Full Physics Formula Sheet PDF

Preview

Summary

Full Physics Formula Sheet Constants: 𝑔 = 9 𝑚/𝑠 2 SI Units: meters ( length ), seconds ( time ), kilograms ( mass ), newtons ( force ), joules ( work and energy ), watts ( power ), 𝑘𝑔 ∗ 𝑚 2 ( moment of inertia ), 𝑘𝑔/𝑚 3 ( density ), pascals ( pressure and strain/stress ) Vector Calculations: Vector ...

Description

Full Physics Formula Sheet Constants: 𝑔 = 9.81 𝑚/𝑠 2 SI Units: meters (length), seconds (time), kilograms (mass), newtons (force), joules (work and energy), watts (power), 𝑘𝑔 ∗ 𝑚 2 (moment of inertia), 𝑘𝑔/𝑚 3 (density), pascals (pressure and strain/stress) Vector Calculations: Vector Components 𝐴

𝐴𝑦

𝐴𝑦 = 𝐴 sin 𝜃

tan 𝜃 =

𝜃

𝐴𝑥 Vector Addition 󰇍 = 𝑅󰇍 𝐴 + 𝐵

𝐴 = √(𝐴𝑥 )2 + (𝐴𝑦 )

𝐴𝑥 = 𝐴 cos 𝜃

𝑅 = (𝐴𝑥 + 𝐵𝑥 , 𝐴𝑦 + 𝐵𝑦 )

𝐴𝑦

𝐴𝑥

𝑅𝑥 = 𝐴𝑥 + 𝐵𝑥

2

𝑅 = √(𝑅𝑥 )2 + (𝑅𝑦 )

𝑅𝑦 = 𝐴𝑦 + 𝐵𝑦

Vector Multiplication 󰇍 tail-to-tail 𝜙 = the angle between 𝐴 and 𝐵 Scalar Dot Product 𝐴 ∗ 𝐵󰇍 = 𝐴𝐵 cos 𝜙 = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦 + 𝐴𝑧 𝐵𝑧 Vector Cross Product

𝑖 󰇍 = 𝐴𝐵 sin 𝜙 = 𝐴𝑥 𝐴 × 𝐵 𝐵𝑥 Kinematics: Constant Acceleration 𝑣𝑓 = 𝑣𝑖 + 𝑎𝑡 Δ𝑥 = 𝑣𝑖 𝑡 +

𝑣𝑓2

=

𝑣𝑖2 1

1

2

𝑎𝑡 2

+ 2𝑎Δ𝑥

Δ𝑥 = (𝑣𝑖 + 𝑣𝑓 )𝑡 2

𝑗 𝐴𝑦 𝐵𝑦

𝑘 𝐴𝑧 = 𝑖(𝐴𝑦 𝐵𝑧 − 𝐴𝑧 𝐵𝑦 ) + 𝑗(𝐴𝑧 𝐵𝑥 − 𝐴𝑥 𝐵𝑧 ) + 𝑘 (𝐴𝑥 𝐵𝑦 − 𝐴𝑦 𝐵𝑥 ) 𝐵𝑧 𝑣𝑓

𝑎

𝑣𝑓

𝑎

𝑣𝑓

𝑎

Projectile Motion range =

2𝑣𝑖2 cos 𝜃 sin 𝜃 𝑔

Circular Motion 𝑎𝑟𝑎𝑑 =

max height =

𝑡

𝑡 𝑡

Δ𝑥

Δ𝑥

2 𝑣𝑖,𝑦

2𝑔

𝑣2 𝑅

Relative Velocity 𝑣𝐴/𝐶 = 𝑣𝐴/𝐵 + 𝑣𝐵/𝑐 Forces: Newtons Laws Σ𝐹 = 0 @ equilibrium with constant velocity

Σ𝐹 = 𝑚𝑎

Δ𝑥

total time =

2𝑣𝑖,𝑦 𝑔

Uniform Circular Motion 𝑣=

2𝜋𝑅 𝑇

𝑎𝑟𝑎𝑑 =

4𝜋2 𝑅 𝑇2

2

𝐹𝐴 𝑜𝑛 𝐵 = −𝐹𝐵 𝑜𝑛 𝐴 Specific Forces Weight = 𝑚𝑔 Apparent weight: 𝑛 = 𝑚(𝑔 + 𝑎𝑦 ) 2

𝑚𝑣 𝐹𝑟𝑎𝑑 = 𝑅 Friction Kinetic friction: 𝑓𝑘 = 𝜇𝑘 𝐹𝑁 Static friction: 𝑓𝑠 ≤ 𝑓𝑠𝑚𝑎𝑥 = 𝜇𝑠 𝐹𝑁 At critical angle, 𝑓𝑠 = 𝑓𝑠𝑚𝑎𝑥 = 𝜇𝑠 𝐹𝑁 In general, 𝜇𝑘 ≤ 𝜇𝑠 Work and Energy: Work 𝑊 = 𝐹𝑠 𝑊 = 𝐹 ∗ 𝑠 = 𝐹𝑠 cos 𝜙 𝑊𝑡𝑜𝑡𝑎𝑙 = 𝑊𝑛𝑒𝑡 = 𝐹𝑛𝑒𝑡 ∗ 𝑠 = Σ𝐹 ∗ 𝑠 Work with Varying Forces Δ𝑊 = 𝐹 ∗ Δ𝑥 𝑥 𝑊 = ∫𝑥 2 𝐹𝑥 𝑑𝑥 1

Kinetic Energy 1

𝐾𝐸 = 2 𝑚𝑣 2

𝐾𝐸 is never negative, only zero at rest, and doesn’t depend on direction Work-Energy Theorem 1

1

𝑊𝑡𝑜𝑡𝑎𝑙 = Δ𝐾𝐸 = 2 𝑚𝑣𝑓2 − 2 𝑚𝑣𝑖2 If 𝑊𝑡𝑜𝑡𝑎𝑙 > 0, the object speeds up If 𝑊𝑡𝑜𝑡𝑎𝑙 < 0, the object slows down If 𝑊𝑡𝑜𝑡𝑎𝑙 = 0, the object remains at a constant speed Power 𝑃𝑎𝑣𝑒 =

Δ𝑊 Δ𝑡

𝑑𝑊

𝑃𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 = 𝑑𝑡 𝑃 = 𝐹 ∗ 𝑣 Conservation of Energy: Potential Energy 𝑈𝑔𝑟𝑎𝑣 = 𝑚𝑔𝑦 Δ𝑈 = 𝑈2 − 𝑈1 = 𝑚𝑔(𝑦2 − 𝑦1 ) 𝑊𝑔𝑟𝑎𝑣 = 𝑚𝑔(𝑦1 − 𝑦2 ) Δ𝑈𝑔𝑟𝑎𝑣 = −𝑊𝑔𝑟𝑎𝑣 = 𝑊𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 Conservation of Mechanical Energy 𝐾1 + 𝑈𝑔𝑟𝑎𝑣,1 = 𝐾2 + 𝑈𝑔𝑟𝑎𝑣,2 𝐸𝑚𝑒𝑐ℎ = 𝐾 + 𝑈 Δ𝐾 + Δ𝑈 = 0 𝑈 = 𝑈𝑔𝑟𝑎𝑣 + 𝑈𝑒𝑙

Nonconservative Forces Conservative forces depend only on the endpoints Nonconservative forces depend on the path If friction is involved, Δ𝐾 + Δ𝑈 ≠ 0 Nonconservative forces change the interna energy of a system Δ𝐾 + Δ𝑈 + Δ𝑈𝑖𝑛𝑡 = 0 𝑊𝑁𝐶 = Δ𝐾𝐸 + Δ𝑃𝐸 Springs: 𝑥 𝑥 is the distance the spring is stretched If 𝑥 > 0, the spring is stretched If 𝑥 < 0, the spring is compressed Force 𝐹𝑒𝑥𝑡 = 𝑘𝑥 𝐹𝑠𝑝𝑟𝑖𝑛𝑔 = −𝑘𝑥 Work 1

𝑊𝑒𝑥𝑡 = 2 𝑘𝑥 2

𝑊𝑠𝑝𝑟𝑖𝑛𝑔 = −

Elastic Potential Energy

1

2

𝑘𝑥 2

1

𝑈𝑒𝑙 = 2 𝑘𝑥 2

Momentum: Momentum 𝑝 = 𝑚𝑣 𝑑𝑝 Σ𝐹 = 𝑑𝑡

Impulse 𝐽 = Σ𝐹 (𝑡2 − 𝑡1 ) = Σ𝐹 Δ𝑡 𝑡 𝐽 = ∫ 2 Σ𝐹 𝑑𝑡 𝑡1

Impulse-Momentum Theorem 𝐽 = Δ𝑝 Types of Collisions If Σ𝐹𝑒𝑥𝑡 = 0, momentum is conserved and total momentum is constant. Elastic collision: the total kinetic energy is conserved (often when objects bounce off each other) 𝑚1 𝑣1𝑖 + 𝑚2 𝑣2𝑖 = 𝑚1 𝑣1𝑓 + 𝑚2 𝑣2𝑓 1

1

1

1

2 2 𝑚1 𝑣1𝑖2 + 2 𝑚2 𝑣2𝑖 + 2 𝑚2 𝑣22𝑓 = 2 𝑚1 𝑣1𝑓 Inelastic collision: the total kinetic energy after the collision is less than before the collision 𝑚1 𝑣1𝑖 + 𝑚2 𝑣2𝑖 = 𝑚1 𝑣1𝑓 + 𝑚2 𝑣2𝑓 Completely inelastic collision: objects stick together 𝑚1 𝑣1𝑖 + 𝑚2 𝑣2𝑖 = (𝑚1 + 𝑚2 )𝑣𝑓 Center of Mass

2

𝑟𝑐𝑚 =

Σ𝑚𝑖 𝑟 𝑖 Σ𝑚𝑖

𝑀𝑣𝑐𝑚 = 𝑃󰇍 where 𝑀 is total mass and 𝑃󰇍 is total momentum of the system Rotations of Rigid Bodies: Angles

180°

𝑟𝑎𝑑 ∗ 𝜋 = 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 1 𝑟𝑒𝑣 = 2𝜋 𝑟𝑎𝑑 Straight Line Motion 𝑎 = constant 𝑣𝑓 = 𝑣𝑖 + 𝑎𝑡 1 Δ𝑥 = 𝑣𝑖 𝑡 + 2 𝑎𝑡 2 𝑣𝑓2 = 𝑣𝑖2 + 2𝑎Δ𝑥 Δ𝑥 =

1 2

(𝑣𝑖 + 𝑣𝑓 )𝑡

Translational 𝑠 = 𝑟𝜃 Δ𝑠 = 𝑟Δ𝜃 𝑣 = 𝑟𝜔 𝑎𝑡𝑎𝑛 = 𝑟𝛼 𝑎𝑟𝑎𝑑 = 1

𝑣2 𝑟

= 𝜔2 𝑟

𝐾 = 2 𝑚𝑣 2

Moment of Inertia 𝐼 = Σ𝑚𝑖 𝑟𝑖2 = ∫ 𝑟 2 𝑑𝑚 𝐼𝑝 = 𝐼𝑐𝑚 + 𝑀𝑑 2 𝐼 = ∫ 𝑟 2 𝜌𝑑𝑉 Moment of Inertia of Common Objects

Dynamics of Rotations: Torque Angular Momentum

Fixed Axis Rotation 𝛼 = constant 𝜔𝑓 = 𝜔𝑖 + 𝛼𝑡 1 Δ𝜃 = 𝜔𝑖 𝑡 + 2 𝛼𝑡 2 𝜔𝑓2 = 𝜔2𝑖 + 2𝛼Δθ 1

Δ𝜃 = 2 (𝜔𝑖 + 𝜔𝑓 )𝑡 Angular 𝜃 Δ𝜃 𝑑𝜃 𝜔 = 𝑑𝑡 𝛼=

𝑑𝜔 𝑑𝑡

1

1

𝐾 = 𝐼𝜔2 = Σ 𝑚𝑖 𝑣𝑖2 2 2

Static Equilibrium: Σ𝐹 = 0 Σ𝜏 = 0 Strain and Stress: Strain, Stress, Elasticity Stress: the force per area that you apply on the object Strain: the fractional change in the size of the object A deformation is elastic if the object returns to its original shape after the force is removed Tensile Stress and Strain Stress due to force applied perpendicular to the object’s surface Tensile stress =

𝐹⊥ 𝐴

Compressive stress =

𝐹⊥ 𝐴

Young’s Modulus = 𝑌 =

Δ𝑙 𝑙0

Tensile strain =

Compressive strain =

𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑒𝑠𝑠

𝑇𝑒𝑛𝑠𝑖𝑙𝑒 𝑠𝑡𝑟𝑎𝑖𝑛

=

𝐹⊥ /𝐴 Δ𝑙/𝑙0

=

𝐹⊥ 𝐴

𝑙

∗ Δ𝑙0

Δ𝑙 𝑙0

Bulk Stress and Strain Stress due to force applied to all sides of the object pushing inwards Pressure = 𝑝 =

𝐹⊥ 𝐴

Bulk stress = Δ𝑝

Bulk strain =

Bulk Modulus = 𝐵 = −Δ𝑝/(Δ𝑉/𝑉0 )

Δ𝑉 𝑉0

Sheer Stress and Strain Stress due to force applied parallel to the object’s surface Sheer stress =

𝐹∥ 𝐴

Sheer strain =

Sheer Modulus = 𝑆 = (𝐹∥ /𝐴)(ℎ/𝑥)

𝑥

ℎ

Compressibility Compressibility, 𝑘, is the reciprocal of the bulk modulus 1

𝑘=𝐵=−

Δ𝑉/𝑉0 Δ𝑝

1

=−𝑉 ∗ 0

Δ𝑉

Δ𝑝

Fluid Mechanics: Density Density of a homogeneous material = 𝜌 = Pressure

𝑚

𝑉

𝑑𝐹

Pressure at a point in a fluid = 𝑝 = 𝑑𝐴⊥ Pressure at depth, ℎ, compared to the surface pressure, 𝑝0 𝑝 = 𝑝0 + 𝜌𝑔ℎ Pascal’s law: pressure applied to an enclosed fluid 𝐹 𝐹 𝑝= 1= 2 𝐴1

𝐴2

Types of Pressure Gauge pressure: the excess pressure above atmospheric pressure Absolute pressure: total pressure...