Moment of inertia

This article is about the moment of inertia of a rotating object. For the moment of inertia dealing with bending of a plane, see second moment of area. 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

Moment of inertia, also called mass moment of inertia or the angular mass, (SI units kg m², Former British units slug ft²), is the rotational analog of mass. Imperial units or the Imperial system is a collection of units first defined in the British Weights and Measures Act of 1824 That is, it is the inertia of a rigid rotating body with respect to its rotation. The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving The moment of inertia plays much the same role in rotational dynamics as mass does in basic dynamics, determining the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. Rotational motion can occur around more than one axis at once and can involve phenomena such as wobbling and Precession. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Do not confuse with Angular frequency The unit for angular velocity is rad/s A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about While a simple scalar treatment of the moment of inertia suffices for many situations, a more advanced tensor treatment allows the analysis of such complicated systems as spinning tops and gyroscope motion. In Physics, a scalar is a simple Physical quantity that is not changed by Coordinate system rotations or translations (in Newtonian mechanics or History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually A gyroscope is a device for measuring or maintaining orientation, based on the principles of Angular momentum.

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.

1 Overview
2 Scalar moment of inertia
3 Moment of inertia tensor
4 See also
5 References
6 External links

Overview

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). In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position

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. In Physics, a scalar is a simple Physical quantity that is not changed by Coordinate system rotations or translations (in Newtonian mechanics or History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually 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 frictionlessly down a 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 diameter 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.

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 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 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). Polar moment of inertia is a quantity used to predict an object's ability to resist torsion, in objects (or segments of objects with an invariant circular Cross-section In Solid mechanics, torsion is the twisting of an object due to an applied Torque.

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

$I \triangleq m r^2\,\!$

The moment of inertia is additive. Point mass is an Idealistic term used to describe either Matter which is infinitely small or an object which can be thought of as infinitely small Thus, for a rigid body consisting of $N$ point masses $m i$ with distances $r i$ to the rotation axis, the total moment of inertia equals the sum of the point-mass moments of inertia:

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

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:

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

where

V is the volume occupied by the object. In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

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. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different

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:

$I = k\cdot M\cdot {R}^2 \,\!$

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. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand )

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. Mass moments of inertia have units of dimension mass × length2

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. In Physics, the parallel axis theorem can be used to determine the Moment of inertia of a Rigid body about any axis given the moment of inertia of the 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:

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

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. The kinetic energy of an object is the extra Energy which it possesses due to its motion In Physics, a rigid body is an idealization of a solid body of finite size in which Deformation is neglected For a system with $N$ point masses $m i$ moving with speeds $v i$ , the rotational kinetic energy $T$ equals

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

where $ω$ is the common angular velocity (in radians per second). The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 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. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

In the special case where the angular momentum vector is parallel to the angular velocity vector, one can relate them by the equation

$L = I\omega \;$

where L is the angular momentum and $ω$ is the angular velocity. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Do not confuse with Angular frequency The unit for angular velocity is rad/s 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 refers to a change in the direction of the axis of a rotating object

When the moment of inertia is constant, one can also relate the torque on an object and its angular acceleration in a similar equation:

$\tau = I\alpha \!$

where $τ$ is the torque and $α$ is the angular acceleration. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about

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. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually It may be calculated with respect to any point in space, although for practical purposes the center of mass is most commonly used.

Definition

For a rigid object of $N$ point masses $m k$ , the moment of inertia tensor is given by

$\mathbf{I} = \begin{bmatrix} I_{xx} & I_{xy} & I_{xz} \\ I_{yx} & I_{yy} & I_{yz} \\ I_{zx} & I_{zy} & I_{zz} \end{bmatrix}$ . History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually

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,

r_k is the distance of mass k from the point about which the tensor is calculated, and

$δ i j$ is the Kronecker delta. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

The diagonal elements are more succinctly written as

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

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

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

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 $I x x$ denotes the moment of inertia around the $x$ -axis when the objects are rotated around the x-axis, $I x y$ 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. In Linear algebra, the outer product typically refers to the tensor product of two vectors. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main

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 $I = m r 2$ (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

${I} = m [n_1,n_2,n_3]\begin{bmatrix} y^2+z^2 & -xy & -xz \\ -y x & x^2+z^2 & -yz \\ -zx & -zy & x^2+y^2 \end{bmatrix} \begin{bmatrix} n_1 \\ n_2\\ n_3 \end{bmatrix}$ . In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables

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 n_i, 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. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

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

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

where the coordinate axes are called the principal axes and the constants $I 1$ , $I 2$ and $I 3$ are called the principal moments of inertia. In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} In Linear algebra, a Square matrix A is called diagonalizable if it is similar to a Diagonal matrix, i The unit vectors along the principal axes are usually denoted as $(\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3})$ .

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. In Geometry, a Platonic solid is a convex Regular polyhedron. 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. Tire Balance, also referred to as tire unbalance or imbalance describes the 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.

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 or unit matrix of size n is the n -by- n Square matrix with ones on the Main In Linear algebra, the outer product typically refers to the tensor product of two vectors.

Other mechanical quantities

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

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

and the angular momentum can be written as a product

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

Taken together, one can express the rotational kinetic energy in terms of the angular momentum $(L 1, L 2, L 3)$ 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. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about 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). In Classical mechanics, Poinsot's construction is a geometrical method for visualizing the torque-free motion of a rotating Rigid body, that is the motion of a rigid

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

References

Goldstein H. The following is a list of moments of inertia. Mass moments of inertia have units of dimension mass × length2 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. In Physics, the parallel axis theorem can be used to determine the Moment of inertia of a Rigid body about any axis given the moment of inertia of the 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 In Physics, 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 In Classical mechanics, Poinsot's construction is a geometrical method for visualizing the torque-free motion of a rotating Rigid body, that is the motion of a rigid (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

External links

Dictionary

-noun

(physics) A measure of a body's resistance to a change in its angular rotation velocity
(engineering) A measure of a body's resistance to bending; second moment of inertia; second moment of area.

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Contents

Overview

Scalar moment of inertia

Definition

Detailed Analysis

Parallel axis theorem

Composite bodies

Equations involving the moment of inertia

Moment of inertia tensor

Definition

Derivation of the tensor components

Reduction to scalar

Principal moments of inertia

Parallel axis theorem

Other mechanical quantities

See also

References

External links

Dictionary

moment of inertia

-noun