List of moments of inertia of inertia From Wikipedia, the f ree f ree encyclopedia
Description Descriptio n
Figure
M ome nt(s ) of ine rtia
Comme nt
Point mass mass m at a distance r f r f rom rom the axis of rotati of rotation. on.
— A point point mass does not have a m mom oment ent of i of inertia round its its own axis, but by using the Parallel axis axi s theorem a mom moment ent of inertia of inertia around a distant axis of rotation of rotation is achieved.
Two point masses, masses, M M and and m, with reduced mass mass and separated by a distance, x distance, x..
—
Rod of length of length L L and mass mass m (Axis of rotati of rotation on at the end of the of the rod)
This expression assum assumes es that the rod is is an inf initely initely thin (but rigid) wire. This is also a special case of the of the thin rectangular plate with axis of rotati of rotation on at the end of the of the plate, with with h = L and w = 0.
[1]
This expression assum assumes es that the rod is is an inf initely initely thin (but rigid) wire. This is a special case of the of the thin rectangular plate with axis axis of rotation of rotation at the center of the of the plate, with w = L and h = 0.
[1]
Rod of length of length L L and mass mass m
Thi Thin circular hoop of radi of radius us r and r and mass mass m
This is a special special case of a of a torus f or b or b=0. (See below.), as well as of a of a thick-walled thick-walled cylindri cylindrical cal tube with open ends, with r 1=r 2 and h=0.
Thi Thin, solid disk of radi of radius us r and r and mass mass m
This is a special special case of the of the solid cylinder, wi with th h=0.
This expression assum assumes es the shell thickness is negligible. It is a special case of the of the thick-walled cylindri cylindrical cal tube f or r or r 1=r 2. Thi Thin cylindrical cylindrical shell with open ends, of radius r and r and mass mass m
[1]
Also, a point mass mass (m (m) at the end of a of a rod of length of length r has r has this same same mom moment ent of inertia of inertia and the value r is r is called the radius of gyration. gyration.
[1]
This is a special special case of the of the thick-walled thick-walled cylindrical cylindrical tube, with r 1=0. (Note: X-Y axis should be swapped f or or a standard right handed f rame) rame)
Solid cylinder cylinder of radius of radius r , height h and mass m
[1][2] [1][2]
With With a density of ρ ρ and the same same geometry geometry Thi Thick-walled cylindrical tube with open ends, of i of inner radius r 1, outer radius radius r 2, length h and mass mass m
or when def ining ining the normaliz normalized ed thickness thickness t n = t /r and r and letting r = r = r 2, then
Sphere (hollow) (hollow) of radius of radius r and r and mass mass m
Ball (solid) of radius of radius r and r and mass mass m
A hollow hollow sphere can be taken to be made m ade up of two of two stacks of inf inf initesimally initesimally thin, circular circular hoops, where the radius dif f fers e rs f rom rom 0 to r (or r (or a single stack, , where the radius dif f fers e rs f rom rom -r to -r to r ).
[1]
A sphere can be taken to be made made up of two of two stacks of inf inf initesimally initesimally thin, solid discs, where the radius dif f fers e rs f rom rom 0 to r (or a single stack, where the radius dif f fers e rs f rom -r to -r to r ). ).
[1]
Also, it can be taken to be made made up of in of inf f initesimally initesimally thin, hollow spheres, where the radius dif f fers e rs f rom rom 0 to r .
[3]
Right circular circular cone with radius r , height h and mass mass m
About a diameter: diameter: Torus of tube of tube radius a, cross-sectional radius b and mass mass m.
—
[3]
About the vertical axis:
[4]
[4]
—
Ellipsoid Ellipsoid (so lid) lid) of sem of semiaxes iaxes a, b, and c with with axis of rotation of rotation a and mass mass m
—
Thi Thin rectangular plate plate of height of height h and of width width w and mass mass m (Axis of rotati of rotation on at the end of the of the plate)
—
Thin rectangular plate of height h and of width w and mass m
[1]
For a similarly oriented cube with sides of length ,
Solid cuboid of height h, width w, and depth d , and mass m
.
Solid cuboid of height D, width W , and length L, and mass m with the longest diagonal as the axis. Plane polygon with vertices ...,
,
,
—
For a cube with sides ,
.
,
and
mass unif ormly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.
This expression assumes that the polygon is star-shaped. The vectors , , , ..., are position vectors of the vertices.
Inf inite disk with mass normally distributed on two axes around the axis of rotation (i.e.
—
Where :
is the mass-density as a
f unction of x and y).
See also Parallel axis theorem Perpendicular axis theorem List of area moments of inertia List of moment of inertia tensors
References 1. ^ a b c d e f g h Raymond A. Serway (1986). Physics f or Scientists and Eng ineers, second ed.. Saunders College Publishing. p. 202. ISB N 0-03-004534- 7. 2. ^ Classical Mechanics - Moment of inertia of a uniform hollow cylinder (http://www.livephysics.com/problems-and-answers/classical-mechanics/find-moment-of-inertia-of-a-uniform-hollow-cylinder.html) . LivePhysics.com. Retrieved on 2008-01-31. 3. ^ a b Ferdinand P. Beer and E. Russell Johnston, Jr (1984). V ector Mechanics f or Engineers, f ourth ed.. Mc Graw-Hill. p. 911. ISB N 0-07-004389- 2. 4. ^ a b Eric W. Weisstein. "Moment of Inertia — Ring" (http://scienceworld.wolfram.com/physics/MomentofInertiaRing.html) . Wolfram Research. http://scienceworld.wolfram.com/physics/Mom entofInertiaRing.html. Retrieved 2010-0325.
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