Moment of inertia

thumb|[[Tightrope walkers use the moment of inertia of a long rod for balance as they walk the rope. Samuel Dixon crossing the Niagara River in 1890.]]

The moment of inertia (also known as mass moment of inertia, angular/rotational mass, second moment of mass, or rotational inertia) is a measure of how difficult it is to change the rotation rate of a rigid body about a given axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis. The term moment of inertia ("momentum inertiae" in Latin) was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Euler's second law.

The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body.

The moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor.

The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw.

Definition

The moment of inertia is defined as the product of mass of section and the square of the distance between the reference axis and the centroid of the section. It is denoted with the symbol I or J. thumb|right|upright|Spinning figure skaters can reduce their moment of inertia by pulling in their arms, allowing them to spin faster due to [[conservation of angular momentum.]]

thumb|right|Video of rotating chair experiment, illustrating moment of inertia. When the spinning professor pulls his arms, his moment of inertia decreases; to conserve angular momentum, his angular velocity increases.

The moment of inertia is also defined as the ratio of the net angular momentum of a system to its angular velocity around a principal axis,

If the shape of the body does not change, then its moment of inertia appears in Newton's law of motion as the ratio of an applied torque on a body to the angular acceleration around a principal axis, that is

<math qid=Q48103 display="block">\tau = I \alpha.</math>

For a simple pendulum, this definition yields a formula for the moment of inertia in terms of the mass of the pendulum and its distance from the pivot point as,

Thus, the moment of inertia of the pendulum depends on both the mass of a body and its geometry, or shape, as defined by the distance to the axis of rotation.

This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses each multiplied by the square of its perpendicular distance to an axis . An arbitrary object's moment of inertia thus depends on the spatial distribution of its mass.

In general, given an object of mass , an effective radius can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is

where is known as the radius of gyration around the axis.

Examples

Simple pendulum

Mathematically, the moment of inertia of a simple pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum, this is found to be the product of the mass of the particle <math>m</math> with the square of its distance <math>r</math> to the pivot, that is

This can be shown as follows:

The force of gravity on the mass of a simple pendulum generates a torque <math>\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}</math> around the axis perpendicular to the plane of the pendulum movement. Here <math>\mathbf{r}</math> is the distance vector from the torque axis to the pendulum center of mass, and <math>\mathbf{F}</math> is the net force on the mass. Associated with this torque is an angular acceleration, <math>\boldsymbol{\alpha}</math>, of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is <math>\mathbf{a} = \boldsymbol{\alpha} \times \mathbf{r}</math>. Since <math>\mathbf F = m \mathbf a</math> the torque equation becomes:

<math display="block">\begin{align}

\boldsymbol{\tau}

&= \mathbf{r} \times \mathbf{F}

= \mathbf{r} \times (m \boldsymbol{\alpha} \times \mathbf{r}) \\

&= m \left(\left(\mathbf{r} \cdot \mathbf{r}\right) \boldsymbol{\alpha} - \left(\mathbf{r} \cdot \boldsymbol{\alpha}\right) \mathbf{r}\right) \\

&= mr^2 \boldsymbol{\alpha}

= I\alpha \mathbf{\hat{k,

\end{align}</math>

where <math>\mathbf{\hat{k</math> is a unit vector perpendicular to the plane of the pendulum. (The second to last step uses the vector triple product expansion with the perpendicularity of <math>\boldsymbol{\alpha}</math> and <math>\mathbf{r}</math>.) The quantity <math>I = mr^2</math> is the moment of inertia of this single mass around the pivot point.

The quantity <math>I = mr^2</math> also appears in the angular momentum of a simple pendulum, which is calculated from the velocity <math>\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}</math> of the pendulum mass around the pivot, where <math>\boldsymbol{\omega}</math> is the angular velocity of the mass about the pivot point. This angular momentum is given by

<math display="block">\begin{align}

\mathbf{L}

&= \mathbf{r} \times \mathbf{p}

= \mathbf{r} \times \left(m\boldsymbol{\omega} \times \mathbf{r}\right) \\

& = m\left(\left(\mathbf{r} \cdot \mathbf{r}\right)\boldsymbol{\omega} - \left(\mathbf{r} \cdot \boldsymbol{\omega}\right)\mathbf{r}\right) \\

&= mr^2 \boldsymbol{\omega}

= I\omega\mathbf{\hat{k,

\end{align}</math>

using a similar derivation to the previous equation.

Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield

<math display="block">E_\text{K} = \frac{1}{2} m \mathbf{v} \cdot \mathbf{v} = \frac{1}{2} \left(mr^2\right)\omega^2 = \frac{1}{2}I\omega^2.</math>

This shows that the quantity <math>I = mr^2</math> is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values <math>mr^2</math> for all of the elements of mass in the body.

Compound pendulums

thumb|left|Pendulums used in Mendenhall [[gravimeter apparatus, from 1897 scientific journal. The portable gravimeter developed in 1890 by Thomas C. Mendenhall provided the most accurate relative measurements of the local gravitational field of the Earth.]]

A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of. The natural frequency (<math>\omega_\text{n}</math>) of a compound pendulum depends on its moment of inertia, <math>I_P</math>,

<math display="block">\omega_\text{n} = \sqrt{\frac{mgr}{I_P,</math>

where <math>m</math> is the mass of the object, <math>g</math> is local acceleration of gravity, and <math>r</math> is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body. The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.

Moment of inertia of area

Moment of inertia of area is also known as the second moment of area and its physical meaning is completely different from the mass moment of inertia.

These calculations are commonly used in civil engineering for structural design of beams and columns. Cross-sectional areas calculated for vertical moment of the x-axis <math>I_{xx}</math> and horizontal moment of the y-axis <math>I_{yy}</math>.

Height (h) and breadth (b) are the linear measures, except for circles, which are effectively half-breadth derived, <math>r</math>

Sectional areas moment formulae:

Square: <math>I_{xx}=I_{yy}=\frac{b^4}{12}</math>
Rectangular: <math>I_{xx}=\frac{bh^3}{12}</math> and; <math>I_{yy}=\frac{hb^3}{12}</math>
Triangular: <math>I_{xx}=\frac{bh^3}{36}</math>
Circular: <math>I_{xx}=I_{yy}=\frac{1}{4} {\pi} r^4=\frac{1}{64} {\pi} d^4</math>

Motion in a fixed plane

Point mass

thumb|right|Four objects with identical masses and radii rolling down a plane without slipping. From back to front: The time to reach the finishing line is longer for objects with a greater moment of inertia. ([[:File:Rolling Racers - Moment of inertia.ogv|OGV version)]]

The moment of inertia about an axis of a body is calculated by summing <math>mr^2</math> for every particle in the body, where <math>r</math> is the perpendicular distance to the specified axis. To see how moment of inertia arises in the study of the movement of an extended body, it is convenient to consider a rigid assembly of point masses. (This equation can be used for axes that are not principal axes provided that it is understood that this does not fully describe the moment of inertia.)

Consider the kinetic energy of an assembly of <math>N</math> masses <math>m_i</math> that lie at the distances <math>r_i</math> from the pivot point <math>P</math>, which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of the individual masses,

E_\text{K} =

\sum_{i=1}^N \frac{1}{2}\,m_i \mathbf{v}_i \cdot \mathbf{v}_i =

\sum_{i=1}^N \frac{1}{2}\,m_i \left(\omega r_i\right)^2 =

\frac12\, \omega^2 \sum_{i=1}^N m_i r_i^2.

</math>

This shows that the moment of inertia of the body is the sum of each of the <math>mr^2</math> terms, that is

Thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia.

The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, except with infinitely many point particles. Thus the limits of summation are removed, and the sum is written as follows:

Another expression replaces the summation with an integral,

<math display="block">I_P = \iiint_{Q} \rho(x, y, z) \left\|\mathbf{r}\right\|^2 dV</math>

Here, the function <math>\rho</math> gives the mass density at each point <math>(x, y, z)</math>, <math>\mathbf{r}</math> is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point <math>(x, y, z)</math> in the solid, and the integration is evaluated over the volume <math>V</math> of the body <math>Q</math>. The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area.

Note on second moment of area: The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused. The moment of inertia of a body with the shape of the cross-section is the second moment of this area about the <math>z</math>-axis perpendicular to the cross-section, weighted by its density. This is also called the polar moment of the area, and is the sum of the second moments about the <math>x</math>- and <math>y</math>-axes. The stresses in a beam are calculated using the second moment of the cross-sectional area around either the <math>x</math>-axis or <math>y</math>-axis depending on the load.

Examples

thumb|right

The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass.

If a system of <math>n</math> particles, <math>P_i, i = 1, \dots, n</math>, are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point <math>\mathbf{R}</math>, and absolute velocities <math>\mathbf{v}_i</math>:

<math display="block">\begin{align}

\Delta\mathbf{r}_i &= \mathbf{r}_i - \mathbf{R}, \\

\mathbf{v}_i &= \boldsymbol{\omega} \times \left(\mathbf{r}_i - \mathbf{R}\right) + \mathbf{V}

= \boldsymbol{\omega} \times \Delta\mathbf{r}_i + \mathbf{V},

\end{align}</math>

where <math>\boldsymbol{\omega}</math> is the angular velocity of the system and <math>\mathbf{V}</math> is the velocity of <math>\mathbf{R}</math>.

For planar movement the angular velocity vector is directed along the unit vector <math>\mathbf{k}</math> which is perpendicular to the plane of movement. Introduce the unit vectors <math>\mathbf{e}_i</math> from the reference point <math>\mathbf{R}</math> to a point <math>\mathbf{r}_i</math>, and the unit vector <math>\mathbf{\hat{t_i = \mathbf{\hat{k \times \mathbf{\hat{e_i</math>, so

<math display="block">\begin{align}

\mathbf{\hat{e_i &= \frac{\Delta\mathbf{r}_i}{\Delta r_i},\quad

\mathbf{\hat{k = \frac{\boldsymbol{\omega{\omega},\quad

\mathbf{\hat{t_i = \mathbf{\hat{k \times \mathbf{\hat{e_i, \\

\mathbf{v}_i &= \boldsymbol{\omega} \times \Delta\mathbf{r}_i + \mathbf{V}

= \omega\mathbf{\hat{k \times \Delta r_i\mathbf{\hat{e_i + \mathbf{V}

= \omega\, \Delta r_i\mathbf{\hat{t_i + \mathbf{V}

\end{align}</math>

This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.

Note on the cross product: When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from the geometric properties of the cross product. For this reason, in this section on planar movement the angular velocity and accelerations of the body are vectors perpendicular to the ground plane, and the cross product operations are the same as used for the study of spatial rigid body movement.

Angular momentum

The angular momentum vector for the planar movement of a rigid system of particles is given by

Let the system of <math>n</math> particles, <math>P_i, i = 1, \dots, n</math> be located at the coordinates <math>\mathbf{r}_i</math> with velocities <math>\mathbf{v}_i</math> relative to a fixed reference frame. For a (possibly moving) reference point <math>\mathbf{R}</math>, the relative positions are

<math display="block">\Delta\mathbf{r}_i = \mathbf{r}_i - \mathbf{R}</math>

and the (absolute) velocities are

<math display="block">\mathbf{v}_i = \boldsymbol{\omega} \times \Delta\mathbf{r}_i + \mathbf{V}_\mathbf{R}</math>

where <math>\boldsymbol{\omega}</math> is the angular velocity of the system, and <math>\mathbf{V_R}</math> is the velocity of <math>\mathbf{R}</math>.

Angular momentum

Note that the cross product can be equivalently written as matrix multiplication by combining the first operand and the operator into a skew-symmetric matrix, <math>\left[\mathbf{b}\right]</math>, constructed from the components of <math>\mathbf{b} = (b_x, b_y, b_z)</math>:

<math display="block">\begin{align}

\mathbf{b} \times \mathbf{y}

&\equiv \left[\mathbf{b}\right] \mathbf{y} \\

\left[\mathbf{b}\right] &\equiv \begin{bmatrix}

0 & -b_z & b_y \\

b_z & 0 & -b_x \\

-b_y & b_x & 0

\end{bmatrix}.

\end{align}</math>

The inertia matrix is constructed by considering the angular momentum, with the reference point <math>\mathbf{R}</math> of the body chosen to be the center of mass <math>\mathbf{C}</math>:

<math display="block">\mathbf{I} = \mathbf{R}\mathbf{I_0}\mathbf{R}^\textsf{T}</math>

Inertia matrix in different reference frames

The use of the inertia matrix in Newton's second law assumes its components are computed relative to axes parallel to the inertial frame and not relative to a body-fixed reference frame. When the body has an axis of symmetry (sometimes called the figure axis or axis of figure) then the other two moments of inertia will be identical and any axis perpendicular to the axis of symmetry will be a principal axis.

A toy top is an example of a rotating rigid body, and the word top is used in the names of types of rigid bodies. When all principal moments of inertia are distinct, the principal axes through center of mass are uniquely specified and the rigid body is called an asymmetric top. If two principal moments are the same, the rigid body is called a symmetric 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 <math>m</math>, meaning it is symmetrical under rotations of about the given axis, that axis is a principal axis. When <math>m > 2</math>, the rigid body is a symmetric 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, for example, a cube or any other Platonic solid.

The motion of vehicles is often described in terms of yaw, pitch, and roll which usually correspond approximately to rotations about the three principal axes. If the vehicle has bilateral symmetry then one of the principal axes will correspond exactly to the transverse (pitch) axis.

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.

Rotating molecules are also classified as asymmetric, symmetric, or spherical tops, and the structure of their rotational spectra is different for each type.

Ellipsoid

thumb|right|An ellipsoid with the semi-principal diameters labelled <math>a</math>, <math>b</math>, and <math>c</math>.

The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body called Poinsot's ellipsoid. Let <math>\boldsymbol{\Lambda}</math> be the inertia matrix relative to the center of mass aligned with the principal axes, then the surface

<math display="block">\mathbf{x}^\mathsf{T}\boldsymbol{\Lambda}\mathbf{x} = 1,</math>

defines an ellipsoid in the body frame. Write this equation in the form,

<math display="block"> \left(\frac{x}{1/\sqrt{I_1\right)^2 + \left(\frac{y}{1/\sqrt{I_2\right)^2 + \left(\frac{z}{1/\sqrt{I_3\right)^2 = 1,</math>

to see that the semi-principal diameters of this ellipsoid are given by

Let a point <math>\mathbf{x}</math> on this ellipsoid be defined in terms of its magnitude and direction, <math>\mathbf{x} = \|\mathbf{x}\|\mathbf{n}</math>, where <math>\mathbf{n}</math> is a unit vector. Then the relationship presented above, between the inertia matrix and the scalar moment of inertia <math>I_\mathbf{n}</math> around an axis in the direction <math>\mathbf{n}</math>, yields

<math display="block">\mathbf{x}^\mathsf{T}\boldsymbol{\Lambda}\mathbf{x} = \|\mathbf{x}\|^2\mathbf{n}^\mathsf{T}\boldsymbol{\Lambda}\mathbf{n} = \|\mathbf{x}\|^2 I_\mathbf{n} = 1. </math>

Thus, the magnitude of a point <math>\mathbf{x}</math> in the direction <math>\mathbf{n}</math> on the inertia ellipsoid is

<math display="block"> \|\mathbf{x}\| = \frac{1}{\sqrt{I_\mathbf{n}.</math>

<!---duplicated above

Parallel axis theorem for the inertia matrix

It is useful to note here that if the moment of inertia matrix or tensor is relative to the center of mass, then it can be determined relative to any other reference point in the body using the parallel axis theorem. If [I] is the moment of inertia matrix in the body frame relative to the center of mass C, then the moment of inertia matrix [I] in the same frame but relative to a different point R is given by

where M is the mass of the body, and [d] is the skew-symmetric matrix obtained from the vector d = R − C.

The tensor form of the parallel axis theorem is given by

<math display="block"> \mathbf{I}_R^B = \mathbf{I}_C^B + M((\mathbf{d} \cdot \mathbf{d}) \mathbf{E} - \mathbf{d} \otimes \mathbf{d}).

</math>

-->

References

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External links

Angular momentum and rigid-body rotation in two and three dimensions
Lecture notes on rigid-body rotation and moments of inertia
The moment of inertia tensor
An introductory lesson on moment of inertia: keeping a vertical pole not falling down (Java simulation)
Tutorial on finding moments of inertia, with problems and solutions on various basic shapes
Notes on mechanics of manipulation: the angular inertia tensor
Easy to use and Free Moment of Inertia Calculator online

Definition

Examples

Simple pendulum

Compound pendulums

Moment of inertia of area

Motion in a fixed plane

Point mass

Examples

Angular momentum

Angular momentum

Inertia matrix in different reference frames

Ellipsoid

Parallel axis theorem for the inertia matrix

See also

References

External links