List of moments of inertia

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In physics and applied mathematics, the mass moment of inertia, usually denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. Mass moments of inertia have units of dimension ML2([mass] × [length]2). It should not be confused with the second moment of area, which is used in bending calculations. The mass moment of inertia is often also known as the rotational inertia, and sometimes as the angular mass.

For simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression. Typically this occurs when the mass density is constant, but in some cases the density can vary throughout the object as well. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. When calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and perpendicular axis theorems.

This article mainly considers symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass unless otherwise specified.

Moments of inertia[edit]

Following are scalar moments of inertia. In general, the moment of inertia is a tensor, see below.

Description Figure Moment(s) of inertia
Point mass m at a distance r from the axis of rotation.

A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved.

PointInertia.svg
Two point masses, M and m, with reduced mass μ and separated by a distance, x about an axis passing through the center of mass of the system and perpendicular to line joining the two particles.
Rod of length L and mass m, rotating about its center.

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.

Moment of inertia rod center.svg   [1]
Rod of length L and mass m, rotating about one end.

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.

Moment of inertia rod end.svg   [1]
Thin circular hoop of radius r and mass m.

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.

Moment of inertia hoop.svg
Thin, solid disk of radius r and mass m.

This is a special case of the solid cylinder, with h = 0. That is a consequence of the perpendicular axis theorem.

Moment of inertia disc.svg
Thin cylindrical shell with open ends, of radius r and mass m.

This expression assumes that 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.
Moment of inertia thin cylinder.png   [1]
Solid cylinder of radius r, height h and mass m.

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).

Moment of inertia solid cylinder.svg   [1]
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m. Moment of inertia thick cylinder h.svg

   [1] [2]
where t = (r2–r1)/r2 is a normalized thickness ratio;

With a density of ρ and the same geometry

Regular tetrahedron of side s and mass m Tetraaxial.gif

[3]

Regular octahedron of side s and mass m Octahedral axis.gif [3]
[3]
Regular dodecahedron of side s and mass m

(where ) [3]

Regular icosahedron of side s and mass m

[3]

Hollow sphere of radius r and mass m.

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).

Moment of inertia hollow sphere.svg   [1]
Solid sphere (ball) of radius r and mass m.

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).

Moment of inertia solid sphere.svg   [1]
Sphere (shell) of radius r2, with centered spherical cavity of radius r1 and mass m.

When the cavity radius r1 = 0, the object is a solid ball (above).

When r1 = r2, , and the object is a hollow sphere.

Spherical shell moment of inertia.png   [1]
Right circular cone with radius r, height h and mass m Moment of inertia cone.svg   [4]
  [4]
Torus with minor radius a, major radius b and mass m. Torus cycles (labeled).png About an axis passing through the center and perpendicular to the diameter:   [5]
About a diameter:   [5]
Ellipsoid (solid) of semiaxes a, b, and c with mass m Ellipsoid 321.png



Thin rectangular plate of height h, width w and mass m
(Axis of rotation at the end of the plate)
Recplaneoff.svg
Thin rectangular plate of height h, width w and mass m
(Axis of rotation at the center)
Recplane.svg   [1]
Solid cuboid of height h, width w, and depth d, and mass m.

For a similarly oriented cube with sides of length ,

Moment of inertia solid rectangular prism.png



Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal.

For a cube with sides , .

Moment of Inertia Cuboid.svg
Triangle with vertices at the origin and at P and Q, with mass m, rotating about an axis perpendicular to the plane and passing through the origin.
Plane polygon with vertices P1, P2, P3, ..., PN and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. Polygon Moment of Inertia.svg
Plane regular polygon with n-vertices and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin. a stands for side length.   [6]
Infinite disk with mass normally distributed on two axes around the axis of rotation with mass-density as a function of x and y:
Gaussian 2D.png
Uniform disk about an axis perpendicular to its edge. [7]

List of 3D inertia tensors[edit]

This list of moment of inertia tensors is given for principal axes of each object.

To obtain the scalar moments of inertia I above, the tensor moment of inertia I is projected along some axis defined by a unit vector n according to the formula:

where the dots indicate tensor contraction and we have used the Einstein summation convention. In the above table, n would be the unit Cartesian basis ex, ey, ez to obtain Ix, Iy, Iz respectively.

Description Figure Moment of inertia tensor
Solid sphere of radius r and mass m Moment of inertia solid sphere.svg
Hollow sphere of radius r and mass m Moment of inertia hollow sphere.svg

Solid ellipsoid of semi-axes a, b, c and mass m Ellipsoide.png
Right circular cone with radius r, height h and mass m, about the apex Moment of inertia cone.svg
Solid cuboid of width w, height h, depth d, and mass m
180x
Slender rod along y-axis of length l and mass m about end
Moment of inertia rod end.svg

Slender rod along y-axis of length l and mass m about center
Moment of inertia rod center.svg

Solid cylinder of radius r, height h and mass m Moment of inertia solid cylinder.svg

Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m Moment of inertia thick cylinder h.svg

See also[edit]

References[edit]

  1. ^ a b c d e f g h i Raymond A. Serway (1986). Physics for Scientists and Engineers, second ed. Saunders College Publishing. 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. ^ a b c d e Satterly, John (1958). "The Moments of Inertia of Some Polyhedra". The Mathematical Gazette. Mathematical Association. 42 (339): 11–13. doi:10.2307/3608345. JSTOR 3608345. 
  4. ^ a b Ferdinand P. Beer and E. Russell Johnston, Jr (1984). Vector Mechanics for Engineers, fourth ed. McGraw-Hill. p. 911. ISBN 0-07-004389-2. 
  5. ^ a b Eric W. Weisstein. "Moment of Inertia — Ring". Wolfram Research. Retrieved 2010-03-25. 
  6. ^ Karel Rektorys (1994). Survey of Applicable Mathematics, second ed., Vol. II. Kluwer Academic Publisher. p. 942. ISBN 0-7923-0681-3. 
  7. ^ http://www.pas.rochester.edu/~ygao/phy141/Lecture15/sld010.htm

External links[edit]