OpenStax University Physics/V1/Formulas (master)

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These formulas print to 4 pages


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

1. Units_and_Measurement:  The base SI units are mass: kg (kilogram); length: m (meter); time: s (second). [1]

Percent error is

Vector components with unit vectors.svg

2. VectorsVector involves components (Ax,Ay,Az) and [2] unit vectors.[3] ▭ 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 ▭  ▭  ▭  ▭  ▭  [4]

3. Motion_Along_a_Straight_Line[5] ▭ Average velocity (instantaneous velocity) ▭ Acceleration . ▭ WLOG set and if . Then , and , [6] ▭ At constant acceleration: . ▭ For free fall, replace (positive up) and , where = 9.81 m/s2 at Earth's surface).

4. Motion_in_Two_and_Three_DimensionsInstantaneous velocity: ▭ , where ▭ Acceleration , where . [7] ▭ Uniform circular motion: position , velocity , and acceleration : Note that if then where . [8] ▭ Relative motion: [9] , [10]

5. Newton's_Laws_of_Motion[11], where is momentum, [12] is the sum of all forces This sum needs only include external forces [13].[14]

Free body1.3.svg

▭ Weight. ▭ normal force[15] [16] ▭ [17] where is the spring constant.

6. 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. [18]▭ Drag equation where Drag coefficient, mass density, area, speed. Holds approximately for large Reynold's number[19]

7. Work_and_Kinetic_EnergyInfinitesimal work[20] leads to the path integral ▭ Work done from A→B by friction gravity and spring ▭ Work-energy theorem: [21] where kinetic energy . ▭ Power.

8. Potential_Energy_and_Conservation_of_EnergyPotential 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:

9. Linear_Momentum_and_Collisions is momentum. ▭ Impulse-momentum theorem . ▭ For 2 particles in 2D where (α,β)=(x,y) ▭ Center of mass: , and ▭  [22]

10. Fixed-Axis_Rotation

Theta in radians minimalistic.svg
Torque lever arm w point of application and line of action.svg
Parallel axis theorem.svg

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 .

11. Angular_MomentumCenter of mass (rolling without slip) ▭ Total angular momentum and net torque: for a single particle. ▭ Precession of a top

12. Static_Equilibrium_and_ElasticityEquilibrium Stress = elastic modulus · strain (analogous to Force = k · Δ x ) ▭ (Young's , Bulk , Shear) modulus:

13. GravitationNewton'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

14. Fluid_MechanicsMass density Pressure 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

15. OscillationsFrequency , 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 ▭ [23]Forced harmonic oscillator (MIT wiki!)] where .

16. Waves[24] 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.

17. SoundPressure 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.

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

Footnotes[edit | edit source]

These will not available as you take your test
  1. [
  2. three orthonormal
  3. Vectors A+B=C.png
  4. ▭  ▭ ▭  ▭ 
  5. Delta as difference in limit of differential calculus.
  6. , where is the average velocity.
  7. ▭ Average values: , and ▭ Free fall time of flight ▭ Trajectory ▭ Range
  8. ▭ Tangential and centripetal acceleration where .
  9. ,
  10. ,
  11. Newton's 2nd Law
  12. is mass, and
  13. because all internal forces cancel by the 3rd law
  14. The 1st law is that velocity is constant if the net force is zero.
  15. is a component of the contact force by the surface. If the only forces are contact and weight,
  16. where is the angle of incline.
  17. Hooke's law
  18. ▭ Ideal angle of banked curve: for curve of radius banked at angle .
  19. , where dynamic viscosity; characteristic length. ▭ Stokes's law models a sphere of radius at small Reynold's number: .
  20. done by force:
  21. The work done on a particle is
  22. ▭ Rocket equation where u is the gas speed WRT the rocket.
  23. [
  24. [