en.wikipedia.org website review
Improve your SEO :: free trial!
en.wikipedia.org is 55% geoptimaliseerd!
SEO Keyword summary for en.wikipedia.org/wiki/moment_of_inertia
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be mathbf
Focus keyword
Short and long tail
Short Tail Keywords mathbf displaystyle delta |
long Tail Keywords (2 words) mathbf r hat k mathbf hat delta mathbf inertia matrix |
long Tail Keywords (3 words) moment of inertia mathbf hat k delta mathbf r center of mass rdisplaystyle mathbf r moments of inertia times delta mathbf |
en.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a en.wikipedia.org/wiki/moment_of_inertia page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
Page title
Title length
moment inertia wikipedia
Meta description
Meta description legth
Meta description SEO
No meta relevance in the description detected !
Content SEO
Number of Words
Spam detected?
Headings
Heading distribution
Heading normalisation
Heading SEO impact
Emphasis (bold and italic)
Emphasis SEO impact
Images
Number of images
Images dimensions
Image alt descriptions
Images SEO impact
wikipedia free encyclopedia displaystyle ifrac lomega textbf fmtextbf icon tau ialpha imr imk boldsymbol mathbf times alpha mmathbf beginalignedboldsymbol mboldsymbol mleftleftmathbf cdot rightboldsymbol leftmathbf rightmathbf rightmr hat endaligned omega beginalignedmathbf leftmboldsymbol rightmleftleftmathbf iomega etextkfrac frac leftmr rightomega textn textnsqrt mgrip ipfrac mgromega mgrt glsqrt lfrac gomega ipmr mathrm rads approx ixx iyy ixxiyyfrac ixxfrac iyyfrac etextksum nfrac mimathbf icdot isum mileftomega riright sum nmiri ipsum imiri ipiiint qrho xyzleftmathbf right rho xyz ell ictextrodiiint dvint sdxleftrho sfrac rightfrac leftfrac mell mrho sell dvsrdrdtheta ictextdisciiint int rrho srdrdtheta mpi ipictextrodmtextrodleftfrac ictextdiscmtextdisclr beginalignedictextsphereint rrtfrac dzint leftr extfrac ztfrac tfrac rightrr expi left rightr textstyle mfrac pii dots beginaligneddelta imathbf iboldsymbol delta idelta riquad quad rimathbf nmidelta itimes leftomega rightleftsum leftsum irighttimes imidelta beginalignedetextkfrac nmimathbf nmileftomega rightcdot irightomega irightfrac nmirightmathbf ndelta iendaligned leftalpha rightalpha bxbybz equiv rightequiv beginbmatrix bzbybz bxbybx endbmatrixendaligned leftboldsymbol leftdelta rightrightleftsum rightendaligned nmileftdelta iright nmileftboldsymbol irightcdot irightrightleftsum irightrightfrac nmileftleftdelta irightboldsymbol leftleftdelta rightrightfrac mathsf tleftdelta irightmathsf nleftmathbf righttimes rightrightmathbf irightleftmathbf irightrightboldsymbol leftleftboldsymbol nmathbf ildots text crossproduct scalar multiplicationsum texttangentialimathbf textcentripetalimathbf textcentripetali texttangentialidelta textcentripetaliboldsymbol texttangentialiboldsymbol beginaligned anticommutativitydelta multiplicationdelta ldots self vector triple productboldsymbol crossproductboldsymbol multiplicationboldsymbol dotproduct commutativityendaligned addition associativityendaligned distributivity over additionsum crossproductsum productsum anticommutativitysum nmiboldsymbol summation distributivityboldsymbol not characteristic particle piendaligned beginalignedsum nmileftbeginbmatrix endbmatrixleftbeginbmatrix endbmatrixbeginbmatrixu endbmatrixrightrightldots matrix multiplication ptsum endbmatrixbeginbmatrixdelta endbmatrixright nmibeginbmatrixdelta endbmatrix endbmatrixsum nmileftmathbf ileftmathbf mleftmathbf rightmathsf tleftmathbf ilmathbf tmathbf iperp irightmathbf trightdelta trightmathbf beginalignedleftdelta leftleftmathbf irightrightcdot irightrightendaligned irightrightequiv irightrighttimes beginalignedleftmathbf irightrightleftleftmathbf beginalignedilsum beginbmatrixi iij stackrel def nmkleftleftmathbf kright ijxikxjkright kleftx beginalignedixx nmkleftyk rightiyy nmkleftxk rightizz beginalignedixyiyx nmkxkykixzizx nmkxkzkiyzizy nmkykzkendaligned ixy iiint vrho otimes rightdxdydz textsf dviiint inmathbf exi endbmatrixbeginbmatrixixxixyixz exiyxiyyiyz exizxizyizzendbmatrixbeginbmatrixsum rightsum nmkxkyksum nmkxkzk exsum nmkykzk nmkxkzksum nmkykzksum rightendbmatrix ptmathbf rightendbmatrixendaligned ixxiyyizzixyixziyz ixyi ixzi iyz rightmleftmathbf imleftmathbf rightmcdot imbeginbmatrixn endbmatrixbeginbmatrixy xyxz exyxx exzxzyx endbmatrixbeginbmatrixn exn bmathbf lambda tboldsymbol sqrt afrac bfrac cfrac dot edit wikidata wikimedia foundation powered mediawiki
Mobile SEO en.wikipedia.org/wiki/moment_of_inertia
Mobile rendering
Mobile optimizations
Responsive design detected (mobile css)
No flash detected !
Mobile improvement
Marketing / lead generation for en.wikipedia.org/wiki/moment_of_inertia
Social Media
Facebook shares | Facebook likes | ||
Facebook comments | Tweets | ||
Google +1 |
Conversion form
Search form
Analytics
Online presence
SERP Preview
SERP Title
SERP Link
SERP Description
Domain Level SEO
Domain name
16 characters long
Domain name SEO Impact
Path name
inertia found in path !
moment found in path !
Structured data
Publisher Markup
Other Structured data
Website configuration
Correct processing of non-existing pages?
Favicon icon found?
Robots.txt found?
Sitemap found?
Navigation and internal links
Navigation
Url seperator
Human readable urls
Number of links
Link SEO Impact
statistics
|
en.m.wikipedia.org |
en.wikipedia.org wikimedia foundation inc
|
w view history
httpsenwikipediaorgwindexphptitlemomentofinertiaoldid1224353249
page information
printable version
|
wiki read
second moment of area
flywheel
siunit
dimension
classical mechanics
second law of motion
history
timeline
applied
celestial
continuum mechanics
dynamics
kinematics
kinetics
statics
statistical mechanics
acceleration
angular momentum tensor
couple
dalemberts principle
energy
kinetic energy
potential
force
frame of reference
inertial
impulse
inertia
mass
mechanical power
work
second moment of mass
momentum
space
speed
time
moment
velocity
virtual work
newtons laws of motion
analytical mechanics
lagrangian
hamiltonian mechanics
routhian mechanics
hamiltonjacobi equation
appells equation of motion
koopmanvon neumann mechanics
damping
displacement
equations of motion
eulers laws of motion
fictitious force
friction
harmonic oscillator
noninertial reference frame
mechanics of planar particle motion
motion
linear
newtons law of universal gravitation
relative velocity
rigid body
newtons laws of motion
eulers equations rigid body dynamics
simple harmonic motion
vibration
newtons law of motion
circular motion
rotating reference frame
centripetal force
centrifugal force
reactive
coriolis force
pendulum
tangential speed
rotational frequency
angular acceleration
angular displacement
frequency
angular velocity
kepler
galileo
christiaan huygens
newton
horrocks
halley
maupertuis
daniel bernoulli
johann bernoulli
leonhard euler
dalembert
clairaut
lagrange
laplace
poisson
hamilton
jacobi
augustinlouis cauchy
routh
liouville
appell
gibbs
koopman
von neumann
tightrope walkers
niagara river
extensive
point mass
perpendicular distance
moment
axis
scalar
symmetric
diagonal
mechanical engineering
inertial mass
si
imperial
compound pendulum
latin
eulers second law
centroid
conservation of angular momentum
figure skaters
tuck position
simple pendulum
radius of gyration
list of moments of inertia
vector triple product expansion
gravimeter
natural
center of oscillation
center of percussion
seconds pendulum
katers pendulum
pendulum
integral
function
beam
rotational symmetry
parallel axis theorem tensor generalization
mechanical system
center of mass
resultant force
precession classical newtonian
angular momentum operator
jacobi identity
tensors
kronecker delta
symmetric tensor
outer product
identity matrix
dot product
transpose
matrix
eigendecomposition
j j sylvester 1852
sylvesters law of inertia
top
360
platonic solid
motion
vehicles
yaw pitch and roll
balancing a tire
rotational spectra
poinsots ellipsoid
ellipsoid
central moment
planar lamina
rotational energy
moment of inertia factor
isbn
oxford university press
si units
s
actergy
m s
distance
position
displacement
m
area
angle
rad
solid angle
rad2 sr
frequency
hz
m s1
kinematic viscosity
specific angular momentum
rotational speed
rotational velocity
angular speed
rads1
rotational acceleration
angular jerk
kg
mass flow rate
kgs1
n s
action
angular impulse
weight
j
n m
yank
power
w
rotatum
glossary of tensor theory
mathematics
coordinate system
differential geometry
dyadic algebra
euclidean geometry
exterior calculus
multilinear algebra
tensor algebra
tensor calculus
physics
engineering
computer vision
electromagnetism
general relativity
transport phenomena
abstract index notation
einstein notation
index notation
multiindex notation
penrose graphical notation
ricci calculus
tetrad index notation
van der waerden notation
voigt notation
tensor intrinsic definition
tensor field
tensor density
tensors in curvilinear coordinates
mixed tensor
antisymmetric tensor
tensor operator
tensor bundle
twopoint tensor
operations
covariant derivative
exterior covariant derivative
exterior derivative
exterior product
hodge star operator
lie derivative
raising and lowering indices
symmetrization
tensor contraction
tensor product
affine connection
basis
cartan formalism physics
connection form
covariance and contravariance of vectors
differential form
dimension
exterior form
fiber bundle
geodesic
levicivita connection
linear map
manifold
matrix
multivector
pseudotensor
spinor
vector space
levicivita symbol
metric tensor
nonmetricity tensor
ricci curvature
riemann curvature tensor
torsion tensor
weyl tensor
spin tensor
cauchy stress tensor
stressenergy tensor
einstein tensor
em tensor
gluon field strength tensor
metric tensor gr
mathematicians
lie cartan
elwin bruno christoffel
albert einstein
carl friedrich gauss
hermann grassmann
tullio levicivita
gregorio riccicurbastro
bernhard riemann
jan arnoldus schouten
woldemar voigt
hermann weyl
|
Links to external pages
Outloing links
ar.wikipedia.org
bn.wikipedia.org
ca.wikipedia.org
cv.wikipedia.org
da.wikipedia.org
de.wikipedia.org
et.wikipedia.org
el.wikipedia.org
eo.wikipedia.org
eu.wikipedia.org
fa.wikipedia.org
fr.wikipedia.org
ga.wikipedia.org
gl.wikipedia.org
ko.wikipedia.org
hy.wikipedia.org
id.wikipedia.org
is.wikipedia.org
it.wikipedia.org
he.wikipedia.org
ka.wikipedia.org
kk.wikipedia.org
lv.wikipedia.org
lt.wikipedia.org
hu.wikipedia.org
mk.wikipedia.org
nl.wikipedia.org
ja.wikipedia.org
no.wikipedia.org
nn.wikipedia.org
uz.wikipedia.org
pl.wikipedia.org
pt.wikipedia.org
ro.wikipedia.org
ru.wikipedia.org
simple.wikipedia.org
sk.wikipedia.org
sl.wikipedia.org
sr.wikipedia.org
fi.wikipedia.org
sv.wikipedia.org
ta.wikipedia.org
tt.wikipedia.org
th.wikipedia.org
tg.wikipedia.org
uk.wikipedia.org
wuu.wikipedia.org
zh-yue.wikipedia.org
zh.wikipedia.org
www.wikidata.org
www.wikidata.org
commons.wikimedia.org
www.oxfordreference.com
www.archive.org
www.archive.org
www.archive.org
www.archive.org
books.google.com
www.bsharp.org
books.google.com
www.archive.org
books.google.com
digital.library.unt.edu
books.google.com
graphics.pixar.com
books.google.com
commons.wikimedia.org
www.lightandmatter.com
www.phy.hk
www.d-nb.info
olduli.nli.org.il
aleph.nkp.cz
foundation.wikimedia.org
foundation.wikimedia.org
www.wikimediafoundation.org
SEO Advice for en.wikipedia.org
In this section we provide pointers on how you can to optimize your web page so it can be found more easily by search engines and how to make it rank higher by optimizing the content of the page itself. For each of the individual criteria the maximum score is 100%. A score below 70% is considered to be indication that the page is not complying with general SEO standards and should be evaluated and/or fixed. Not every factor is weighted the same and some are not as important as others. Relatively unimportant factors like meta keywords are not included in the overall score.
Item | Factor | Pointers | |
---|---|---|---|
PageTitle | 100% | Far too many sites lack a page title. A page title is the first thing that shows in the search results so always use the title element. | |
Title relevance | 87% | A title should reflect the contents of a site. This site has a 67 % match | |
Title Length | 30% | Limit your title to anywhere between 40 and 70 characters. Your title was 30 characters long | |
Meta Description | 0% | A meta description is the second element that shows in the search results so always use the meta description. | |
Meta description length | 0% | The meta description should be between 145 and 160 characters. This meta description is 1 characters long. | |
Meta description relevance | 0% | Meta Description should reflect the contents of a site. This site has a 0 % match | |
Number of internal links | 30% | Linking to internal pages makes pages easier to find for search engines. Try to keep the number of links on your page roughly below 100. There are 484 internal links on this page. | |
Folder structure | 100% | We found a folder structure in the links on your page. A good folder structure makes a site easier to navigate. We found 3 level 1 folders and 6 folders above or in the first level of navigation. | |
Headings | 41% | Headers should reflect the contents of a site. This site has a 18 % match | |
Links | 6% | Link anchors should to some degree reflect the contents of a site. This site has a 3 % match | |
Image alt tags | 17% | Image alt tags should to some degree reflect the contents of a site. This site has a 6 % match | |
Bold and italic | 18% | Bold and italic tags should reflect the contents of a site to some degree. This site has a 6 % match | |
Html ratio | 30% | Try to keep the html / text ratio as low as possible. More html means longer loading times. Layout should be handled in a serpate css file | |
Image descriptions | 96% | 95.674300254453 % of all images have been described via the "alt" attribute. Describing images with relevant text may lead to better results in the search engines. | |
Page errors | 100% | Pages with no errors display significantly faster on most browsers. We detected 0 errors and warnings | |
WordCount | 20% | An ideal page contains between 400 and 600 words.This page contains 15864 words | |
Server response time | 100% | A fast server speeds up a website. This server responds 75.43% faster then average | |
Gzip compression | 30% | This site does not use Gzip compression. Pages may not display as fast as they could | |
Keywords in Domainname | 30% | There are no important keywords in your domain name | |
Keywords in domain path | 100% | There are important keywords in the domain path | |
Structured Data | 100% | Structured data makes it easier for search engines to index your website | |
Inline css | 0% | Do not use inline css declarations. Inline css will slow down the rendering of the website. We detected 893 inline style declarations ( <a style="color:green">) with a size of 29922 bytes | |
Excessive use of the same words | 100% | There is no indication that there are one or more keywords that are used excessively. | |
Frames or iframes | 100% | Perfect, detected not (i)frames on your webpagina | |
Flash | 100% | Perfect, we detected no flash objects on your page | |
Css | 30% | We detected too much (2) CSS files on your page. Css files block the loading of a webpage. | |
Javascript | 100% | Perfect, we did not detect too many blocking JavaScript files | |
Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
Most important heading | 100% | Perfect, we detected a correct use of the most important (h1) heading! | |
Normalized headings | 40% | We dit not font a normalized heading structure. A heading 2 (h2) for example should be followed by a heading of an equal level (h2), a child heading (h3) or even a aprent heading (h1). |
How would you like to have SEO advice for all your pages ?? Start your SEO Dashboard and optimize your website!
en.wikipedia.org images and descriptions
241 images found at en.wikipedia.org Images can improve the user experience for a website by making a pag visually appealing Images can also add extra keyword relevance to a webpage by using alt tags. Images can also slow down a website. If the width and height for a picture is not specified for a browser know in advance how large the image is. A browser must first load the picture and see before it knows how much space should be on the page. Upon reservation In the meantime, the browser can do little but wait. When the height and width for the plate are given in the HTML code, a browser just continues to build for a page while the images load in the background.
http://en.wikipedia.org/static/images/icons/wikipedia.png height: 50 width: 50 description: no alt description found |
|
http://en.wikipedia.org/static/images/mobile/copyright/wikipedia-wordmark-en.svg height: height attribute not set width: width attribute not set description: wikipedia |
|
http://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en.svg height: 13 width: 117 description: the free encyclopedia |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/d/da/%d0%9c%d0%b0%d1%85%d0%be%d0%b2%d0%b8%d0%ba.jpg/220px-%d0%9c%d0%b0%d1%85%d0%be%d0%b2%d0%b8%d0%ba.jpg height: 293 width: 220 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a21fa504541c8dd9a46cddd2ad2b0d657b53b9 height: height attribute not set width: width attribute not set description: {\displaystyle i={\frac {l}{\omega }}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/407b5ff4426df509faccc60e389708981f2f864a height: height attribute not set width: width attribute not set description: {\displaystyle {\textbf {f}}=m{\textbf {a}}} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/6/6f/stylised_atom_with_three_bohr_model_orbits_and_stylised_nucleus.svg/14px-stylised_atom_with_three_bohr_model_orbits_and_stylised_nucleus.svg.png height: 16 width: 14 description: icon |
|
https://upload.wikimedia.org/wikipedia/en/thumb/9/96/symbol_category_class.svg/16px-symbol_category_class.svg.png height: 16 width: 16 description: no alt description found |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/c/c0/samuel_dixon_niagara.jpg/220px-samuel_dixon_niagara.jpg height: 197 width: 220 description: no alt description found |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/1/10/chinese_shenyang_j-11_from_below_in_august_2014.jpg/200px-chinese_shenyang_j-11_from_below_in_august_2014.jpg height: 150 width: 200 description: no alt description found |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/dg800.jpg/200px-dg800.jpg height: 150 width: 200 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd9d0ea2911509b014b72a7b536acb7376cb455 height: height attribute not set width: width attribute not set description: {\displaystyle mr^{2}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538 height: height attribute not set width: width attribute not set description: {\displaystyle r} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc height: height attribute not set width: width attribute not set description: {\displaystyle m} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/6/68/cup_of_russia_2010_-_yuko_kawaguti_%282%29.jpg/170px-cup_of_russia_2010_-_yuko_kawaguti_%282%29.jpg height: 277 width: 170 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8923fa0326cf99e7f8673f232f03d88168c2adba height: height attribute not set width: width attribute not set description: {\displaystyle i={\frac {l}{\omega }}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/239cf0a87708ed6f007d62c2ac3beba3757a5267 height: height attribute not set width: width attribute not set description: {\displaystyle \tau =i\alpha .} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3ce730f104d2f33db72caec81b55ac055d27c2 height: height attribute not set width: width attribute not set description: {\displaystyle i=mr^{2}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c42b27cbda5ef2dc6613130fe6864c50d1e86d62 height: height attribute not set width: width attribute not set description: {\displaystyle i=mk^{2},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5ddf1c828a8e6e0b551ec549a92c5e6a1bc870 height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {f} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {r} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {f} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a585d2bb19071162720ea56a7b087dab3ec17156 height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\alpha }}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdd0a5e8578a9907f25949aea4be9e0498c89f1 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c60fab89e8c3193952047dc565bcf8d233d115 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {f} =m\mathbf {a} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c06ecb68b3dbe91c5ebdf6860a39fb6ca55344c3 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\mathbf {r} \times \mathbf {f} =\mathbf {r} \times (m{\boldsymbol {\alpha }}\times \mathbf {r} )\\&=m\left(\left(\mathbf {r} \cdot \mathbf {r} \right){\boldsymbol {\alpha }}-\left(\mathbf {r} \cdot {\boldsymbol {\alpha }}\right)\mathbf {r} \right)\\&=mr^{2}{\boldsymbol {\alpha }}=i\alpha \mathbf {\hat {k}} ,\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/5733741b1fa48a5c01d20c7538b5850d20e63528 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {\hat {k}} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/dc416bcccf5f63d22af13befd74e5c653ab82f1b height: height attribute not set width: width attribute not set description: {\displaystyle i=mr^{2}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/73cbd54f9e6b846701d83f12d500a98204999f10 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb8af7a2f64af348e559652b6b1f0d2415ba444 height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\omega }}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6f5702415bb2709789189919ac2795dc4f6e66 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {l} &=\mathbf {r} \times \mathbf {p} =\mathbf {r} \times \left(m{\boldsymbol {\omega }}\times \mathbf {r} \right)\\&=m\left(\left(\mathbf {r} \cdot \mathbf {r} \right){\boldsymbol {\omega }}-\left(\mathbf {r} \cdot {\boldsymbol {\omega }}\right)\mathbf {r} \right)\\&=mr^{2}{\boldsymbol {\omega }}=i\omega \mathbf {\hat {k}} ,\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/eef50cb4c5eb698b7e9734da6833d9727c6de6dd height: height attribute not set width: width attribute not set description: {\displaystyle e_{\text{k}}={\frac {1}{2}}m\mathbf {v} \cdot \mathbf {v} ={\frac {1}{2}}\left(mr^{2}\right)\omega ^{2}={\frac {1}{2}}i\omega ^{2}.} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/8/83/mendenhall_gravimeter_pendulums.jpg/220px-mendenhall_gravimeter_pendulums.jpg height: 210 width: 220 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a4a52c506ed0db21f87107570016f15b7b606365 height: height attribute not set width: width attribute not set description: {\displaystyle \omega _{\text{n}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/62cf73d06c6f73b7ce35ac8ee4e3bc2042c5384c height: height attribute not set width: width attribute not set description: {\displaystyle i_{p}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/84766ee4c4f1b99c2efab4acf11b56499a0a36b0 height: height attribute not set width: width attribute not set description: {\displaystyle \omega _{\text{n}}={\sqrt {\frac {mgr}{i_{p}}}},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77 height: height attribute not set width: width attribute not set description: {\displaystyle g} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a height: height attribute not set width: width attribute not set description: {\displaystyle p} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560 height: height attribute not set width: width attribute not set description: {\displaystyle t} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/85426a7473f7d944e188ba5d27b4469cf5df1821 height: height attribute not set width: width attribute not set description: {\displaystyle i_{p}={\frac {mgr}{\omega _{\text{n}}^{2}}}={\frac {mgrt^{2}}{4\pi ^{2}}},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8 height: height attribute not set width: width attribute not set description: {\displaystyle l} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/af13f8f7fc5c36205648dfcc8e1ae74d68d1ee13 height: height attribute not set width: width attribute not set description: {\displaystyle \omega _{\text{n}}={\sqrt {\frac {g}{l}}}={\sqrt {\frac {mgr}{i_{p}}}},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae9270c69ece90445dd11b9d6a65d045a3dd092 height: height attribute not set width: width attribute not set description: {\displaystyle l={\frac {g}{\omega _{\text{n}}^{2}}}={\frac {i_{p}}{mr}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9165b16312df8cb79a9f7b2fa7146744de9390b6 height: height attribute not set width: width attribute not set description: {\displaystyle \pi \ \mathrm {rad/s} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6daff307d9615e49828965aef674f12679ebc65f height: height attribute not set width: width attribute not set description: {\displaystyle l={\frac {g}{\omega _{\text{n}}^{2}}}\approx {\frac {9.81\ \mathrm {m/s^{2}} }{(3.14\ \mathrm {rad/s} )^{2}}}\approx 0.99\ \mathrm {m} .} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/23d06dfedcfe6aa332c557ec72c913472732de65 height: height attribute not set width: width attribute not set description: {\displaystyle i_{xx}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/198e446f7fc446d3d3b304e92b140035f47cc49a height: height attribute not set width: width attribute not set description: {\displaystyle i_{yy}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/67fdc4d2bce83b575ff5dced8d85260f2cfae548 height: height attribute not set width: width attribute not set description: {\displaystyle i_{xx}=i_{yy}={\frac {b^{4}}{12}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/08976567f069338477eaccafb5ee93e0687df349 height: height attribute not set width: width attribute not set description: {\displaystyle i_{xx}={\frac {bh^{3}}{12}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/bedf8a6de357b801e28bafc29ba043bbbaa26ef4 height: height attribute not set width: width attribute not set description: {\displaystyle i_{yy}={\frac {hb^{3}}{12}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/825edfd104a201f253f42278e6bf1ead0b93626a height: height attribute not set width: width attribute not set description: {\displaystyle i_{xx}={\frac {bh^{3}}{36}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/cb55282b2d84df46ff89bed82f6e5d311e8d11ac height: height attribute not set width: width attribute not set description: {\displaystyle i_{xx}=i_{yy}={\frac {1}{4}}{\pi }r^{4}={\frac {1}{64}}{\pi }d^{4}} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/2/2e/rolling_racers_-_moment_of_inertia.gif/220px-rolling_racers_-_moment_of_inertia.gif height: 124 width: 220 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3 height: height attribute not set width: width attribute not set description: {\displaystyle n} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2 height: height attribute not set width: width attribute not set description: {\displaystyle m_{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f height: height attribute not set width: width attribute not set description: {\displaystyle r_{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0b6efec911bdb60f83cfb6eac131cf2020523f height: height attribute not set width: width attribute not set description: {\displaystyle e_{\text{k}}=\sum _{i=1}^{n}{\frac {1}{2}}\,m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}=\sum _{i=1}^{n}{\frac {1}{2}}\,m_{i}\left(\omega r_{i}\right)^{2}={\frac {1}{2}}\,\omega ^{2}\sum _{i=1}^{n}m_{i}r_{i}^{2}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/442d042dcc10bf6d82bc5350bfd55e94b189ac5d height: height attribute not set width: width attribute not set description: {\displaystyle i_{p}=\sum _{i=1}^{n}m_{i}r_{i}^{2}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/93175cdea33a864ce6e24a5274dfb2559d1d8c27 height: height attribute not set width: width attribute not set description: {\displaystyle i_{p}=\sum _{i}m_{i}r_{i}^{2}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1d00a7b361ccf90592790e69cd9cf7a4ddcde92c height: height attribute not set width: width attribute not set description: {\displaystyle i_{p}=\iiint _{q}\rho (x,y,z)\left\|\mathbf {r} \right\|^{2}dv} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64 height: height attribute not set width: width attribute not set description: {\displaystyle \rho } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4 height: height attribute not set width: width attribute not set description: {\displaystyle (x,y,z)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845 height: height attribute not set width: width attribute not set description: {\displaystyle v} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed height: height attribute not set width: width attribute not set description: {\displaystyle q} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98 height: height attribute not set width: width attribute not set description: {\displaystyle z} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4 height: height attribute not set width: width attribute not set description: {\displaystyle x} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d height: height attribute not set width: width attribute not set description: {\displaystyle y} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/moment_of_inertia_rod_center.svg/220px-moment_of_inertia_rod_center.svg.png height: 244 width: 220 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632 height: height attribute not set width: width attribute not set description: {\displaystyle s} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e height: height attribute not set width: width attribute not set description: {\displaystyle \ell } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd13f837ca28f34ac6a919ee3386f501835ba56 height: height attribute not set width: width attribute not set description: {\displaystyle i_{c,{\text{rod}}}=\iiint _{q}\rho \,x^{2}\,dv=\int _{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}\rho \,x^{2}s\,dx=\left.\rho s{\frac {x^{3}}{3}}\right|_{-{\frac {\ell }{2}}}^{\frac {\ell }{2}}={\frac {\rho s}{3}}\left({\frac {\ell ^{3}}{8}}+{\frac {\ell ^{3}}{8}}\right)={\frac {m\ell ^{2}}{12}},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/18bc768a06a7b72c521f993077f4322bd7ddca75 height: height attribute not set width: width attribute not set description: {\displaystyle m=\rho s\ell } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33 height: height attribute not set width: width attribute not set description: {\displaystyle r} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/652ca455423e22aee5155d5b0bd094a2bf89f9b4 height: height attribute not set width: width attribute not set description: {\displaystyle dv=sr\,dr\,d\theta } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4f39f04dac6c9332ab53576c52c2eeb811283d height: height attribute not set width: width attribute not set description: {\displaystyle i_{c,{\text{disc}}}=\iiint _{q}\rho \,r^{2}\,dv=\int _{0}^{2\pi }\int _{0}^{r}\rho r^{2}sr\,dr\,d\theta =2\pi \rho s{\frac {r^{4}}{4}}={\frac {1}{2}}mr^{2},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dca17cc43732e0d5266a98ebe1fdde0a678a2c height: height attribute not set width: width attribute not set description: {\displaystyle m=\pi r^{2}\rho s} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/516751c9cd8925d27978349304a410048a9111fc height: height attribute not set width: width attribute not set description: {\displaystyle i_{p}=i_{c,{\text{rod}}}+m_{\text{rod}}\left({\frac {l}{2}}\right)^{2}+i_{c,{\text{disc}}}+m_{\text{disc}}(l+r)^{2},} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/1/19/moment_of_inertia_solid_sphere.svg/220px-moment_of_inertia_solid_sphere.svg.png height: 203 width: 220 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/27d27233c3e28eccb66564e293ed5d94755b7179 height: height attribute not set width: width attribute not set description: {\displaystyle x^{2}+y^{2}+z^{2}=r^{2},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/80e0877d0975c009e3be64c0af3de7cbd74d4468 height: height attribute not set width: width attribute not set description: {\displaystyle r(z)^{2}=x^{2}+y^{2}=r^{2}-z^{2}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c113f6f3a943142103436b3636511e8961e9c369 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}i_{c,{\text{sphere}}}&=\int _{-r}^{r}{\tfrac {1}{2}}\pi \rho r(z)^{4}\,dz=\int _{-r}^{r}{\tfrac {1}{2}}\pi \rho \left(r^{2}-z^{2}\right)^{2}\,dz\\[1ex]&={\tfrac {1}{2}}\pi \rho \left[r^{4}z-{\tfrac {2}{3}}r^{2}z^{3}+{\tfrac {1}{5}}z^{5}\right]_{-r}^{r}\\[1ex]&=\pi \rho \left(1-{\tfrac {2}{3}}+{\tfrac {1}{5}}\right)r^{5}\\[1ex]&={\tfrac {2}{5}}mr^{2},\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9257cdb7884a0f213fa63a3cde2009cd4804a05c height: height attribute not set width: width attribute not set description: {\textstyle m={\frac {4}{3}}\pi r^{3}\rho } |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/9/95/rollingvsinertia.gif/400px-rollingvsinertia.gif height: 246 width: 400 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b height: height attribute not set width: width attribute not set description: {\displaystyle n} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a53f94e233ce5d3d83d79a26c5111d08c2255f94 height: height attribute not set width: width attribute not set description: {\displaystyle p_{i},i=1,\dots ,n} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {r} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {v} _{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/763631f93bfbb2d477b0f4bc32f6f276b38989c3 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\delta \mathbf {r} _{i}&=\mathbf {r} _{i}-\mathbf {r} ,\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {r} \right)+\mathbf {v} ={\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}+\mathbf {v} ,\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c0048514530d0c0fb8a7beb795110815a818784d height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {v} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea699cbc1f843f2e855577d57529430ec33a1ed height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {k} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ba77c5a75ef9e230d1a36183785477a2eb3c5c1e height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} _{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {r} _{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d8be043bb3ba35151a191103bb68449759673fb3 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b9fa841cd3da6473a5c2721cc37981bc44ebf0f3 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {\hat {e}} _{i}&={\frac {\delta \mathbf {r} _{i}}{\delta r_{i}}},\quad \mathbf {\hat {k}} ={\frac {\boldsymbol {\omega }}{\omega }},\quad \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i},\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}+\mathbf {v} =\omega \mathbf {\hat {k}} \times \delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {v} =\omega \,\delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {v} \end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9b74f7da5a5a9e9fcf7dd09ebbfaae0ae67076e8 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {l} &=\sum _{i=1}^{n}m_{i}\delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\omega \,\delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {v} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\delta r_{i}^{2}\right)\omega \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {v} .\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {c} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2142ef52e531692cc9869f02413e2ccdf95df6c4 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\delta r_{i}\mathbf {\hat {e}} _{i}&=\mathbf {r} _{i}-\mathbf {c} ,\\\sum _{i=1}^{n}m_{i}\,\delta r_{i}\mathbf {\hat {e}} _{i}&=0,\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed60543babbd8e72ba50197ad76e78856500230 height: height attribute not set width: width attribute not set description: {\displaystyle i_{\mathbf {c} }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0d74ff1b42e1668c88e66632b6cf03e2aa94cea8 height: height attribute not set width: width attribute not set description: {\displaystyle i_{\mathbf {c} }=\sum _{i}m_{i}\,\delta r_{i}^{2},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/140a33b8857435797ce58aaec89d066d8e796faf height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {l} =i_{\mathbf {c} }\omega \mathbf {\hat {k}} .} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/f/fe/lever_shear_flywheel.jpg/220px-lever_shear_flywheel.jpg height: 105 width: 220 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2857fa9af9745d1cd6110f7c5781d32ec9d55b98 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}e_{\text{k}}&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i},\\&={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left(\omega \,\delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {v} \right)\cdot \left(\omega \,\delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {v} \right),\\&={\frac {1}{2}}\omega ^{2}\left(\sum _{i=1}^{n}m_{i}\,\delta r_{i}^{2}\mathbf {\hat {t}} _{i}\cdot \mathbf {\hat {t}} _{i}\right)+\omega \mathbf {v} \cdot \left(\sum _{i=1}^{n}m_{i}\,\delta r_{i}\mathbf {\hat {t}} _{i}\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v} \cdot \mathbf {v} .\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4679b9b32ae721e111ca1734fa8b31aa1bd189c6 height: height attribute not set width: width attribute not set description: {\displaystyle e_{\text{k}}={\frac {1}{2}}i_{\mathbf {c} }\omega ^{2}+{\frac {1}{2}}m\mathbf {v} \cdot \mathbf {v} .} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/johndeered.jpg/220px-johndeered.jpg height: 165 width: 220 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d878e318dbc371b1e6fa920e9af0db290bc5356d height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {f} &=\sum _{i=1}^{n}m_{i}\mathbf {a} _{i},\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}\delta \mathbf {r} _{i}\times m_{i}\mathbf {a} _{i},\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3 height: height attribute not set width: width attribute not set description: {\displaystyle p_{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e689c5f3f580c43874d5a86ca01005643bedc633 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} _{i}={\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i}+{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}+\mathbf {a} .} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f01cfee86a741d3d25c6510aea329ab584019060 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {\hat {e}} _{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a914cf61ff0a1ca9fefa05e4bf3ac1a4f13340c5 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {a} _{i}&=\alpha \mathbf {\hat {k}} \times \delta r_{i}\mathbf {\hat {e}} _{i}-\omega \mathbf {\hat {k}} \times \omega \mathbf {\hat {k}} \times \delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {a} \\&=\alpha \delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {a} .\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/06402c0d2f280037bc077adef38e371ef4ade20d height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}\,\delta r_{i}\mathbf {\hat {e}} _{i}\times \left(\alpha \delta r_{i}\mathbf {\hat {t}} _{i}-\omega ^{2}\delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {a} \right)\\&=\left(\sum _{i=1}^{n}m_{i}\,\delta r_{i}^{2}\right)\alpha \mathbf {\hat {k}} +\left(\sum _{i=1}^{n}m_{i}\,\delta r_{i}\mathbf {\hat {e}} _{i}\right)\times \mathbf {a} ,\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d5d5871aade530539c6fde126bd6d2212e356c48 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {e}} _{i}=\mathbf {0} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0c88af56868802fdc05b30d048d9fda9fb22ec99 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {\hat {e}} _{i}\times \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2050af209e999c252e5be940b7d1eba7ff935acd height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\tau }}=i_{\mathbf {c} }\alpha \mathbf {\hat {k}} .} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9b70b1347bdd9d18f140f45ea7e4db3666d38a height: height attribute not set width: width attribute not set description: {\displaystyle \delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {r} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/40f8e4a3717df9525ccdad8aeac3a14f57f3d37b height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {v} _{i}={\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}+\mathbf {v} _{\mathbf {r} }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1b1f71d137ed03b340503b5e49c353f1f67aff27 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {v_{r}} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce9643510bd433e0ed612edd5d3a9ed7faecfec height: height attribute not set width: width attribute not set description: {\displaystyle \left[\mathbf {b} \right]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/5684411608489b3650c74432ef2c035f76bc411c height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {b} =(b_{x},b_{y},b_{z})} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b4782eb664e98a6d680dd96b77601ff9bc12ab4a height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {b} \times \mathbf {y} &\equiv \left[\mathbf {b} \right]\mathbf {y} \\\left[\mathbf {b} \right]&\equiv {\begin{bmatrix}0&-b_{z}&b_{y}\\b_{z}&0&-b_{x}\\-b_{y}&b_{x}&0\end{bmatrix}}.\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/24a7ac3c1aa4822b810baba705b84177c6994928 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {l} &=\sum _{i=1}^{n}m_{i}\,\delta \mathbf {r} _{i}\times \mathbf {v} _{i}\\&=\sum _{i=1}^{n}m_{i}\,\delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}+\mathbf {v} _{\mathbf {r} }\right)\\&=\left(-\sum _{i=1}^{n}m_{i}\,\delta \mathbf {r} _{i}\times \left(\delta \mathbf {r} _{i}\times {\boldsymbol {\omega }}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\,\delta \mathbf {r} _{i}\times \mathbf {v} _{\mathbf {r} }\right),\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/98d7a4dee205ebb34cdcd994634c778207154cc0 height: height attribute not set width: width attribute not set description: {\displaystyle =\mathbf {c} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee5960e77611f673ab2f2050f356dbf0e403132 height: height attribute not set width: width attribute not set description: {\displaystyle [\delta \mathbf {r} _{i}]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/690eb2a602890211d9c1cd09e601f81b3f8bc9df height: height attribute not set width: width attribute not set description: {\displaystyle \delta \mathbf {r} _{i}=\mathbf {r} _{i}-\mathbf {c} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/26ff4f4ab9aa3fff11085b8828fc71a4af7de700 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {l} =\left(-\sum _{i=1}^{n}m_{i}\left[\delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}=\mathbf {i} _{\mathbf {c} }{\boldsymbol {\omega }},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3bdba483ab8231b911ab724380a9ff07d66e2 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i_{c}} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6766e8a9ce07ff8f32ec86aaa929f1e5a3ce1a21 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} _{\mathbf {c} }=-\sum _{i=1}^{n}m_{i}\left[\delta \mathbf {r} _{i}\right]^{2},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f56981fba8d261103bebdb3250107b39273d9b9c height: height attribute not set width: width attribute not set description: {\displaystyle e_{\text{k}}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}={\frac {1}{2}}\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}+\mathbf {v} _{\mathbf {c} }\right)\cdot \left({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}+\mathbf {v} _{\mathbf {c} }\right),} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e2206bca6f4db61956bc2fcf0b5eafc78afac83a height: height attribute not set width: width attribute not set description: {\displaystyle e_{\text{k}}={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}\right)\cdot \left({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}\right)\right)+\left(\sum _{i=1}^{n}m_{i}\mathbf {v} _{\mathbf {c} }\cdot \left({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\mathbf {v} _{\mathbf {c} }\cdot \mathbf {v} _{\mathbf {c} }\right).} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9580a37f69d773e0424723a284a69e2c8baf9b4b height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{i=1}^{n}m_{i}\delta \mathbf {r} _{i}=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2c246c5646e149b67a600084c164f154ec7dda89 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}e_{\text{k}}&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left(\left[\delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\cdot \left(\left[\delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v} _{\mathbf {c} }\cdot \mathbf {v} _{\mathbf {c} }\\&={\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\left({\boldsymbol {\omega }}^{\mathsf {t}}\left[\delta \mathbf {r} _{i}\right]^{\mathsf {t}}\left[\delta \mathbf {r} _{i}\right]{\boldsymbol {\omega }}\right)\right)+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v} _{\mathbf {c} }\cdot \mathbf {v} _{\mathbf {c} }\\&={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \left(-\sum _{i=1}^{n}m_{i}\left[\delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}+{\frac {1}{2}}\left(\sum _{i=1}^{n}m_{i}\right)\mathbf {v} _{\mathbf {c} }\cdot \mathbf {v} _{\mathbf {c} }.\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1d4ef82b66824a7c74937a1db0d8b3923d2fb6bc height: height attribute not set width: width attribute not set description: {\displaystyle e_{\text{k}}={\frac {1}{2}}{\boldsymbol {\omega }}\cdot \mathbf {i} _{\mathbf {c} }{\boldsymbol {\omega }}+{\frac {1}{2}}m\mathbf {v} _{\mathbf {c} }^{2}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd height: height attribute not set width: width attribute not set description: {\displaystyle m} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/93ab161ceb777c444827961be31c4ae628dc3c03 height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\tau }}=\sum _{i=1}^{n}\left(\mathbf {r_{i}} -\mathbf {r} \right)\times m_{i}\mathbf {a} _{i},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0a01879ce830ef8790aa7dc9f3665d6727f3af3a height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} _{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1c521614a9d5e3931efc3e3787e74e75c6ae88b3 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} _{\mathbf {r} }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/688586f53034c01682cb7296456c421309003d31 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} _{i}={\boldsymbol {\alpha }}\times \left(\mathbf {r} _{i}-\mathbf {r} \right)+{\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {r} \right)\right)+\mathbf {a} _{\mathbf {r} }.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9b83ab9af0efe0f59204a493b10b5eb453594c19 height: height attribute not set width: width attribute not set description: {\displaystyle \left[\delta \mathbf {r} _{i}\right]=\left[\mathbf {r} _{i}-\mathbf {c} \right]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f51f6499f20fd09caa9defcf2a02e5c0226ceab7 height: height attribute not set width: width attribute not set description: {\displaystyle (\mathbf {r} _{i}-\mathbf {c} )\times } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/b93d7ec560fb15fbaaba36a2fd0869da807fd92e height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\tau }}=\left(-\sum _{i=1}^{n}m_{i}\left[\delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}\left[\delta \mathbf {r} _{i}\right]^{2}\right){\boldsymbol {\omega }}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff50ef07ccacc8425a7948fc373deac860775a9 height: height attribute not set width: width attribute not set description: {\displaystyle \delta \mathbf {r} _{i}\times \left({\boldsymbol {\omega }}\times \left({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}\right)\right)+{\boldsymbol {\omega }}\times \left(\left({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}\right)\times \delta \mathbf {r} _{i}\right)=0,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f0c3e85512d4f04a615d04c2c28e855fbf2120 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}(\mathbf {r_{i}} -\mathbf {r} )\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}\delta \mathbf {r} _{i}\times (m_{i}\mathbf {a} _{i})\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times \mathbf {a} _{i}]\;\ldots {\text{ cross-product scalar multiplication}}\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times (\mathbf {a} _{{\text{tangential}},i}+\mathbf {a} _{{\text{centripetal}},i}+\mathbf {a} _{\mathbf {r} })]\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times (\mathbf {a} _{{\text{tangential}},i}+\mathbf {a} _{{\text{centripetal}},i}+0)]\\\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3b03c5c91424d1a87643f1a178ed471303f07c07 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {a} _{\mathbf {r} }=0} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/92a8c5e71419b6391fce82b173f4ca48bf2896af height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times \mathbf {a} _{{\text{tangential}},i}+\delta \mathbf {r} _{i}\times \mathbf {a} _{{\text{centripetal}},i}]\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times \mathbf {v} _{{\text{tangential}},i})]\\{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))]\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1faaf09eaed98f82dc96bf237ef4bdd7921d0f88 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}0&=\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\times \delta \mathbf {r} _{i})+({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\times (\delta \mathbf {r} _{i}\times {\boldsymbol {\omega }})\\&=\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\times \delta \mathbf {r} _{i})+({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\times -({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\;\ldots {\text{ cross-product anticommutativity}}\\&=\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\times \delta \mathbf {r} _{i})+-[({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})]\;\ldots {\text{ cross-product scalar multiplication}}\\&=\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\times \delta \mathbf {r} _{i})+-[0]\;\ldots {\text{ self cross-product}}\\0&=\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))+{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\times \delta \mathbf {r} _{i})\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/64290822854e10ddee72f763b816c319e88c257a height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))&=-[{\boldsymbol {\omega }}\times (({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\times \delta \mathbf {r} _{i})]\\&=-[({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})({\boldsymbol {\omega }}\cdot \delta \mathbf {r} _{i})-\delta \mathbf {r} _{i}({\boldsymbol {\omega }}\cdot ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))]\;\ldots {\text{ vector triple product}}\\&=-[({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})({\boldsymbol {\omega }}\cdot \delta \mathbf {r} _{i})-\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot ({\boldsymbol {\omega }}\times {\boldsymbol {\omega }}))]\;\ldots {\text{ scalar triple product}}\\&=-[({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})({\boldsymbol {\omega }}\cdot \delta \mathbf {r} _{i})-\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot (0))]\;\ldots {\text{ self cross-product}}\\&=-[({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})({\boldsymbol {\omega }}\cdot \delta \mathbf {r} _{i})]\\&=-[{\boldsymbol {\omega }}\times (\delta \mathbf {r} _{i}({\boldsymbol {\omega }}\cdot \delta \mathbf {r} _{i}))]\;\ldots {\text{ cross-product scalar multiplication}}\\&={\boldsymbol {\omega }}\times -(\delta \mathbf {r} _{i}({\boldsymbol {\omega }}\cdot \delta \mathbf {r} _{i}))\;\ldots {\text{ cross-product scalar multiplication}}\\\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))&={\boldsymbol {\omega }}\times -(\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))\;\ldots {\text{ dot-product commutativity}}\\\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/870182ea378ff573ef3abb446dfb868374252fd2 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i}))]\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+{\boldsymbol {\omega }}\times -(\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }}))]\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{0-\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})-{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})]-\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\;\ldots \;{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})-{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})=0\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{[{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})-\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }})]-{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})\}]\;\ldots {\text{ addition associativity}}\\\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/85585a7db797b2d9747d752c2c5b54578da2dd71 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})-\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-{\boldsymbol {\omega }}\times {\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})]\;\ldots {\text{ cross-product distributivity over addition}}\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})-\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})({\boldsymbol {\omega }}\times {\boldsymbol {\omega }})]\;\ldots {\text{ cross-product scalar multiplication}}\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})-\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}-(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})(0)]\;\ldots {\text{ self cross-product}}\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{{\boldsymbol {\omega }}(\delta \mathbf {r} _{i}\cdot \delta \mathbf {r} _{i})-\delta \mathbf {r} _{i}(\delta \mathbf {r} _{i}\cdot {\boldsymbol {\omega }})\}]\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times ({\boldsymbol {\alpha }}\times \delta \mathbf {r} _{i})+{\boldsymbol {\omega }}\times \{\delta \mathbf {r} _{i}\times ({\boldsymbol {\omega }}\times \delta \mathbf {r} _{i})\}]\;\ldots {\text{ vector triple product}}\\&=\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times -(\delta \mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{\delta \mathbf {r} _{i}\times -(\delta \mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ cross-product anticommutativity}}\\&=-\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times {\boldsymbol {\alpha }})+{\boldsymbol {\omega }}\times \{\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ cross-product scalar multiplication}}\\&=-\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+-\sum _{i=1}^{n}m_{i}[{\boldsymbol {\omega }}\times \{\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times {\boldsymbol {\omega }})\}]\;\ldots {\text{ summation distributivity}}\\{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times {\boldsymbol {\omega }})]\;\ldots \;{\boldsymbol {\omega }}{\text{ is not characteristic of particle }}p_{i}\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {u} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ef0bb9597377649b0d39c7b8e75fa2f880df2e23 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}-\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times \mathbf {u} )]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\delta r_{3,i}&\delta r_{2,i}\\\delta r_{3,i}&0&-\delta r_{1,i}\\-\delta r_{2,i}&\delta r_{1,i}&0\end{bmatrix}}\left({\begin{bmatrix}0&-\delta r_{3,i}&\delta r_{2,i}\\\delta r_{3,i}&0&-\delta r_{1,i}\\-\delta r_{2,i}&\delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\right)\right)\;\ldots {\text{ cross-product as matrix multiplication}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}\left({\begin{bmatrix}0&-\delta r_{3,i}&\delta r_{2,i}\\\delta r_{3,i}&0&-\delta r_{1,i}\\-\delta r_{2,i}&\delta r_{1,i}&0\end{bmatrix}}{\begin{bmatrix}-\delta r_{3,i}\,u_{2}+\delta r_{2,i}\,u_{3}\\+\delta r_{3,i}\,u_{1}-\delta r_{1,i}\,u_{3}\\-\delta r_{2,i}\,u_{1}+\delta r_{1,i}\,u_{2}\end{bmatrix}}\right)\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\delta r_{3,i}(+\delta r_{3,i}\,u_{1}-\delta r_{1,i}\,u_{3})+\delta r_{2,i}(-\delta r_{2,i}\,u_{1}+\delta r_{1,i}\,u_{2})\\+\delta r_{3,i}(-\delta r_{3,i}\,u_{2}+\delta r_{2,i}\,u_{3})-\delta r_{1,i}(-\delta r_{2,i}\,u_{1}+\delta r_{1,i}\,u_{2})\\-\delta r_{2,i}(-\delta r_{3,i}\,u_{2}+\delta r_{2,i}\,u_{3})+\delta r_{1,i}(+\delta r_{3,i}\,u_{1}-\delta r_{1,i}\,u_{3})\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-\delta r_{3,i}^{2}\,u_{1}+\delta r_{1,i}\delta r_{3,i}\,u_{3}-\delta r_{2,i}^{2}\,u_{1}+\delta r_{1,i}\delta r_{2,i}\,u_{2}\\-\delta r_{3,i}^{2}\,u_{2}+\delta r_{2,i}\delta r_{3,i}\,u_{3}+\delta r_{2,i}\delta r_{1,i}\,u_{1}-\delta r_{1,i}^{2}\,u_{2}\\+\delta r_{3,i}\delta r_{2,i}\,u_{2}-\delta r_{2,i}^{2}\,u_{3}+\delta r_{3,i}\delta r_{1,i}\,u_{1}-\delta r_{1,i}^{2}\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\delta r_{2,i}^{2}+\delta r_{3,i}^{2})\,u_{1}+\delta r_{1,i}\delta r_{2,i}\,u_{2}+\delta r_{1,i}\delta r_{3,i}\,u_{3}\\+\delta r_{2,i}\delta r_{1,i}\,u_{1}-(\delta r_{1,i}^{2}+\delta r_{3,i}^{2})\,u_{2}+\delta r_{2,i}\delta r_{3,i}\,u_{3}\\+\delta r_{3,i}\delta r_{1,i}\,u_{1}+\delta r_{3,i}\delta r_{2,i}\,u_{2}-(\delta r_{1,i}^{2}+\delta r_{2,i}^{2})\,u_{3}\end{bmatrix}}\\[6pt]&=-\sum _{i=1}^{n}m_{i}{\begin{bmatrix}-(\delta r_{2,i}^{2}+\delta r_{3,i}^{2})&\delta r_{1,i}\delta r_{2,i}&\delta r_{1,i}\delta r_{3,i}\\\delta r_{2,i}\delta r_{1,i}&-(\delta r_{1,i}^{2}+\delta r_{3,i}^{2})&\delta r_{2,i}\delta r_{3,i}\\\delta r_{3,i}\delta r_{1,i}&\delta r_{3,i}\delta r_{2,i}&-(\delta r_{1,i}^{2}+\delta r_{2,i}^{2})\end{bmatrix}}{\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}\\&=-\sum _{i=1}^{n}m_{i}[\delta r_{i}]^{2}\mathbf {u} \\[6pt]-\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times \mathbf {u} )]&=\left(-\sum _{i=1}^{n}m_{i}[\delta r_{i}]^{2}\right)\mathbf {u} \;\ldots \;\mathbf {u} {\text{ is not characteristic of }}p_{i}\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/24eed2971eccd0e0b63f32600e1d3fc6254dca49 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=-\sum _{i=1}^{n}m_{i}[\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times {\boldsymbol {\alpha }})]+{\boldsymbol {\omega }}\times -\sum _{i=1}^{n}m_{i}\delta \mathbf {r} _{i}\times (\delta \mathbf {r} _{i}\times {\boldsymbol {\omega }})]\\&=\left(-\sum _{i=1}^{n}m_{i}[\delta r_{i}]^{2}\right){\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(-\sum _{i=1}^{n}m_{i}[\delta r_{i}]^{2}\right){\boldsymbol {\omega }}\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6817a680bf02967f5ab5788badea07e5b205cb42 height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\tau }}=\mathbf {i} _{\mathbf {c} }{\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \mathbf {i} _{\mathbf {c} }{\boldsymbol {\omega }},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/fb6d262f5376a22cca507b03419910ddd33a5fc7 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i_{r}} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/93e62233b3acff4fc919fc24e8d6241551543087 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} _{\mathbf {r} }=-\sum _{i=1}^{n}m_{i}\left[\mathbf {r} _{i}-\mathbf {r} \right]^{2}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3acacbb57b62840187254b0c59ca86dd7baa40fe height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {r} =(\mathbf {r} -\mathbf {c} )+\mathbf {c} =\mathbf {d} +\mathbf {c} ,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec3b626fc045b6ff579316e29978fccfed884c2 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {d} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0063409d605348215928372e62f294287f06fb98 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} _{\mathbf {r} }=-\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\left(\mathbf {c} +\mathbf {d} \right)]^{2}=-\sum _{i=1}^{n}m_{i}[\left(\mathbf {r} _{i}-\mathbf {c} \right)-\mathbf {d} ]^{2}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/68eee7fefe0e716823fd5d80384303b8aef58b99 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} _{\mathbf {r} }=-\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {c} ]^{2}\right)+\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {c} ]\right)[\mathbf {d} ]+[\mathbf {d} ]\left(\sum _{i=1}^{n}m_{i}[\mathbf {r} _{i}-\mathbf {c} ]\right)-\left(\sum _{i=1}^{n}m_{i}\right)[\mathbf {d} ]^{2}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a5785dcf9605fb6ace7c1a8c8b4d6b365af48366 height: height attribute not set width: width attribute not set description: {\displaystyle [\mathbf {d} ]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1c59b0848c4b1d69fff75c4257005225cd42c74d height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} _{\mathbf {r} }=\mathbf {i} _{\mathbf {c} }-m[\mathbf {d} ]^{2},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3ed54432828b43d6ae23e9a2e9c009c7336024 height: height attribute not set width: width attribute not set description: {\displaystyle -m\left[\mathbf {r} \right]^{2}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/27fed5e06e85cd3d74847c333b7a32cbd6a31fe7 height: height attribute not set width: width attribute not set description: {\displaystyle m\left[\mathbf {r} \right]^{\mathsf {t}}\left[\mathbf {r} \right]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4c4e0b7a62e35392117d1331b60c9e3276b5c9 height: height attribute not set width: width attribute not set description: {\displaystyle [\mathbf {r} ]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1b57d141754c4d3852d6d0bd08d2e16166630c height: height attribute not set width: width attribute not set description: {\displaystyle i_{l}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/78bb7c062959fd615347939341b32bf66de221c6 height: height attribute not set width: width attribute not set description: {\displaystyle i_{l}=\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{n}m_{i}\left[\delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \mathbf {i} _{\mathbf {r} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {t}}\mathbf {i} _{\mathbf {r} }\mathbf {\hat {k}} ,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7d7d2c2d4f498f8ec004f0a3efa2f82f833893 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {l} (t)=\mathbf {r} +t\mathbf {\hat {k}} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ab5ca8f4f58653e5a4ec0e9571b9c5e1d76b74 height: height attribute not set width: width attribute not set description: {\displaystyle \delta \mathbf {r} _{i}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a67ee90abc7d893471fed7c8d37bdb9b73c8a967 height: height attribute not set width: width attribute not set description: {\displaystyle \delta \mathbf {r} _{i}^{\perp }=\delta \mathbf {r} _{i}-\left(\mathbf {\hat {k}} \cdot \delta \mathbf {r} _{i}\right)\mathbf {\hat {k}} =\left(\mathbf {e} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {t}}\right)\delta \mathbf {r} _{i},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/96d0817ff96d9ee9243ddc37f688e5385c0b406e height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {t}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a515c7495954477be07dee4f29d4c4c4ec50cf98 height: height attribute not set width: width attribute not set description: {\displaystyle \left[\mathbf {\hat {k}} \right]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/10d1281a399f50084c4d71e516f5d56b1cf5426f height: height attribute not set width: width attribute not set description: {\displaystyle \left[\mathbf {\hat {k}} \right]\mathbf {y} =\mathbf {\hat {k}} \times \mathbf {y} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d92be0b3adbb50fbe61b31dc0237f0fc102476a0 height: height attribute not set width: width attribute not set description: {\displaystyle -\left[\mathbf {\hat {k}} \right]^{2}\equiv \left|\mathbf {\hat {k}} \right|^{2}\left(\mathbf {e} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {t}}\right)=\mathbf {e} -\mathbf {\hat {k}} \mathbf {\hat {k}} ^{\mathsf {t}},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9d205a6da935ecfa8f5e13feb1a930f1d032022a height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\left|\delta \mathbf {r} _{i}^{\perp }\right|^{2}&=\left(-\left[\mathbf {\hat {k}} \right]^{2}\delta \mathbf {r} _{i}\right)\cdot \left(-\left[\mathbf {\hat {k}} \right]^{2}\delta \mathbf {r} _{i}\right)\\&=\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\right)\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8db6d7de03a2fc68aaec686046142b6d435307 height: height attribute not set width: width attribute not set description: {\displaystyle \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\right)\equiv \left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right),} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/62111ff66a6922e2705a7eb60478088077b7d9bf height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}&\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\right)\cdot \left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\right)\\={}&\left(\left(\mathbf {\hat {k}} \times \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\right)\times \mathbf {\hat {k}} \right)\cdot \left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\\={}&\left(\mathbf {\hat {k}} \times \delta \mathbf {r} _{i}\right)\cdot \left(-\delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left(\delta \mathbf {r} _{i}\times \delta \mathbf {r} _{i}\times \mathbf {\hat {k}} \right)\\={}&-\mathbf {\hat {k}} \cdot \left[\delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} .\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4a4a61db87467a4b76294e68bba221a2810cdf height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}i_{l}&=\sum _{i=1}^{n}m_{i}\left|\delta \mathbf {r} _{i}^{\perp }\right|^{2}\\&=-\sum _{i=1}^{n}m_{i}\mathbf {\hat {k}} \cdot \left[\delta \mathbf {r} _{i}\right]^{2}\mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \left(-\sum _{i=1}^{n}m_{i}\left[\delta \mathbf {r} _{i}\right]^{2}\right)\mathbf {\hat {k}} \\&=\mathbf {\hat {k}} \cdot \mathbf {i} _{\mathbf {r} }\mathbf {\hat {k}} =\mathbf {\hat {k}} ^{\mathsf {t}}\mathbf {i} _{\mathbf {r} }\mathbf {\hat {k}} ,\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5e3b1ba705c5adda8381cb22580d26d64968f8 height: height attribute not set width: width attribute not set description: {\displaystyle m_{k}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/a4440832e4252456d6421e7afcabbe27163e6e7c height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} ={\begin{bmatrix}i_{11}&i_{12}&i_{13}\\i_{21}&i_{22}&i_{23}\\i_{31}&i_{32}&i_{33}\end{bmatrix}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a54c4961303ffc219b1afcf4bb0968299293ba height: height attribute not set width: width attribute not set description: {\displaystyle i_{ij}\ {\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{n}m_{k}\left(\left\|\mathbf {r} _{k}\right\|^{2}\delta _{ij}-x_{i}^{(k)}x_{j}^{(k)}\right)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20 height: height attribute not set width: width attribute not set description: {\displaystyle i} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0 height: height attribute not set width: width attribute not set description: {\displaystyle j} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f45b48b2f71d22de6ec450c1c9a18a5319389157 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {r} _{k}=\left(x_{1}^{(k)},x_{2}^{(k)},x_{3}^{(k)}\right)} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/fa75d04c11480d976e1396951e02cbb3c4f71568 height: height attribute not set width: width attribute not set description: {\displaystyle \delta _{ij}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8a458c8aeb096ce732abf346ae8edf3e4f53a126 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/d751b7aa4b938bc2ce2003fbb5cefc0d644d0b88 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}i_{xx}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{n}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right),\\i_{yy}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{n}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right),\\i_{zz}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{n}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right),\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9b72c7aeab55d09a27e566f1f3422831f6a46e70 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}i_{xy}=i_{yx}\ &{\stackrel {\mathrm {def} }{=}}\ -\sum _{k=1}^{n}m_{k}x_{k}y_{k},\\i_{xz}=i_{zx}\ &{\stackrel {\mathrm {def} }{=}}\ -\sum _{k=1}^{n}m_{k}x_{k}z_{k},\\i_{yz}=i_{zy}\ &{\stackrel {\mathrm {def} }{=}}\ -\sum _{k=1}^{n}m_{k}y_{k}z_{k}.\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/600241c202cafa4422df4a4106d081f21bafe545 height: height attribute not set width: width attribute not set description: {\displaystyle i_{xy}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8afa4148f931bd25c09a32c2a8919107a0c57bb9 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} =\iiint _{v}\rho (x,y,z)\left(\|\mathbf {r} \|^{2}\mathbf {e} _{3}-\mathbf {r} \otimes \mathbf {r} \right)\,dx\,dy\,dz,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/9f31a01f85d97dfab0d0f7c4ce489c5aeb21e411 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {r} \otimes \mathbf {r} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/0f32ba1c18fa5966e4dcd2275a7f7e3bc21f6dc0 height: height attribute not set width: width attribute not set description: {\displaystyle [\mathbf {r} ]\mathbf {x} =\mathbf {r} \times \mathbf {x} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/3f468dc8f2e4246bf78b1916390083f90998be5c height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} =\iiint _{v}\rho (\mathbf {r} )[\mathbf {r} ]^{\textsf {t}}[\mathbf {r} ]\,dv=-\iiint _{q}\rho (\mathbf {r} )[\mathbf {r} ]^{2}\,dv} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {n} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4665b70c28077a426fbd1f1af3f17b6d97ced4e7 height: height attribute not set width: width attribute not set description: {\displaystyle i_{n}=\mathbf {n} \cdot \mathbf {i} \cdot \mathbf {n} ,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/71d87c9988135efda49f4cdfc0ffced4872a6497 height: height attribute not set width: width attribute not set description: {\displaystyle i_{12}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/8121bd2cb30b298966841de1aaa32a05199d4372 height: height attribute not set width: width attribute not set description: {\displaystyle i_{12}=\mathbf {e} _{1}\cdot \mathbf {i} \cdot \mathbf {e} _{2},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/300757855c441dbf4e1e9d36f1cd88dad29e5e8a height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} ={\begin{bmatrix}i_{11}&i_{12}&i_{13}\\[1.8ex]i_{21}&i_{22}&i_{23}\\[1.8ex]i_{31}&i_{32}&i_{33}\end{bmatrix}}={\begin{bmatrix}i_{xx}&i_{xy}&i_{xz}\\[1.8ex]i_{yx}&i_{yy}&i_{yz}\\[1.8ex]i_{zx}&i_{zy}&i_{zz}\end{bmatrix}}={\begin{bmatrix}\sum _{k=1}^{n}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right)&-\sum _{k=1}^{n}m_{k}x_{k}y_{k}&-\sum _{k=1}^{n}m_{k}x_{k}z_{k}\\[1ex]-\sum _{k=1}^{n}m_{k}x_{k}y_{k}&\sum _{k=1}^{n}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right)&-\sum _{k=1}^{n}m_{k}y_{k}z_{k}\\[1ex]-\sum _{k=1}^{n}m_{k}x_{k}z_{k}&-\sum _{k=1}^{n}m_{k}y_{k}z_{k}&\sum _{k=1}^{n}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right)\end{bmatrix}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/78eb9b0b4686b23e51350e74bb24511fce107f93 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}i_{xy}=i_{yx}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{n}m_{k}x_{k}y_{k},\\i_{xz}=i_{zx}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{n}m_{k}x_{k}z_{k},\\i_{yz}=i_{zy}\ &{\stackrel {\mathrm {def} }{=}}\ \sum _{k=1}^{n}m_{k}y_{k}z_{k},\\[3pt]\mathbf {i} ={\begin{bmatrix}i_{11}&i_{12}&i_{13}\\[1.8ex]i_{21}&i_{22}&i_{23}\\[1.8ex]i_{31}&i_{32}&i_{33}\end{bmatrix}}&={\begin{bmatrix}i_{xx}&-i_{xy}&-i_{xz}\\[1.8ex]-i_{yx}&i_{yy}&-i_{yz}\\[1.8ex]-i_{zx}&-i_{zy}&i_{zz}\end{bmatrix}}={\begin{bmatrix}\sum _{k=1}^{n}m_{k}\left(y_{k}^{2}+z_{k}^{2}\right)&-\sum _{k=1}^{n}m_{k}x_{k}y_{k}&-\sum _{k=1}^{n}m_{k}x_{k}z_{k}\\[1ex]-\sum _{k=1}^{n}m_{k}x_{k}y_{k}&\sum _{k=1}^{n}m_{k}\left(x_{k}^{2}+z_{k}^{2}\right)&-\sum _{k=1}^{n}m_{k}y_{k}z_{k}\\[1ex]-\sum _{k=1}^{n}m_{k}x_{k}z_{k}&-\sum _{k=1}^{n}m_{k}y_{k}z_{k}&\sum _{k=1}^{n}m_{k}\left(x_{k}^{2}+y_{k}^{2}\right)\end{bmatrix}}.\end{aligned}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/70c1a51151db53f8d6d1bfd1caa8e9bb39032fd4 height: height attribute not set width: width attribute not set description: {\displaystyle (i_{xx},i_{yy},i_{zz},i_{xy},i_{xz},i_{yz})} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/7b725b8d733f9b219cd025c9d9762636accf14aa height: height attribute not set width: width attribute not set description: {\displaystyle (i_{12}=i_{xy},i_{13}=i_{xz},i_{23}=i_{yz})} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/67c52210a9cd9cb7311678251ef334c7fa585287 height: height attribute not set width: width attribute not set description: {\displaystyle (i_{12}=-i_{xy},i_{13}=-i_{xz},i_{23}=-i_{yz})} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb84e133d15551d660800ec29b44783ff36e19d height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {\hat {n}} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4c98a7ad70f08dcf4a55d75e8641aac2bfc41bad height: height attribute not set width: width attribute not set description: {\displaystyle \left|\mathbf {x} -\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)\mathbf {\hat {n}} \right|} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/aa89b424310b0d9c04ee3f7e8a73651d86d71c9c height: height attribute not set width: width attribute not set description: {\displaystyle i=mr^{2}=m\left(\mathbf {x} -\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)\mathbf {\hat {n}} \right)\cdot \left(\mathbf {x} -\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)\mathbf {\hat {n}} \right)=m\left(\mathbf {x} ^{2}-2\mathbf {x} \left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)\mathbf {\hat {n}} +\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)^{2}\mathbf {\hat {n}} ^{2}\right)=m\left(\mathbf {x} ^{2}-\left(\mathbf {x} \cdot \mathbf {\hat {n}} \right)^{2}\right).} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/541b6e25570a7f9da63fd8e6a7034b712ce2d0f3 height: height attribute not set width: width attribute not set description: {\displaystyle i=m\left(\mathbf {x} ^{\textsf {t}}\mathbf {x} -\mathbf {\hat {n}} ^{\textsf {t}}\mathbf {x} \mathbf {x} ^{\textsf {t}}\mathbf {\hat {n}} \right)=m\cdot \mathbf {\hat {n}} ^{\textsf {t}}\left(\mathbf {x} ^{\textsf {t}}\mathbf {x} \cdot \mathbf {e_{3}} -\mathbf {x} \mathbf {x} ^{\textsf {t}}\right)\mathbf {\hat {n}} ,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb30784bc5347a3bef70bf77ea6413725e1d4d7 height: height attribute not set width: width attribute not set description: {\displaystyle i=m{\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}{\begin{bmatrix}y^{2}+z^{2}&-xy&-xz\\[0.5ex]-yx&x^{2}+z^{2}&-yz\\[0.5ex]-zx&-zy&x^{2}+y^{2}\end{bmatrix}}{\begin{bmatrix}n_{1}\\[0.7ex]n_{2}\\[0.7ex]n_{3}\end{bmatrix}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/98d9ee8c31336f45453465113151c97830827f2c height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} _{0}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/22568fa28674e243955b8fdddced7496b8e6487d height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} =\mathbf {i} _{0}+m[(\mathbf {r} \cdot \mathbf {r} )\mathbf {e} _{3}-\mathbf {r} \otimes \mathbf {r} ]} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/de29098f5a34ee296a505681a0d5e875070f2aea height: height attribute not set width: width attribute not set description: {\displaystyle \otimes } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/076ae8537b9617353da64ffafefc3980e78248ac height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} =\mathbf {r} \mathbf {i_{0}} \mathbf {r} ^{\textsf {t}}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/5375b412570a299355aebd42dc45bc0c46cd3a20 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} _{\mathbf {c} }^{b}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f199d57a899bdf9d2d69ade12203d8c11d56c851 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} =\mathbf {a} \mathbf {y} ,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/bb25a040b592282dc2a254c3117e792c3c81161f height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {y} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/932ee4752bbfe56b40201f525520bdbb5e368df1 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} _{\mathbf {c} }=\mathbf {a} \mathbf {i} _{\mathbf {c} }^{b}\mathbf {a} ^{\mathsf {t}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {q} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/e0dd88f811f247adeb58a4fd0174e95f11eefca2 height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\lambda }}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/54e2e55f1753f6bd594ab4eae2793aad38b6bf50 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {i} _{\mathbf {c} }^{b}=\mathbf {q} {\boldsymbol {\lambda }}\mathbf {q} ^{\mathsf {t}},} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/7144289c4fcf135450cf15f6957847a8921cb2f6 height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\lambda }}={\begin{bmatrix}i_{1}&0&0\\0&i_{2}&0\\0&0&i_{3}\end{bmatrix}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/03f18d041b2df30adef07164dbf285878893dedc height: height attribute not set width: width attribute not set description: {\displaystyle i_{1}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3506ae39df854f347365bae6f326ef4f565be5 height: height attribute not set width: width attribute not set description: {\displaystyle i_{2}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/becba5d3350c4dd244f3cda48eb13439f21ed350 height: height attribute not set width: width attribute not set description: {\displaystyle i_{3}} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/44e4ce1c04edd8f9602e60f0ec4457b7ac12fcd4 height: height attribute not set width: width attribute not set description: {\displaystyle m>2} |
|
https://upload.wikimedia.org/wikipedia/commons/thumb/6/6d/triaxial_ellipsoid.jpg/220px-triaxial_ellipsoid.jpg height: 123 width: 220 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc height: height attribute not set width: width attribute not set description: {\displaystyle a} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3 height: height attribute not set width: width attribute not set description: {\displaystyle b} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455 height: height attribute not set width: width attribute not set description: {\displaystyle c} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/25e1b955758b9a28b1e5f06b7a3ae54ad78fdb3f height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} ^{\mathsf {t}}{\boldsymbol {\lambda }}\mathbf {x} =1,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/4f77feed11fc0b809a4fd3946eb966b0d41af0ec height: height attribute not set width: width attribute not set description: {\displaystyle i_{1}x^{2}+i_{2}y^{2}+i_{3}z^{2}=1,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/24cb74ab9d03623f89fb144e09f2afe23ca1243c height: height attribute not set width: width attribute not set description: {\displaystyle \left({\frac {x}{1/{\sqrt {i_{1}}}}}\right)^{2}+\left({\frac {y}{1/{\sqrt {i_{2}}}}}\right)^{2}+\left({\frac {z}{1/{\sqrt {i_{3}}}}}\right)^{2}=1,} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/914038d07d02704f45ee854f8f72eb6cc1916320 height: height attribute not set width: width attribute not set description: {\displaystyle a={\frac {1}{\sqrt {i_{1}}}},\quad b={\frac {1}{\sqrt {i_{2}}}},\quad c={\frac {1}{\sqrt {i_{3}}}}.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/50714d8205fc8894156c1c6a0f709d9850a131a2 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} =\|\mathbf {x} \|\mathbf {n} } |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/c72ebb3350fb95aa9983a893859493a811ef446b height: height attribute not set width: width attribute not set description: {\displaystyle i_{\mathbf {n} }} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/330523c115ea17084e9707aa306b7f76fcfab6b7 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} ^{\mathsf {t}}{\boldsymbol {\lambda }}\mathbf {x} =\|\mathbf {x} \|^{2}\mathbf {n} ^{\mathsf {t}}{\boldsymbol {\lambda }}\mathbf {n} =\|\mathbf {x} \|^{2}i_{\mathbf {n} }=1.} |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/14d0e64404638a3496991377b877dc052a11981c height: height attribute not set width: width attribute not set description: {\displaystyle \|\mathbf {x} \|={\frac {1}{\sqrt {i_{\mathbf {n} }}}}.} |
|
https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/commons-logo.svg/30px-commons-logo.svg.png height: 40 width: 30 description: no alt description found |
|
https://wikimedia.org/api/rest_v1/media/math/render/svg/ad59b9876301e8fb75b9ddbf08de594b87251d3b height: height attribute not set width: width attribute not set description: {\displaystyle {\dot {m}}} |
|
https://upload.wikimedia.org/wikipedia/en/thumb/8/8a/oojs_ui_icon_edit-ltr-progressive.svg/10px-oojs_ui_icon_edit-ltr-progressive.svg.png height: 10 width: 10 description: edit this at wikidata |
|
https://login.wikimedia.org/wiki/special:centralautologin/start?type=1x1 height: 1 width: 1 description: no alt description found |
|
http://en.wikipedia.org/static/images/footer/wikimedia-button.png height: 31 width: 88 description: wikimedia foundation |
|
http://en.wikipedia.org/static/images/footer/poweredby_mediawiki_88x31.png height: 31 width: 88 description: powered by mediawiki |
How are images contributing to your SEO site-wise ? Your leading content tool has the awnsers!