Utente:Andrea And/Sandbox/3

Descrizione	Figura	Momento di inerzia	Commento
Massa puntiforme m a distanza r dall'asse di rotazione.		${\displaystyle I=mr^{2}}$	— Un massa puntiforme non ha momento di inerzia intorno al proprio asse, ma usando il Parallel axis theorem a moment of inertia around a distant axis of rotation is achieved.
Due masse puntiformi, M e m, con massa ridotta ${\displaystyle \mu }$ e separato da una distanza, x.		${\displaystyle I={\frac {Mm}{M\!+\!m}}x^{2}=\mu x^{2}}$	—
Rod of length L and mass m (Axis of rotation at the end of the rod)		${\displaystyle I_{\mathrm {end} }={\frac {mL^{2}}{3}}\,\!}$  [1]	This expression assumes that the rod is an infinitely thin (but rigid) wire. This is also a special case of the thin rectangular plate with axis of rotation at the end of the plate, with h = L and w = 0.
Rod of length L and mass m		${\displaystyle I_{\mathrm {center} }={\frac {mL^{2}}{12}}\,\!}$  [1]	This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a special case of the thin rectangular plate with axis of rotation at the center of the plate, with w = L and h = 0.
Thin circular hoop of radius r and mass m		${\displaystyle I_{z}=mr^{2}\!}$ ${\displaystyle I_{x}=I_{y}={\frac {mr^{2}}{2}}\,\!}$	This is a special case of a torus for b=0. (See below.), as well as of a thick-walled cylindrical tube with open ends, with r1=r2 and h=0.
Thin, solid disk of radius r and mass m		${\displaystyle I_{z}={\frac {mr^{2}}{2}}\,\!}$ ${\displaystyle I_{x}=I_{y}={\frac {mr^{2}}{4}}\,\!}$	This is a special case of the solid cylinder, with h=0.
Thin cylindrical shell with open ends, of radius r and mass m		${\displaystyle I=mr^{2}\,\!}$  [1]	This expression assumes the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for r1=r2. Also, a point mass (m) at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration.
Solid cylinder of radius r, height h and mass m		${\displaystyle I_{z}={\frac {mr^{2}}{2}}\,\!}$  [1] ${\displaystyle I_{x}=I_{y}={\frac {1}{12}}m\left(3r^{2}+h^{2}\right)}$	This is a special case of the thick-walled cylindrical tube, with r1=0. (Note: X-Y axis should be swapped for a standard right handed frame)
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m		${\displaystyle I_{z}={\frac {1}{2}}m\left({r_{1}}^{2}+{r_{2}}^{2}\right)}$  [1][2] ${\displaystyle I_{x}=I_{y}={\frac {1}{12}}m\left[3\left({r_{2}}^{2}+{r_{1}}^{2}\right)+h^{2}\right]}$ or when defining the normalized thickness tn = t/r and letting r = r2, then ${\displaystyle I_{z}=mr^{2}\left(1-t_{n}+{\frac {1}{2}}{t_{n}}^{2}\right)}$	With a density of ρ and the same geometry ${\displaystyle I_{z}={\frac {1}{2}}\pi \rho h\left({r_{2}}^{4}-{r_{1}}^{4}\right)}$ ${\displaystyle I_{x}=I_{y}={\frac {1}{12}}\pi \rho h\left(3({r_{2}}^{4}-{r_{1}}^{4})+h^{2}({r_{2}}^{2}-{r_{1}}^{2})\right)}$
Sphere (hollow) of radius r and mass m		${\displaystyle I={\frac {2mr^{2}}{3}}\,\!}$  [1]	A hollow sphere can be taken to be made up of two stacks of infinitesimally thin, circular hoops, where the radius differs from 0 to r (or a single stack, , where the radius differs from -r to r).
Ball (solid) of radius r and mass m		${\displaystyle I={\frac {2mr^{2}}{5}}\,\!}$  [1]	A sphere can be taken to be made up of two stacks of infinitesimally thin, solid discs, where the radius differs from 0 to r (or a single stack, where the radius differs from -r to r). Also, it can be taken to be made up of infinitesimally thin, hollow spheres, where the radius differs from 0 to r.
Right circular cone with radius r, height h and mass m		${\displaystyle I_{z}={\frac {3}{10}}mr^{2}\,\!}$  [3] ${\displaystyle I_{x}=I_{y}={\frac {3}{5}}m\left({\frac {r^{2}}{4}}+h^{2}\right)\,\!}$  [3]	—
Torus of tube radius a, cross-sectional radius b and mass m.		About a diameter: ${\displaystyle {\frac {1}{8}}\left(4a^{2}+5b^{2}\right)m}$  [4] About the vertical axis: ${\displaystyle \left(a^{2}+{\frac {3}{4}}b^{2}\right)m}$  [4]	—
Ellipsoid (solid) of semiaxes a, b, and c with axis of rotation a and mass m		${\displaystyle I_{a}={\frac {m(b^{2}+c^{2})}{5}}\,\!}$	—
Thin rectangular plate of height h and of width w and mass m (Axis of rotation at the end of the plate)		${\displaystyle I_{e}={\frac {mh^{2}}{3}}+{\frac {mw^{2}}{12}}\,\!}$	—
Thin rectangular plate of height h and of width w and mass m		${\displaystyle I_{c}={\frac {m(h^{2}+w^{2})}{12}}\,\!}$  [1]	—
Solid cuboid of height h, width w, and depth d, and mass m		${\displaystyle I_{h}={\frac {1}{12}}m\left(w^{2}+d^{2}\right)}$ ${\displaystyle I_{w}={\frac {1}{12}}m\left(h^{2}+d^{2}\right)}$ ${\displaystyle I_{d}={\frac {1}{12}}m\left(h^{2}+w^{2}\right)}$	For a similarly oriented cube with sides of length ${\displaystyle s}$, ${\displaystyle I_{CM}={\frac {ms^{2}}{6}}\,\!}$.
Solid cuboid of height D, width W, and length L, and mass m with the longest diagonal as the axis.		${\displaystyle I={\frac {m\left(W^{2}D^{2}+L^{2}D^{2}+L^{2}W^{2}\right)}{6\left(L^{2}+W^{2}+D^{2}\right)}}}$	For a cube with sides ${\displaystyle s}$, ${\displaystyle I={\frac {ms^{2}}{6}}\,\!}$.
Plane polygon with vertices ${\displaystyle {\vec {P}}_{1}}$, ${\displaystyle {\vec {P}}_{2}}$, ${\displaystyle {\vec {P}}_{3}}$, ..., ${\displaystyle {\vec {P}}_{N}}$ and mass ${\displaystyle m}$ uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.		${\displaystyle I={\frac {m}{6}}{\frac {\sum \limits _{n=1}^{N-1}\|{\vec {P}}_{n+1}\times {\vec {P}}_{n}\|(({\vec {P}}_{n+1}\cdot {\vec {P}}_{n+1})+({\vec {P}}_{n+1}\cdot {\vec {P}}_{n})+({\vec {P}}_{n}\cdot {\vec {P}}_{n}))}{\sum \limits _{n=1}^{N-1}\|{\vec {P}}_{n+1}\times {\vec {P}}_{n}\|}}}$	This expression assumes that the polygon is star-shaped. The vectors ${\displaystyle {\vec {P}}_{1}}$, ${\displaystyle {\vec {P}}_{2}}$, ${\displaystyle {\vec {P}}_{3}}$, ..., ${\displaystyle {\vec {P}}_{N}}$ are position vectors of the vertices.
Infinite disk with mass normally distributed on two axes around the axis of rotation (i.e. ${\displaystyle \rho (x,y)={\tfrac {m}{2\pi ab}}\,e^{-((x/a)^{2}+(y/b)^{2})/2}}$ Where : ${\displaystyle \rho (x,y)}$ is the mass-density as a function of x and y).		${\displaystyle I=m(a^{2}+b^{2})\,\!}$	—

See also

• Parallel axis theorem
• Perpendicular axis theorem
• List of area moments of inertia
• List of moment of inertia tensors

References

1. Raymond A. Serway, Physics for Scientists and Engineers, second ed., Saunders College Publishing, 1986, p. 202, ISBN 0-03-004534-7.
2. ^ Classical Mechanics - Moment of inertia of a uniform hollow cylinder. LivePhysics.com. Retrieved on 2008-01-31.
3. Ferdinand P. Beer and E. Russell Johnston, Jr, Vector Mechanics for Engineers, fourth ed., McGraw-Hill, 1984, p. 911, ISBN 0-07-004389-2.
4. Eric W. Weisstein, Moment of Inertia — Ring, su scienceworld.wolfram.com, Wolfram Research. URL consultato il 25 marzo 2010.