OpenStax University Physics/V1/Equations

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      Equations for Electromagnetism available at Quizbank/Electricity and Magnetism (calculus based)/Equations


Equations lifted from chapter summaries in https://cnx.org/contents/1Q9uMg_a@9.7:Gofkr9Oy@10/Preface

Do not edit this page. Instead copy and paste the source wikitext somewhere else. See these two examples:
WSU Lake University Physics V1 equations
Formulas (in compact form)
Volume 2 equations are complete

Introduction

metric prefixes
da h k M G T P E Z Y
deca hecto kilo mega giga tera peta exa zetta yotta
1E+01 1E+02 1E+03 1E+06 1E+09 1E+12 1E+15 1E+18 1E+21 1E+24
d c m µ n p f a z y
deci centi milli micro nano pico femto atto zepto yocto
1E-01 1E-02 1E-03 1E-06 1E-09 1E-12 1E-15 1E-18 1E-21 1E-24


Units_and_Measurement[edit]

The base SI units are mass: kg (kilogram); length: m (meter); time: s (second). Percent error is


Vectors[edit]

Vector involves components (Ax,Ay,Az) and three orthonormal unit vectors.

Vectors A+B=C.png
Vector components with unit vectors.svg

▭ If , then Ax+Bx=Cx, etc, and vector subtraction is defined by .

▭ The two-dimensional displacement from the origin is . The magnitude is . The angle (phase) is .

▭ Scalar multiplication

▭ Any vector divided by its magnitude is a unit vector and has unit magnitude: where

▭ Dot product and

▭ Cross product where is any cyclic permutation of , i.e., (α,β,γ) represents either (x,y,z) or (y,z,x) or (z,x,y).

▭ Cross-product magnitudes obey where is the angle between and , and by the right hand rule.

▭ Vector identities

▭ 

▭ 

▭ 

▭ 

▭ 

▭ 

▭ 

▭ 

▭ 


Motion_Along_a_Straight_Line[edit]

Delta as difference in limit of differential calculus.

▭ Average velocity (instantaneous velocity)

▭ Acceleration .

▭ WLOG set and if . Then , and , , where is the average velocity.

▭ At constant acceleration: .

▭ For free fall, replace (positive up) and , where = 9.81 m/s2 at Earth's surface).


Motion_in_Two_and_Three_Dimensions[edit]

Instantaneous velocity: , where

▭ Acceleration , where .

▭ Average values: , and

▭ Free fall time of flight ▭ Trajectory ▭ Range

▭ Uniform circular motion: where



▭ Tangential and centripetal acceleration where .

▭ Relative motion: , , ,


Newton's_Laws_of_Motion[edit]

Newton's 2nd Law , where is momentum, is mass, and is the sum of all forces This sum needs only include external forces because all internal forces cancel by the 3rd law . The 1st law is that velocity is constant if the net force is zero.

Free body1.3.svg

▭ Weight.

▭ normal force is a component of the contact force by the surface. If the only forces are contact and weight, where is the angle of incline.

▭ Hooke's law where is the spring constant.


Applications_of_Newton's_Laws[edit]

: friction, coefficient of (static,kinetic) friction, normal force.

▭ Centripetal force for uniform circular motion. Angular velocity is measured in radians per second.

▭ Ideal angle of banked curve: for curve of radius banked at angle .

▭ Drag equation where Drag coefficient, mass density, area, speed. Holds approximately for large Reynold's number , where dynamic viscosity; characteristic length.

▭ Stokes's law models a sphere of radius at small Reynold's number: .


Work_and_Kinetic_Energy[edit]

Infinitesimal work done by force: leads to the path integral

▭ Work done from A→B by friction gravity and spring

▭ Work-energy theorem: The work done on a particle is where kinetic energy .

▭ Power.


Potential_Energy_and_Conservation_of_Energy[edit]

Potential Energy: ; PE at WRT is

(gravitational PE Earth's surface. (ideal spring)

▭ Conservative force: . In 2D, is conservative if and only if

▭ Mechanical energy is conserved if no non-conservative forces are present:


Linear_Momentum_and_Collisions[edit]

is momentum.

▭ Impulse-momentum theorem .

▭ For 2 particles in 2D where (α,β)=(x,y)

▭ Center of mass: , and

▭ 

▭ Rocket equation where u is the gas speed WRT the rocket.


Fixed-Axis_Rotation[edit]

Theta in radians minimalistic.svg

is angle in radians, is angular velocity;

Torque lever arm w point of application and line of action.svg

▭  is tangential speed. Angular acceleration is . is the tangential acceleration.

▭ Constant angular acceleration is average angular velocity.

▭ 

▭ 

▭ Total acceleration is centripetal plus tangential:

▭ Rotational kinetic energy is where is the Moment of inertia.

Parallel axis theorem.svg

▭ parallel axis theorem

▭ Restricting ourselves to fixed axis rotation, is the distance from a fixed axis; the sum of torques, requires only one component, summed as .

▭ Work done by a torque is . The Work-energy theorem is .

▭ Rotational power .

I=∫r2dm for a hoop, disk, cylinder, box, plate, rod, and spherical shell or solid can be found from this figure.


Angular_Momentum[edit]

Center of mass (rolling without slip)

▭ Total angular momentum and net torque: for a single particle.

▭ Precession of a top


Static_Equilibrium_and_Elasticity[edit]

Equilibrium Stress = elastic modulus · strain (analogous to Force = k · Δ x )

▭ (Young's , Bulk , Shear) modulus:


Gravitation[edit]

Newton's law of gravity

▭ Earth's gravity

▭ Gravitational PE beyond Earth

▭ Energy conservation

▭ Escape velocity

▭ Orbital speed

▭ Orbital period

▭ Energy in circular orbit

▭ Conic section

▭ Kepler's third law

▭ Schwarzschild radius


Fluid_Mechanics[edit]

Mass density Pressure

Pressure is the weight per unit area of the fluid above a point.
  • The buoyant force equals the weight of the displaced fluid. If is the weight of a cylindrical object, the displaced volume is and:

    and ▭

Pressure vs depth/height (constant density)

Absolute vs gauge pressure

Pascal's principle: depends only on depth, not on orientation of A.

Volume flow rate

Continuity equation

Bernoulli's principle

Viscosity where F is the force applied by a fluid that is moving along a distance L from an area A.

Poiseuille equation where is "resistance" for a pipe of radius and length .


Oscillations[edit]

Frequency , period and angular frequency

▭ Simple harmonic motion also models the x-component of uniform circular motion.

▭ For positive:

▭ Mass-spring

▭ Energy

▭ Simple pendulum

▭ Physical pendulum and measures from pivot to CM.

▭ Torsional pendulum

▭ Damped harmonic oscillator where and

▭ Forced harmonic oscillator (MIT wiki!) where .


Waves[edit]

Wave speed (phase velocity) where is wavenumber.

▭ Wave and pulse speed of a stretched string where is tension and is linear mass density.

▭ Speed of a compression wave in a fluid

▭ Periodic travelling wave travels in the positive/negative direction. The phase is and the amplitude is .

▭ The resultant of two waves with identical amplitude and frequency where is the phase shift.

▭ This wave equation is linear in

▭ Power in a tranverse stretched string wave .

▭ Intensity of a plane wave in a spherical wave.

▭ Standing wave For symmetric boundary conditions , or equivalently where is the fundamental frequency.


Sound[edit]

Pressure and displacement fluctuations in a sound wave and

▭ Speed of sound in a fluid , ▭ in a solid , ▭ in an idal gas , ▭ in air

▭ Decreasing intensity spherical wave

▭ Sound intensity ▭  ...level

▭ Resonance tube One end closed: ▭ Both ends open:

▭ Beat frequency


▭ (nonrelativisticDoppler effect where is the speed of sound, is the velocity of the source, and is the velocity of the observer.

▭ Angle of shock wave where is the speed of sound, is the speed of the source, and is the Mach number.