OpenStax University Physics/V1/Equations (master)

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Equations (master) | Formulas (master) | Equations | Formulas | College Physics

Equations inspired by the Chapter Summaries of OpenStax University Physics Volume 1. Instructors who wish to base their course notes on Wikiversity should not use this version, but instead copy this much more user-friendly that contains easily understood transclusions to this "master". A four-page summary suitable for use during in-class exams is available in two different versions: "master" (online viewing) and "compact". The "compact" version is also available in this pdf form.


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

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


Vectors

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

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Motion_Along_a_Straight_Line

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

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

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

: 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

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

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

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

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

Center of mass (rolling without slip)

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

▭ Precession of a top


Static_Equilibrium_and_Elasticity

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

▭ (Young's , Bulk , Shear) modulus:


Gravitation

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

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

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

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

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.