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Encyclopedia > Moment of inertia

The symbols I and sometimes J are usually used to refer to the moment of inertia.

Moment of inertia was introduced by Euler in his book a Theoria motus corporum solidorum seu rigidorum in 1730. In this book, he discussed at length moment of inertia and many concepts, such as principal axis of inertia, related to the moment of inertia. Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

The moment of inertia of an object about a given axis describes how difficult it is to change its angular motion about that axis. For example, consider two discs (A and B) of the same mass. Disc A has a larger radius than disc B. Assuming that there is uniform thickness and mass distribution, it requires more effort to accelerate disc A (change its angular velocity) because its mass is distributed further from its axis of rotation: mass that is further out from that axis must, for a given angular velocity, move more quickly than mass closer in. In this case, disc A has a larger moment of inertia than disc B.

Divers minimizing their moments of inertia in order to increase their rates of rotation.

The moment of inertia of an object can change if its shape changes. A figure skater who begins a spin with arms outstretched provides a striking example. By pulling in her arms, she reduces her moment of inertia, causing her to spin faster (by the conservation of angular momentum). Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... This box:      This gyroscope remains upright while spinning due to its angular momentum. ...

The moment of inertia has two forms, a scalar form I (used when the axis of rotation is known) and a more general tensor form that does not require knowing the axis of rotation. The scalar moment of inertia I (often called simply the "moment of inertia") allows a succinct analysis of many simple problems in rotational dynamics, such as objects rolling down inclines and the behavior of pulleys. For instance, while a block of any shape will slide down a frictionless decline at the same rate, rolling objects may descend at different rates, depending on their moments of inertia. A hoop will descend more slowly than a solid disk of equal mass and radius because more of its mass is located far from the axis of rotation, and thus needs to move faster if the hoop rolls at the same angular velocity. However, for (more complicated) problems in which the axis of rotation can change, the scalar treatment is inadequate, and the tensor treatment must be used (although shortcuts are possible in special situations). Examples requiring such a treatment include gyroscopes, tops, and even satellites, all objects whose alignment can change. See scalar for an account of the broader concept also used in mathematics and computer science. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

The moment of inertia can also be called the mass moment of inertia (especially by mechanical engineers) to avoid confusion with the second moment of area, which is sometimes called the moment of inertia (especially by structural engineers) and denoted by the same symbol I. The easiest way to differentiate these quantities is through their units. In addition, the moment of inertia should not be confused with the polar moment of inertia, which is a measure of an object's ability to resist torsion (twisting). The second moment of area, also known as the area moment of inertia or second moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection which are directly proportional. ... The former Weights and Measures office in Middlesex, England. ... Polar moment of inertia is a quantity used to predict an objects ability to resist torsion, in objects (or segments of objects) with an invariant circular cross-section and no significant warping or out-of-plane deformation. ... Look up torsion in Wiktionary, the free dictionary. ...

## Scalar moment of inertia

### Definition

A simple definition of the moment of inertia of any object, be it a point mass or a 3D-structure, is given by:

$I = int r^2 ,dm$

where

m is the mass,
and r is the (perpendicular) distance of the point mass to the axis of rotation.

#### Detailed Analysis

The (scalar) moment of inertia of a point mass rotating about a known axis is defined by A point mass in physics is an idealisation of a body whose dimensions can be neglected compared to the distances of its movement. ...

$I triangleq m r^2,!$

The moment of inertia is additive. Thus, for a rigid body consisting of N point masses mi with distances ri to the rotation axis, the total moment of inertia equals the sum of the point-mass moments of inertia: In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ...

$I triangleq sum_{i=1}^{N} {m_{i} r_{i}^2},!$

For a solid body described by a continuous mass density function ρ(r), the moment of inertia about a known axis can be calculated by integrating the square of the distance (weighted by the mass density) from a point in the body to the rotation axis: This article is about the concept of integrals in calculus. ...

$I triangleq iiint_V r^2 ,rho(boldsymbol{r}),dV !$

where

V is the volume occupied by the object.
ρ is the spatial density function of the object, and
$boldsymbol{r} equiv (r,theta,phi),(x,y,z), or (r,theta,z)$are coordinates of a point inside the body.
Diagram for the calculation of a disk's moment of inertia. Here k is 1/2 and r is the radius used in determining the moment.

Based on dimensional analysis alone, the moment of inertia of a non-point object must take the form: For other uses, see Density (disambiguation). ... MetaPost image of a disc, with dimensions and axes specified. ... MetaPost image of a disc, with dimensions and axes specified. ... Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. ...

$I = kcdot Mcdot {R}^2 ,!$

where

M is the mass
R is the radius of the object from the center of mass (in some cases, the length of the object is used instead.)
k is a dimensionless constant called the inertia constant that varies with the object in consideration.

Inertial constants are used to account for the differences in the placement of the mass from the center of rotation. Examples include:

• k = 1, thin ring or thin-walled cylinder around its center,
• k = 2/5, solid sphere around its center
• k = 1/2, solid cylinder or disk around its center.

For more examples, see the List of moments of inertia. The following is a list of moments of inertia. ...

### Parallel axis theorem

Main article: Parallel axis theorem

Once the moment of inertia has been calculated for rotations about the center of mass of a rigid body, one can conveniently recalculate the moment of inertia for all parallel rotation axes as well, without having to resort to the formal definition. If the axis of rotation is displaced by a distance R from the center of mass axis of rotation (e.g. spinning a disc about a point on its periphery, rather than through its center,) the displaced and center-moment of inertia are related as follows: 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. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

$I_{mathrm{displaced}} = I_{mathrm{center}} + M R^{2} ,!$

This theorem is also known as the parallel axes rule and is a special case of Steiner's parallel-axis theorem.

### Composite bodies

If a body can be decomposed (either physically or conceptually) into several constituent parts, then the moment of inertia of the body about a given axis is obtained by summing the moments of inertia of each constituent part around the same given axis[1].

### Equations involving the moment of inertia

The rotational kinetic energy of a rigid body can be expressed in terms of its moment of inertia. For a system with N point masses mi moving with speeds vi, the rotational kinetic energy T equals The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ...

$T = sum_{i=1}^{N} frac{1}{2} m_{i} v_{i}^{2},! = sum_{i=1}^{N} frac{1}{2} m_{i} (omega r_{i})^{2} = frac{1}{2} sum_{i=1}^{N} m_{i} r_{i}^{2} omega^{2} = frac{1}{2} I omega^{2}$

where ω is the common angular velocity (in radians per second). The final formula $T=frac{1}{2} I omega^{2},!$ also holds for a continuous distribution of mass with a generalisation of the above derivation from a discrete summation to an integration. For the musical group, see Radian (band). ... This article is about the concept of integrals in calculus. ...

In the special case where the angular momentum vector is parallel to the angular velocity vector, one can relate them by the equation This box:      This gyroscope remains upright while spinning due to its angular momentum. ... This article is about vectors that have a particular relation to the spatial coordinates. ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ...

$L = Iomega ;$

where L is the angular momentum and ω is the angular velocity. However, this equation does not hold in many cases of interest, such as the torque-free precession of a rotating object, although its more general tensor form is always correct. Precession redirects here. ...

When the moment of inertia is constant, one can also relate the torque on an object and its angular acceleration in a similar equation: For other senses of this word, see torque (disambiguation). ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ...

$tau = Ialpha !$

where τ is the torque and α is the angular acceleration.

## Moment of inertia tensor

For the same object, different axes of rotation will have different moments of inertia about those axes. In general, the moments of inertia are not equal unless the object is symmetric about all axes. The moment of inertia tensor is a convenient way to summarize all moments of inertia of an object with one quantity. It may be calculated with respect to any point in space, although for practical purposes the center of mass is most commonly used. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

### Definition

For a rigid object of N point masses mk, the moment of inertia tensor is given by In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

$mathbf{I} = begin{bmatrix} I_{xx} & I_{xy} & I_{xz} I_{yx} & I_{yy} & I_{yz} I_{zx} & I_{zy} & I_{zz} end{bmatrix}$.

Its components are defined as

$I_{ij} stackrel{mathrm{def}}{=} sum_{k=1}^{N} m_{k} (r_k^{2}delta_{ij} - r_{ki}r_{kj}),!$

where

i, j equal 1, 2, or 3 for x, y, and z, respectively,
rk is the distance of mass k from the point about which the tensor is calculated, and
δij is the Kronecker delta.

The diagonal elements are more succinctly written as In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

$I_{xx} stackrel{mathrm{def}}{=} sum_{k=1}^{N} m_{k} (y_{k}^{2}+z_{k}^{2}),,!$
$I_{yy} stackrel{mathrm{def}}{=} sum_{k=1}^{N} m_{k} (x_{k}^{2}+z_{k}^{2}),,!$
$I_{zz} stackrel{mathrm{def}}{=} sum_{k=1}^{N} m_{k} (x_{k}^{2}+y_{k}^{2}),,!$

while the off-diagonal elements, also called the products of inertia, are

$I_{xy} = I_{yx} stackrel{mathrm{def}}{=} -sum_{k=1}^{N} m_{k} x_{k} y_{k},,!$
$I_{xz} = I_{zx} stackrel{mathrm{def}}{=} -sum_{k=1}^{N} m_{k} x_{k} z_{k},,!$ and
$I_{yz} = I_{zy} stackrel{mathrm{def}}{=} -sum_{k=1}^{N} m_{k} y_{k} z_{k},,!$

Here Ixx denotes the moment of inertia around the x-axis when the objects are rotated around the x-axis, Ixy denotes the moment of inertia around the y-axis when the objects are rotated around the x-axis, and so on.

These quantities can be generalized to an object with continuous density in a similar fashion to the scalar moment of inertia. One then has

$mathbf{I}=iiint_V rho(x,y,z)left( r^2 mathbf{E}_{3} - mathbf{r}otimes mathbf{r}right), dx,dy,dz,$

where $mathbf{r}otimes mathbf{r}$ is their outer product, E3 is the 3 × 3 identity matrix, and V is a region of space completely containing the object. Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ...

### Derivation of the tensor components

The distance r of a particle at $mathbf{x}$ from the axis of rotation passing through the origin in the $mathbf{hat{n}}$ direction is $|mathbf{x}-(mathbf{x} cdot mathbf{hat{n}}) mathbf{hat{n}}|$. By using the formula I = mr2 (and some simple vector algebra) it can be seen that the moment of inertia of this particle (about the axis of rotation passing through the origin in the $mathbf{hat{n}}$ direction) is $I=m(|mathbf{x}|^2 (mathbf{hat{n}} cdot mathbf{hat{n}})-(mathbf{x} cdot mathbf{hat{n}})^2).$ This is a quadratic form in $mathbf{hat{n}}$ and, after a bit more algebra, this leads to a tensor formula for the moment of inertia In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

${I} = m [n_1,n_2,n_3]begin{bmatrix} y^2+z^2 & -xy & -xz -y x & x^2+z^2 & -yz -zx & -zy & x^2+y^2 end{bmatrix} begin{bmatrix} n_1 n_2 n_3 end{bmatrix}$.

This is exactly the formula given below for the moment of inertia in the case of a single particle. For multiple particles we need only recall that the moment of inertia is additive in order to see that this formula is correct.

### Reduction to scalar

For any axis $hat{mathrm{n}}$, represented as a column vector with elements ni, the scalar form I can be calculated from the tensor form I as

$I = mathbf{hat{n}^{T}} mathbf{I}, mathbf{hat{n}} = sum_{j=1}^{3} sum_{k=1}^{3} n_{j} I_{jk} n_{k} .$

The range of both summations correspond to the three Cartesian coordinates. Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

The following equivalent expression avoids the use of transposed vectors which are not supported in maths libraries because internally vectors and their transpose are stored as the same linear array,

$I = mathbf{{I}^{T}} mathbf{hat{n}} cdot mathbf{hat{n}}.$

However it should be noted that although this equation is mathematically equivalent to the equation above for any matrix, inertia tensors are symmetrical. This means that it can be further simplified to:

$I = mathbf{{I}} mathbf{hat{n}} cdot mathbf{hat{n}}.$

### Principal moments of inertia

Since the moment of inertia tensor is real and symmetric, it is possible to find a Cartesian coordinate system in which it is diagonal, having the form In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...

$mathbf{I} = begin{bmatrix} I_{1} & 0 & 0 0 & I_{2} & 0 0 & 0 & I_{3} end{bmatrix}$

where the coordinate axes are called the principal axes and the constants I1, I2 and I3 are called the principal moments of inertia. The unit vectors along the principal axes are usually denoted as $(mathbf{e}_{1}, mathbf{e}_{2}, mathbf{e}_{3})$. In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

When all principal moments of inertia are distinct, the principal axes are uniquely specified. If two principal moments are the same, the rigid body is called a symmetrical top and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a spherical top (although it need not be spherical) and any axis can be considered a principal axis, meaning that the moment of inertia is the same about any axis.

The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order m, i.e., is symmetrical under rotations of 360°/m about a given axis, the symmetry axis is a principal axis. When m > 2, the rigid body is a symmetrical top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, e.g., a cube or any other Platonic solid. A practical example of this mathematical phenomenon is the routine automotive task of balancing a tire, which basically means adjusting the distribution of mass of a car wheel such that its principal axis of inertia is aligned with the axle so the wheel does not wobble. In geometry, a Platonic solid is a convex regular polyhedron. ... Tire weight Tire Balance, also referred to as tire imbalance and tire unbalance, describes the unsymmetrical distribution of mass within an automobile tire and/or the wheel to which it is attached. ...

### Parallel axis theorem

Once the moment of inertia tensor has been calculated for rotations about the center of mass of the rigid body, there is a useful labor-saving method to compute the tensor for rotations offset from the center of mass. In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

If the axis of rotation is displaced by a vector R from the center of mass, the new moment of inertia tensor equals

$mathbf{I}^{mathrm{displaced}} = mathbf{I}^{mathrm{center}} + M left[ left(mathbf{R} cdot mathbf{R}right) mathbf{E}_{3} - mathbf{R} otimes mathbf{R} right]$

where M is the total mass of the rigid body, E3 is the 3 × 3 identity matrix, and $otimes$ is the outer product. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ...

### Other mechanical quantities

Using the tensor I, the kinetic energy can be written as a quadratic form

$T = frac{1}{2} boldsymbolomega^T mathbf{I}, boldsymbolomega = frac{1}{2} I_{1} omega_{1}^{2} + frac{1}{2} I_{2} omega_{2}^{2} + frac{1}{2} I_{3} omega_{3}^{2}$

and the angular momentum can be written as a product

$mathbf{L} = mathbf{I}, boldsymbolomega = omega_{1} I_{1} mathbf{e}_{1} + omega_{2} I_{2} mathbf{e}_{2} + omega_{3} I_{3} mathbf{e}_{3}$

Taken together, one can express the rotational kinetic energy in terms of the angular momentum (L1,L2,L3) in the principal axis frame as

$T = frac{L_{1}^{2}}{2I_{1}} + frac{L_{2}^{2}}{2I_{2}} + frac{L_{3}^{2}}{2I_{3}}.,!$

The rotational kinetic energy and the angular momentum are constants of the motion (conserved quantities) in the absence of an overall torque. The angular velocity ω is not constant; even without a torque, the endpoint of this vector may move in a plane (see Poinsot's construction). For other senses of this word, see torque (disambiguation). ... Poinsots construction is a geometrical method for visualizing the torque-free motion of a rotating rigid body. ...

See the article on the rigid rotor for more ways of expressing the kinetic energy of a rigid body. The rigid rotor is a mechanical model that is used to explain rotating systems. ... In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ...

The following is a list of moments of inertia. ... This list of moment of inertia tensors is given for principal axes of each object. ... The rotational energy or angular kinetic energy is the kinetic energy due to the rotation of an object and is part of its total kinetic energy. ... 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. ... In physics, the perpendicular axis theorem (or plane figure theorem) can be used to determine the moment of inertia of a rigid object that lies entirely within a plane, about an axis at right angles to the plane, given the moments of inertia of the object about two perpendicular axes... The stretch rule states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to the axis of rotation, without changing the distribution of mass. ...

## References

• Goldstein H. (1980) Classical Mechanics, 2nd. ed., Addison-Wesley. ISBN 0-201-02918-9
• Landau LD and Lifshitz EM. (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover).
• Marion JB and Thornton ST. (1995) Classical Dynamics of Systems and Particles, 4th. ed., Thomson. ISBN 0-03-097302-3
• Symon KR. (1971) Mechanics, 3rd. ed., Addison-Wesley. ISBN 0-201-07392-7
• Tenenbaum, RA. (2004) Fundamentals of Applied Dynamics, Springer. ISBN 0-387-00887-X

Results from FactBites:

 Moment of Inertia (0 words) Moment of inertia is a measurement of the 'rotatibility' of an object. The moment of inertia of mass around an axis of rotation (assuming an infinitely stiff connection) is the mass multiplied by the square of the distance to the rotation axis. The additional moment of inertia generated by a stabilizer depends on three factors: mass, distance of mass to the bow rotation axis and the stiffness of the stabilizer rod.
 NationMaster - Encyclopedia: Moment of inertia (2632 words) 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. Moment of inertia is a measurement of the 'rotatibility' of an object. The moment of inertia of mass around an axis of rotation (assuming an infinitely stiff connection) is the mass multiplied by the square of the distance to the rotation axis.
More results at FactBites »

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