0 + 𝜌𝑔ℎ

Gauge Pressure

Volume flow rate

Chapter 15

Acceleration Kinetic and Potential Energy Angular frequency

14.25

14.29

Period Torsion pendulum

14.25 Simple Pendulum 14.24 Physical Pendulum Damping force displacement Angular frequency Mechanical Energy

1 𝑓=𝑇 𝑥 = 𝑥𝑚 cos(𝜔𝑡 + 𝜙) 𝜔=

2𝜋 = 2𝜋𝑓 𝑇

𝑣 = −𝜔𝑥𝑚 sin(𝜔𝑡 + 𝜙)

𝑎 = −𝜔2 𝑥𝑚 cos(𝜔𝑡 + 𝜙) 1 1 𝐾 = 𝑚𝑣 2 𝑈 = 𝑘𝑥 2 2 2 𝜔 =√

𝑘 𝑚

𝑚 𝑇 = 2𝜋√ 𝑘 𝑇 = 2𝜋√

𝑇 = 2𝜋√

𝐼 𝑘

𝐿

𝑔

𝐼 𝑇 = 2𝜋√ 𝑚𝑔𝐿 𝐹𝑑 = −𝑏𝑣 𝑏𝑡

𝑥(𝑡) = 𝑥𝑚 𝑒 − 2𝑚cos(𝜔′ 𝑡 + 𝜙) 𝜔′ = √

𝑏2 𝑘 − 𝑚 4𝑚2

1 2 −𝑏𝑡 𝑒 𝑚 𝐸(𝑡) ≈ 𝑘𝑥𝑚 2

15.2 15.3 15.5 15.6 15.7

15.12 15.13 15.23

15.28

15.29

15.42 15.43 15.44

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 = 2𝜋𝑓 𝜔= 𝑇

𝑣=

𝜔 𝜆 = = 𝜆𝑓 𝑘 𝑇

𝑃𝑎𝑣𝑔 =

1 𝜇𝑣𝜔2 𝑦𝑚2 2

Traveling Wave Form 16.2 16.5

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

𝑦(𝑥, 𝑡) = ℎ(𝑘𝑥 ± 𝜔𝑡) 𝜏 𝑣 = √𝜇

1 1 𝑦 ′ (𝑥, 𝑡) = [2𝑦𝑚 cos ( 𝜙)] sin (𝑘𝑥 − 𝜔𝑡 + 𝜙) 2 2

𝑦 ′ (𝑥, 𝑡) = [2𝑦𝑚 sin(𝑘𝑥)]cos(𝜔𝑡) 𝑓= =𝑛 𝑣

𝜆

𝑣

2𝐿

for n=1,2,…

16.17 16.26

16.51 16.60 16.66

Chapter 17 Sound Waves Speed of sound wave displacement Change in pressure Pressure amplitude

Standing Waves Patterns in Pipes

𝐵 𝑣 = √𝜌

17.3

Δ𝑝 = Δ𝑝𝑚 sin(𝑘𝑥 − 𝜔𝑡)

𝑠 = 𝑠𝑚 cos(𝑘𝑥 − 𝜔𝑡)

17.12

Δ𝑝𝑚 = (𝑣𝜌𝜔)𝑠𝑚

17.13

Standing wave frequency (open at both ends) Standing wave frequency (open at one end)

17.14

beats

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 Interference

Phase difference Fully Constructive Interference Full Destructive interference Mechanical Energy

𝜙=

1 2 −𝑏𝑡 𝑒 𝑚 𝐸(𝑡) ≈ 𝑘𝑥𝑚 2

Sound Intensity Intensity Intensity -uniform in all directions

Intensity level in decibels Mechanical Energy

𝐼=

𝐼=

𝑃

𝐴

1 𝜌𝑣𝜔2 𝑠𝑚2 2

𝐼=

𝑃𝑠 4𝜋𝑟 2

𝐼 𝛽 = (10𝑑𝐵) log ( ) 𝐼𝑜 1 2 −𝑏𝑡 𝑒 𝑚 𝐸(𝑡) ≈ 𝑘𝑥𝑚 2

17.21 17.22 17.23 17.24 17.25 15.44

17.26 17.27 17.29 17.29 15.44

𝑓= 𝑓=

𝑣

𝜆

𝑣

𝜆

= =

𝑛𝑣 2𝐿 𝑛𝑣

4𝐿

Half-angle 𝜃 of Mach cone

17.39

for n=1,3,5

17.41

𝑓𝑏𝑒𝑎𝑡 = 𝑓1 − 𝑓2 Doppler Effect

Source Moving toward stationary observer Source Moving away from stationary observer Observer moving toward stationary source Observer moving away from stationary source

for n=1,2,3

𝑓′ = 𝑓 𝑓′ = 𝑓

𝑣 𝑣 − 𝑣𝑠

𝑣 𝑣 + 𝑣𝑠

𝑣 + 𝑣𝐷 𝑣 𝑣 − 𝑣𝐷 𝑓′ = 𝑓 𝑣 𝑓′ = 𝑓

Shockwave

𝑠𝑖𝑛𝜃 =

𝑣 𝑣𝑠

17.46

17.53 17.54

17.49 17.51

17.57

Chapter 18 Temperature Scales

5 𝑇𝐶 = 9 (𝑇𝐹 − 32) 9 𝑇𝐹 = 𝑇𝐶 + 32 5

Fahrenheit to Celsius Celsius to Fahrenheit

𝑇 = 𝑇𝐶 + 273.15

Celsius to Kelvin

First Law of Thermodynamics

18.8 18.8 18.7

First Law of 18.26 ∆𝐸𝑖𝑛𝑡 = 𝐸𝑖𝑛𝑡,𝑓 − 𝐸𝑖𝑛𝑡,𝑖 = 𝑄 − 𝑊 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

∆𝐿 = 𝐿𝛼∆𝑇

Linear Thermal Expansion

∆𝑉 = 𝑉𝛽∆𝑇

Volume Thermal Expansion

𝑄 = 𝐶(𝑇𝑓 − 𝑇𝑖 ) 𝑄 = 𝑐𝑚(𝑇𝑓 − 𝑇𝑖 )

Applications of First Law 18.9

Q=0 ∆𝐸𝑖𝑛𝑡 = −𝑊

Adiabatic (no heat flow)

18.10

W=0 ∆𝐸𝑖𝑛𝑡 = 𝑄 ∆𝐸𝑖𝑛𝑡 = 0 Q=W 𝑄 = 𝑊 = ∆𝐸𝑖𝑛𝑡 = 0

(constant volume)

Heat Heat and temperature change Heat and phase change

𝑄 = 𝐿𝑚

18.13 18.14

Power (Conducted) Rate objects absorbs energy Power from radiation

𝑃𝑐𝑜𝑛𝑑 =

𝜎 = 5.6704 × 10