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Encyclopedia > List of moments of inertia

The following is a list of moments of inertia. Mass moments of inertia have units of dimension mass × length2. It is the rotational analogue to mass. It should not be confused with the second moment of area (area moment of inertia), which is used in bending calculations. The following moments of inertia assume constant density throughout the object. Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2) quantifies the rotational inertia of a rigid body, i. ... 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. ... 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. ...

Description Figure Moment(s) of inertia Comment
Thin cylindrical shell with open ends, of radius r and mass m $I = m r^2 ,!$ This expression assumes the shell thickness is negligible. It is a special case of the next object for r1=r2.
Thick cylinder with open ends, of inner radius r1, outer radius r2, length h and mass m $I_z = frac{1}{2} mleft({r_2}^2 - {r_1}^2right)$
$I_x = I_y = frac{1}{12} mleft[3left({r_1}^2 + {r_2}^2right)+h^2right]$
or when defining the normalized thickness tn = t/r and letting r = r2,
then $I_z = mr^2left(1-t_n+frac{1}{2}t_n^2right)$
Solid cylinder of radius r, height h and mass m $I_z = frac{m r^2}{2},!$
$I_x = I_y = frac{1}{12} mleft(3r^2+h^2right)$
This is a special case of the previous object for r1=0.
Thin, solid disk of radius r and mass m $I_z = frac{m r^2}{2},!$
$I_x = I_y = frac{m r^2}{4},!$
This is a special case of the previous object for h=0.
Solid sphere of radius r and mass m $I = frac{2 m r^2}{5},!$
Hollow sphere of radius r and mass m $I = frac{2 m r^2}{3},!$
Right circular cone with radius r, height h and mass m $I_z = frac{3}{10}mr^2 ,!$
$I_x = I_y = frac{3}{5}mleft(frac{r^2}{4}+h^2right) ,!$
Solid cuboid of height h, width w, and depth d, and mass m $I_h = frac{1}{12} mleft(w^2+d^2right)$
$I_w = frac{1}{12} mleft(h^2+d^2right)$
$I_d = frac{1}{12} mleft(h^2+w^2right)$
For a similarly oriented cube with sides of length s, $I_{CM} = frac{m s^2}{6},!$.
Rod of length L and mass m $I_{mathrm{center}} = frac{m L^2}{12} ,!$ This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a special case of the previous object for w=L and h=d=0.
Rod of length L and mass m $I_{mathrm{end}} = frac{m L^2}{3} ,!$ This expression assumes that the rod is an infinitely thin (but rigid) wire.
Torus of tube radius a, cross-sectional radius b and mass m. About a diameter: $frac{1}{8}left(4a^2 + 5b^2right)m$
About the vertical axis: $left(a^2 + frac{3}{4}b^2right)m$
Thin, solid, polygon shaped plate with vertices $vec{P}_{1}$, $vec{P}_{2}$, $vec{P}_{3}$, ..., $vec{P}_{N}$ and mass m. $I=frac{m}{6}frac{sum_{n=1}^{N}||vec{P}_{n+1}timesvec{P}_{n}||(vec{P}^{2}_{n+1}+vec{P}_{n+1}cdotvec{P}_{n}+vec{P}_{n}^{2})}{sum_{n=1}^{N}||vec{P}_{n+1}timesvec{P}_{n}||}$

Results from FactBites:

 NodeWorks - Encyclopedia: Moment of inertia (463 words) Moment of inertia is to rotational motion as mass is to linear motion. It should not be confused with the second moment of inertia, also known as the second moment of area and area moment of inertia, which is a property of a shape that is used to predict its resistance to bending. In general, an object's moment of inertia depends on its shape and the distribution of mass within that shape: the greater the concentration of material away from the object's centroid, the larger the moment of inertia.
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