NationMaster - Encyclopedia: List of area moments of inertia

The following is list of area moments of inertia. The area moment of inertia or second moment of area has a unit of dimension length⁴, and should not be confused with the mass moment of inertia. Each is with respect to a horizontal axis through the centroid of the given shape, unless otherwise specified. The second moment of area, also known as the second moment of inertia and the area moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection. ... The definition, agreement and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. ... Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², English units lbs ft2) quantifies the rotational inertia of a rigid body, i. ... Centroid of a triangle In geometry, the centroid or barycenter of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. ...

Description

Figure

Area moment of inertia

Comment

Reference

a filled circular area of radius $r ,$

$I_0 = frac{pi r^4}{4} ,$

^[1]

an annulus of inner radius

r 1

and outer radius

r 2

$I_0 = frac{pi}{4} left({r_2}^4-{r_1}^4right)$

a filled circular sector of angle $theta ,$ in radians and radius $r ,$ with respect to an axis through the centroid of the sector and the centre of the circle

$I_0 = left(theta -sinthetaright)frac{r^4}{8} ,$

a filled semicircle with radius $r ,$ with respect to a horizontal line passing through the centroid of the area

$I_0 = left(frac{pi}{8} - frac{8}{9pi}right)r^4 ,$

^[2]

a filled semicircle as above but with respect to an axis collinear with the base

$I = frac{pi r^4}{8} ,$

This is a consequence of the parallel axes rule and the fact that the distance between these two axes is $frac{4r}{3pi} ,$

^[2]

a filled semicircle as above but with respect to a vertical axis through the centroid

$I_0 = frac{pi r^4}{8} ,$

^[2]

a filled quarter circle with radius $r ,$ entirely in the 1st quadrant of the Cartesian coordinate system

$I = frac{pi r^4}{16} ,$

^[3]

a filled quarter circle as above but with respect to a horizontal or vertical axis through the centroid

$I_0 = left(frac{pi}{16}-frac{4}{9pi}right)r^4 ,$

This is a consequence of the parallel axes rule and the fact that the distance between these two axes is $frac{4r}{3pi} ,$

^[3]

a filled ellipse whose radius along the

x

-axis is $a ,$ and whose radius along the

y

-axis is $b ,$

$I_0 = frac{pi}{4} ab^3 ,$

a filled rectangular area with a base width of $b ,$ and height $h ,$

$I_0 = frac{bh^3}{12} ,$

^[4]

a filled rectangular area as above but with respect to an axis collinear with the base

$I = frac{bh^3}{3} ,$

This is a trivial result from the parallel axes rule

^[4]

a filled triangular area with a base width of $b ,$ and height

h

$I_0 = frac{bh^3}{36} ,$

^[5]

a filled triangular area as above but with respect to an axis collinear with the base

$I = frac{bh^3}{12} ,$

This is a consequence of the parallel axes rule and the fact that the distance between these two axes is always $frac{h}{3} ,$

^[5]

a filled regular hexagon with a side length of $a ,$

$I_0 = frac{5sqrt{3}}{16}a^4 ,$

The result is valid for both a horizontal and a vertical axis through the centroid.

Image File history File links Area_moment_of_inertia_of_a_circle. ... An annulus In mathematics, an annulus (the Latin word for little ring, with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. ... Image File history File links Area_moment_of_inertia_of_a_circular_area. ... A circular sector or circle sector also known as a pie piece is the portion of a circle enclosed by two radii and an arc. ... Some common angles, measured in radians. ... Image File history File links Area_moment_of_inertia_of_a_circular_sector. ... Image File history File links Area_moment_of_inertia_of_a_semicircle_2. ... Image File history File links Area_moment_of_inertia_of_a_semicircle. ... The parallel axes rule can be used to determine the moment of inertia of a rigid object about any axis, given the moment of inertia of the object about the parallel axis through the objects center of mass and the perpendicular distance between the axes. ... Image File history File links Area_moment_of_inertia_of_a_semicircle_3. ... Fig. ... Image File history File links Area_moment_of_inertia_of_a_quartercircle. ... Image File history File links Area_moment_of_inertia_of_a_quartercircle_2. ... For other uses, see Ellipse (disambiguation). ... Image File history File links Area_moment_of_inertia_of_an_ellipsis. ... Image File history File links Area_moment_of_inertia_of_a_rectangle. ... Image File history File links Area_moment_of_inertia_of_a_rectangle_2. ... Image File history File links Area_moment_of_inertia_of_a_triangle. ... Image File history File links Area_moment_of_inertia_of_a_triangle_2. ... Image File history File links Area_moment_of_inertia_of_a_regular_hexagon. ...

See also

References