Moment of inertia

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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.

Moment of inertia, also called mass moment of inertia, rotational inertia, or the angular mass, (SI units kg·m²) is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass, the inertia of a rigid rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, determining the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. The symbol I and sometimes J are usually used to refer to the moment of inertia.

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 gyroscopic motion.

The concept was introduced by Euler in his book a Theoria motus corporum solidorum seu rigidorum in 1730.^[1] In this book, he discussed the moment of inertia and many related concepts, such as the principal axis of inertia.

Contents

• 1 Overview
• 2 Scalar moment of inertia
  • 2.1 Definition
    • 2.1.1 Detailed analysis
  • 2.2 Parallel axis theorem
  • 2.3 Composite bodies
  • 2.4 Equations involving the moment of inertia
• 3 Moment of inertia tensor
  • 3.1 Definition
  • 3.2 Derivation of the tensor components
  • 3.3 Reduction to scalar
  • 3.4 Principal moments of inertia
  • 3.5 Parallel axis theorem
  • 3.6 Rotational symmetry
  • 3.7 Comparison with covariance matrix
• 4 See also
• 5 Notes
• 6 References
• 7 External links

[edit] Overview

The moment of inertia of an object about a given axis describes howdifficult it is to change its angular motion about that axis.Therefore, it encompasses not just how much mass the object hasoverall, but how far each bit of mass is from the axis. The farther outthe object's mass is, the more rotational inertia the object has, andthe more force is required to change its rotation rate. For example,consider two hoops, A and B, made of the same material and of equalmass. Hoop A is larger in diameter but thinner than B. It requires moreeffort to accelerate hoop A (change its angular velocity) because itsmass is distributed farther from its axis of rotation: mass that isfarther out from that axis must, for a given angular velocity, movemore quickly than mass closer in. So in this case, hoop A has a largermoment of inertia than hoop B.

The moment of inertia of an object can change if its shape changes.A figure skater who begins a spin with arms outstretched provides astriking example. By pulling in her arms, she reduces her moment ofinertia, causing her to spin faster (by the conservation of 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. Forinstance, while a block of any shape will slide down a frictionlessdecline at the same rate, rolling objects may descend at differentrates, depending on their moments of inertia. A hoop will descend moreslowly than a solid disk of equal mass and radius because more of itsmass is located far from the axis of rotation, and thus needs to movefaster 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 beused (although shortcuts are possible in special situations). Examplesrequiring such a treatment include gyroscopes, tops, and evensatellites, all objects whose alignment can change.

The moment of inertia (I) is also 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 bystructural engineers). The easiest way to differentiate thesequantities is through their units (kg.m² vs m⁴). In addition, moment of inertia should not be confused with polar moment of inertia, which is a measure of an object's ability to resist torsion (twisting) only.

[edit] Scalar moment of inertia

[edit] Definition

A simple definition of the moment of inertia (with respect to a given axis of rotation) of any object, be it a point mass or a 3D-structure, is given by:

where m is mass and r is the perpendicular distance to the axis of rotation.

[edit] Detailed analysis

The (scalar) moment of inertia of a point mass rotating about a known axis is defined by

The moment of inertia is additive. 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:

The mass distribution along the axis of rotation has no effect on the moment of inertia.

For a solid body described by a 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:

where

V is the volume occupied by the object.

ρ is the spatial density function of the object, and

r = (r,θ,φ), (x,y,z), or (r,θ,z) is the vector (orthogonal to the axis of rotation) between the axis of rotation and the point in the body.

Based on dimensional analysis alone, the moment of inertia of a non-point object must take the form:

where

M is the mass

L is a length dimension taken from the centre of mass (in some cases, the length of the object is used instead.)

c is a dimensionless constant called the inertial constant that varies with the object in consideration.

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

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

When c is 1, the length (L) is called the radius of gyration.

For more examples, see the List of moments of inertia.

[edit] Parallel axis theorem

Once the moment of inertia has been calculated for rotations about the center of massof a rigid body, one can conveniently recalculate the moment of inertiafor all parallel rotation axes as well, without having to resort to theformal definition. If the axis of rotation is displaced by a distance rfrom the center of mass axis of rotation (e.g., spinning a disc about apoint on its periphery, rather than through its center,) the displacedand center-moment of inertia are related as follows:

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

[edit] Composite bodies

If a body can be decomposed (either physically or conceptually) intoseveral constituent parts, then the moment of inertia of the body abouta given axis is obtained by summing the moments of inertia of eachconstituent part around the same given axis.^[2]

[edit] 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 m_i moving with speeds v_i, the rotational kinetic energy T equals

where ω is the common angular velocity (in radians per second). The final expression I ω² / 2 also holds for a mass density function with a generalization of the above derivation from a discrete summation to an integration.

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

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.

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

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

[edit] Moment of inertia tensor

For the same object, different axes of rotation will have differentmoments of inertia about those axes. In general, the moments of inertiaare not equal unless the object is symmetric about all axes. The moment of inertia tensoris a convenient way to summarize all moments of inertia of an objectwith one quantity. It may be calculated with respect to any point inspace, although for practical purposes the center of mass is mostcommonly used.

[edit] Definition

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

,

where

and I₁₂ = I₂₁, I₁₃ = I₃₁, and I₂₃ = I₃₂. (Thus I is a symmetric tensor.)

The diagonal elements of I are called the principal moments of inertia; the scalars I_ij with are called the products of inertia.

Here I_xx denotes the moment of inertia around the x-axis when the objects are rotated around the x-axis, I_xy 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 distributedmass, described by a mass density function, in a similar fashion to thescalar moment of inertia. One then has

where is their outer product, E₃ is the 3 × 3 identity matrix, and V is a region of space completely containing the object.

[edit] Derivation of the tensor components

The distance r of a particle at from the axis of rotation passing through the origin in the direction is . By using the formula I = mr²(and some simple vector algebra) it can be seen that the moment ofinertia of this particle (about the axis of rotation passing throughthe origin in the direction) is This is a quadratic form in and, after a bit more algebra, this leads to a tensor formula for the moment of inertia

.

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

[edit] Reduction to scalar

For any axis , represented as a column vector with elements n_i, the scalar form I can be calculated from the tensor form I as

The range of both summations correspond to the three Cartesian coordinates.

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

However it should be noted that although this equation ismathematically equivalent to the equation above for any matrix, inertiatensors are symmetrical. This means that it can be further simplifiedto:

[edit] Principal moments of inertia

By the spectral theorem, 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

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 (e₁, e₂, e₃). This result was first shown by J. J. Sylvester (1852), and is a form of Sylvester's law of inertia.

When all principal moments of inertia are distinct, the principalaxes are uniquely specified. If two principal moments are the same, therigid body is called a symmetrical top and there is no uniquechoice for the two corresponding principal axes. If all three principalmoments are the same, the rigid body is called a spherical top(although it need not be spherical) and any axis can be considered aprincipal axis, meaning that the moment of inertia is the same aboutany 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 twosymmetry axes that are not parallel or perpendicular to each other, itis 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 wheelsuch that its principal axis of inertia is aligned with the axle so thewheel does not wobble.

[edit] 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

where m is the total mass of the rigid body, E₃ is the 3 × 3 identity matrix, and is the outer product.

[edit] Rotational symmetry

Using the above equation to express all moments of inertia in termsof integrals of variables either along or perpendicular to the axis ofsymmetry usually simplifies the calculation of these momentsconsiderably.

[edit] Comparison with covariance matrix

The moment of inertia tensor about the center of mass of a 3 dimensional rigid body is related to the covariance matrix of a trivariate random vector whose probability density function is proportional to the pointwise density of the rigid body by:^{[citation needed]}

where n is the number of points.

The structure of the moment-of-inertia tensor comes from the fact that it is to be used as a bilinear form on rotation vectors in the form

Each element of mass has a kinetic energy of

The velocity of each element of mass is where r is a vector from the center of rotation to that element of mass. The cross product can be converted to matrix multiplication so that

and similarly

Thus

plugging in the definition of the term leads directly to the structure of the moment tensor.

[edit] See also