# moment of inertia

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## Moment of inertia

A relation between the area of a surface or the mass of a body to the position of a line. The analogous positive number quantities, moment of inertia of area and moment of inertia of mass, are involved in the analysis of problems of statics and dynamics respectively.

The moment of inertia of a figure (area or mass) about a line is the sum of the products formed by multiplying the magnitude of each element (of area or of mass) by the square of its distance from the line. The moment of inertia of a figure is the sum of moments of inertia of its parts.

For a body of mass distributed continuously within volume *V*, the movement of inertia of the mass about the *X* axis is given by either or , where *dm* is the mass included in volume element *dV* at whose position the mass per unit volume is ρ (see illustration). Similarly and .

The moments of inertia of a figure about lines which intersect at a common point are generally unequal. The moment is greatest about one line and least about another line perpendicular to the first one. A set of three orthogonal lines consisting of these two and a line perpendicular to both are the principal axes of inertia of the figure relative to that point. If the point is the figure's centroid, the axes are the central principal axes of inertia. The moments of inertia about principal axes are principal moments of inertia. *See* Product of inertia, Radius of gyration

## moment of inertia

Symbol:*I*. A property of any rotating body by which it resists any attempt to make it stop or change speed. If the body is rotating with angular velocity ω, the kinetic energy of the body is ½

*I*ω

^{2}and its angular momentum is

*I*ω.

## Moment of Inertia

a quantity that characterizes the mass distribution of a body and that is, together with the mass, a measure of the inertia of the body during nontranslational motion. In mechanics a distinction is made between (1) axial moments of inertia and (2) products of inertia. The quantity defined by the equation

is called the axial moment of inertia of the body with respect to the *z*-axis; in this equation, the w, are the masses of the points of the body, the *m _{i}* are the distances of the points from the

*z*-axis, ρ is the mass density, and

*V*is the volume of the body. The quantity

*I*is a measure of the body’s inertia when the body rotates about the axis. The axial moment of inertia also may be expressed in terms of the linear quantity

_{z}*k*—the radius of gyration—according to the formula

*I*=

_{z}*Mk*, where

^{2}*M*is the mass of the body. The dimensions of the moment of inertia are

*L*, and the units of measurement are kg ⋅ m

^{2}M^{2}or g ⋅ cm

^{2}.

The quantities defined by the equations

(2) *I _{xy}* = Σ

*m*,

_{i}x_{i}y_{i}*I*= Σ

_{yz}*m*,

_{i}y_{i}z_{i}*I*

_{zx}or by the corresponding volume integrals are called the products of inertia with respect to a system of rectangular axes *x, y, z* at point *O*. These quantities are characteristics of the dynamic unbalance of the masses. For example, when a body rotates about the *z*-axis, the forces of the pressure on the bearings that support the axis depend on the values of *I _{xz}* and

*I*.

_{yz}The moments of inertia with respect to parallel axes *z* and *z*′ are related by the equation

(3) *I _{z}* =

*I*+

_{z′}*Md*

^{2}

where z′ is an axis that passes through the center of mass of the body and *d* is the distance between the axes (Huygens’ theorem).

The moment of inertia with respect to any axis *Ol* that has direction cosines a, α β , and γ and that passes through the origin *O* is found according to the formula

(4) *I _{ol}* =

*I*+

_{x}α^{2}*I*+

_{y}β^{2}*I*− 2

_{z}γ^{2}*I*α β − 2

_{xy}*I*β γ − 2

_{xy}*I*γ α

_{zx}Knowing the six quantities *I _{x}, I_{y}, I_{z}, I_{xy}, I_{yz}*, and

*I*, we can successively calculate, using formulas (4) and (3), the entire set of moments and products of inertia of a body with respect to any axis. These six quantities define the inertia tensor of the body. Through each point of the body, we can draw three mutually perpendicular axes—called the principal axes of inertia—for which

_{zx}*I*=

_{x}y*I*=

_{yz}*I*= 0. Then, the moment of inertia of the body with respect to any axis can be determined if the principal axes of inertia and the moments of inertia with respect to the principal axes are known.

_{zx}The moments of inertia of bodies of complex shape usually are determined experimentally. The concept of moment of inertia is used extensively in solving many problems of mechanics and engineering.

### REFERENCES

*Kratkii fiziko-tekhnicheskii spravochnik*, vol. 2. Editor in chief, K. P. lakovlev. Moscow, 1960. Pages 94–101.

Favorin, M. V.

*Momenty inertsii tel: Spravochnik*. Moscow, 1970.

Gernet, M. M., and V. F. Ratobyl’skii.

*Opredelenie momenlov inertsii*. Moscow, 1969.

S. M. TARG