These formulas print to 4 pages
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
![](//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Vector_components_with_unit_vectors.svg/160px-Vector_components_with_unit_vectors.svg.png)
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:
![](//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Theta_in_radians_minimalistic.svg/95px-Theta_in_radians_minimalistic.svg.png)
![](//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Torque_lever_arm_w_point_of_application_and_line_of_action.svg/175px-Torque_lever_arm_w_point_of_application_and_line_of_action.svg.png)
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)
![{\displaystyle v_{CM}=r\omega ,\;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/427223f3e6b583f6cb56877d7fa13e994bbf00cc)
▭ 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 ![{\displaystyle \rho _{1}A_{1}v_{1}=\rho _{2}A_{2}v_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02bfd5a86d73148f7b220b81fda912092b8b6969)
15. Oscillations:
Frequency
,
period
and
angular frequency
▭ Simple harmonic motion
also models the x-component of uniform circular motion.
▭ For
positive:
▭ Mass-spring
▭ Energy ![{\displaystyle E_{Tot}={\tfrac {1}{2}}kx^{2}+{\tfrac {1}{2}}mv^{2}={\tfrac {1}{2}}kA^{2}\Rightarrow }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1dcfd99460c27ff1060da483602f780f1bc16b8)
▭ Simple pendulum
▭ Physical pendulum ![{\displaystyle \tau =-MgL\sin \theta \approx -MgL\theta \Rightarrow \;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc9b5883c7365184c43f71a82f4dc28ef345d96)
and
measures from pivot to CM.
▭ Torsional pendulum ![{\displaystyle \tau =-\kappa \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce76d86399e028f62811508aeba4512cd89cfc9)
▭ Damped harmonic oscillator ![{\displaystyle m{\tfrac {d^{2}x}{dt^{2}}}=-kx-b{\tfrac {dx}{dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8d6533de86816adf80ea3b2f35f66fc589f1dd)
where
and
▭ [23]Forced harmonic oscillator (MIT wiki!)] ![{\displaystyle m{\tfrac {d^{2}x}{dt^{2}}}=-kx-b{\tfrac {dx}{dt}}+F_{0}\sin \omega t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfd1033a12f5ab3b30ed918655a0d1cca98b532)
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.
I=∫r2dm for a hoop, disk, cylinder, box, plate, rod, and spherical shell or solid can be found from this figure.
These will not available as you take your test
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- ↑ [http://wiki.ubc.ca/index.php?title=Uncertainty_and_Error&oldid=81540
- ↑ three orthonormal
- ↑
- ↑
▭
▭ ▭ ▭
- ↑ Delta as difference
in limit of differential calculus.
- ↑ , where
is the average velocity.
- ↑
▭ Average values:
, and
▭ Free fall time of flight
▭ Trajectory ▭ Range
- ↑ ▭ Tangential and centripetal acceleration
where .
- ↑
,
- ↑
,
- ↑ Newton's 2nd Law
- ↑
is mass, and
- ↑ because all internal forces cancel by the 3rd law
- ↑ The 1st law is that velocity is constant if the net force is zero.
- ↑ 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
- ↑ ▭ Ideal angle of banked curve:
for curve of radius banked at angle .
- ↑
, where dynamic viscosity; characteristic length. ▭ Stokes's law models a sphere of radius at small Reynold's number: .
- ↑ done by force:
- ↑ The work done on a particle is
- ↑ ▭ Rocket equation
where u is the gas speed WRT the rocket.
- ↑ [https://scripts.mit.edu/~srayyan/PERwiki/index.php?title=Module_3_--_Damped_and_Driven_Harmonic_Oscillations&oldid=7055
- ↑ [http://wiki.ubc.ca/index.php?title=Waves_and_the_Doppler_Effect&oldid=218637
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