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
The base SI units are mass: kg (kilogram); length: m (meter); time: s (second). Percent error is
▭ 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 | edit source]
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 | edit source]
Instantaneous velocity: , where
▭ Acceleration , where .
▭ Average values:
▭ Free fall time of flight
▭ Uniform circular motion: where
▭ Tangential and centripetal acceleration where .
▭ Relative 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.
▭ 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 | edit source]
: 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: .
Infinitesimal work done by force: leads to the
▭ Work done from A→B by friction gravity and spring
▭ Work-energy theorem: The work done on a particle is where kinetic energy .
Potential_Energy_and_Conservation_of_Energy[edit | edit source]
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 | edit source]
▭ Impulse-momentum theorem .
▭ For 2 particles in 2D
▭ Center of mass: , and
▭ Rocket equation where u is the gas speed WRT the rocket.
is angle in radians, is angular velocity;
▭ 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
▭ 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 .
dm for a hoop, disk, cylinder, box, plate, rod, and spherical shell or solid can be found
from this figure.
Center of mass
(rolling without slip)
▭ Total angular momentum and net torque: for a single particle.
▭ Precession of a top
▭ (Young's , Bulk , Shear) modulus:
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
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 .
▭ Simple harmonic motion also models the x-component of uniform circular motion.
▭ For positive:
▭ Simple pendulum
▭ Physical pendulum and measures from pivot to CM.
▭ Torsional pendulum
▭ Damped harmonic oscillator where and
▭ Forced harmonic oscillator (MIT wiki!) where .
▭ 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.
Pressure and displacement fluctuations in a sound wave
▭ Speed of sound in a fluid ,
▭ in a solid ,
▭ in an idal gas ,
▭ in air
▭ Decreasing intensity spherical wave
▭ Sound intensity
▭ Resonance tube One end closed:
▭ Both ends open:
▭ Beat frequency
▭ (nonrelativistic) Doppler 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.