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
|
1. Units_and_Measurement:
The base SI units are mass: kg (kilogram); length: m (meter); time: s (second). [1]
Percent error is
2. Vectors: Vector 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_Dimensions:
Instantaneous 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]
▭ 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_Energy:
Infinitesimal 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_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:
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:
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_Momentum:
Center of mass
(rolling without slip)
▭ Total angular momentum and net torque: for a single particle.
▭ Precession of a top
12. Static_Equilibrium_and_Elasticity: Equilibrium
Stress = elastic modulus · strain (analogous to Force = k · Δ x )
▭ (Young's , Bulk , Shear) modulus:
13. 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
14. Fluid_Mechanics:
Mass 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. 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
▭ [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. 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
▭ (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.