Dalībnieks:Edgars2007/Inerces moments

en:Moment of inertia

en:List of moments of inertia

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Inerces moments ir moments, kas raksturo ķermeņa inerci rotācijas kustībā ap rotācijas asi. Tas ir atkarīgs no ķermeņa masas sadalījuma attiecībā pret rotācijas asi.

Materiāla punkta inerces moments ir tā masas reizinājums ar attāluma līdz rotācijas asij kvadrātu:

$I=mr^{2}$.

Inerces momenti

Apraksts	Figūra	Inerces moments
Masas punkts m, kas atrodas r lielā attālumā no rotācijas ass		$I=mr^{2}$
Divi masas punkti M un m ar samazinātu masu $\mu$, kas atrodas viens no otra attālumā x		$I={\frac {Mm}{M\!+\!m}}x^{2}=\mu x^{2}$
Stienis, kura garums ir L un masa m (rotācijas ass atrodas stieņa galā)		$I_{\mathrm {gals} }={\frac {mL^{2}}{3}}\,\!$
Stienis, kura garums ir L un masa m		$I_{\mathrm {centrs} }={\frac {mL^{2}}{12}}\,\!$
Gredzens ar rādiusu r un masu m		$I_{z}=mr^{2}\!$ $I_{x}=I_{y}={\frac {mr^{2}}{2}}\,\!$
Plāns disks vai cilindrs (aizpildīts) ar rādiusu r un masu m		$I_{z}={\frac {mr^{2}}{2}}\,\!$ $I_{x}=I_{y}={\frac {mr^{2}}{4}}\,\!$
Plāns cilindrs ar atvērtiem gadiem ar rādiusu r un masu m		$I=mr^{2}\,\!$
Ciets cilindrs ar rādiusu r, augstumu h un masu m		$I_{z}={\frac {mr^{2}}{2}}\,\!$   $I_{x}=I_{y}={\frac {1}{12}}m\left(3r^{2}+h^{2}\right)$
Cilindriska caurule ar biezām sienām ar atvērtiem galiem, ar iekšējo rādiusu r1, ārējo rādiusu r2, garumu h un masu m		$I_{z}={\frac {1}{2}}m\left({r_{1}}^{2}+{r_{2}}^{2}\right)$   $I_{x}=I_{y}={\frac {1}{12}}m\left[3\left({r_{2}}^{2}+{r_{1}}^{2}\right)+h^{2}\right]$ or when defining the normalized thickness tn = t/r and letting r = r2, then $I_{z}=mr^{2}\left(1-t_{n}+{\frac {1}{2}}{t_{n}}^{2}\right)$
Tetraedrs ar sānu s un masu m		$I_{solid}={\frac {3ms^{2}}{7}}\,\!$ $I_{hollow}={\frac {4ms^{2}}{7}}\,\!$
Oktaedrs (caurs) ar sānu s un masu m		$I_{z}=I_{x}=I_{y}={\frac {5ms^{2}}{9}}\,\!$
Oktaedrs (aizpildīts) ar sānu s un masu m		$I_{z}=I_{x}=I_{y}={\frac {ms^{2}}{6}}\,\!$
Sphere (hollow) ar rādiusu r un masu m		$I={\frac {2mr^{2}}{3}}\,\!$
Lode (aizpildīta) ar rādiusu r un masu m		$I={\frac {2mr^{2}}{5}}\,\!$
Sphere (shell) of radius r2, with centered spherical cavity of radius r1 and mass m		$I={\frac {2m}{5}}\left[{\frac {{r_{2}}^{5}-{r_{1}}^{5}}{{r_{2}}^{3}-{r_{1}}^{3}}}\right]\,\!$
Right circular cone with radius r, height h and mass m		$I_{z}={\frac {3}{10}}mr^{2}\,\!$   $I_{x}=I_{y}={\frac {3}{5}}m\left({\frac {r^{2}}{4}}+h^{2}\right)\,\!$
Torus of tube radius a, cross-sectional radius b and mass m.		About a diameter: ${\frac {1}{8}}\left(4a^{2}+5b^{2}\right)m$   About the vertical axis: $\left(a^{2}+{\frac {3}{4}}b^{2}\right)m$
Ellipsoid (solid) of semiaxes a, b, and c with axis of rotation a and mass m		$I_{a}={\frac {m(b^{2}+c^{2})}{5}}\,\!$
Thin rectangular plate of height h and of width w and mass m (Axis of rotation at the end of the plate)		$I_{e}={\frac {mh^{2}}{3}}+{\frac {mw^{2}}{12}}\,\!$
Thin rectangular plate of height h and of width w and mass m		$I_{c}={\frac {m(h^{2}+w^{2})}{12}}\,\!$
Solid cuboid of height h, width w, and depth d, and mass m		$I_{h}={\frac {1}{12}}m\left(w^{2}+d^{2}\right)$ $I_{w}={\frac {1}{12}}m\left(h^{2}+d^{2}\right)$ $I_{d}={\frac {1}{12}}m\left(h^{2}+w^{2}\right)$
Solid cuboid of height D, width W, and length L, and mass m with the longest diagonal as the axis.		$I={\frac {m\left(W^{2}D^{2}+L^{2}D^{2}+L^{2}W^{2}\right)}{6\left(L^{2}+W^{2}+D^{2}\right)}}$
Plane polygon with vertices ${\vec {P}}_{1}$, ${\vec {P}}_{2}$, ${\vec {P}}_{3}$, ..., ${\vec {P}}_{N}$ and mass $m$ uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.		$I={\frac {m}{6}}{\frac {\sum \limits _{n=1}^{N-1}\|{\vec {P}}_{n+1}\times {\vec {P}}_{n}\|(({\vec {P}}_{n+1}\cdot {\vec {P}}_{n+1})+({\vec {P}}_{n+1}\cdot {\vec {P}}_{n})+({\vec {P}}_{n}\cdot {\vec {P}}_{n}))}{\sum \limits _{n=1}^{N-1}\|{\vec {P}}_{n+1}\times {\vec {P}}_{n}\|}}$
Infinite disk with mass normally distributed on two axes around the axis of rotation (i.e. $\rho (x,y)={\tfrac {m}{2\pi ab}}\,e^{-((x/a)^{2}+(y/b)^{2})/2}$ Where : $\rho (x,y)$ is the mass-density as a function of x and y).		$I=m(a^{2}+b^{2})\,\!$

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