Moment of inertia

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

Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI unit kilogram metre squared kg m²) quantifies the rotational inertia of an object, i.e. its inertia with respect to rotational motion, in a manner somewhat analogous to how mass quantifies the inertia of an object with respect to translational motion. The symbols <math>I</math> and sometimes <math>J</math> are usually used to refer to the moment of inertia.

The moment of inertia is a mathematical construct that was created for the sole purpose of making the physics of rotation easier to understand and write. In many cases, a rotational equation involving moment of inertia has a similar structure to its linear counterpart. (For examples, see the applications section.)

Moment of inertia should not be confused with the second moment of area, which is sometimes called the moment of inertia (especially by structural engineers) and use the same symbol <math>I</math>. Because of this, some people (especially mechanical engineers) refer to the mass moment of inertia to avoid confusion.

Contents 1 Mathematical definition 2 Types of moment of inertia 3 Applications of moment of inertia 3.1 Torque 3.2 Kinetic Energy 4 Inertia tensor 5 See also 6 External links

Mathematical definition

For a point mass, the moment of inertia (around a given axis) is defined as:

<math>I = m r^2\,</math>

where

m is its mass,

and r is its perpendicular distance from the axis of rotation.

For a system with <math>N</math> point masses, each with mass <math>m_i</math> and distance <math>r_i</math>, moment of inertia is defined as the sum of the moments of inertia:

<math>I = \sum_{i=1}^N {m_i r_i^2}</math>

If the system isn't comprised of point masses, as in the case of a solid body, we need an infinite sum of the moments of all the points which make up the system. Since a solid body has an (approximately) continuous mass-density, integrating the moments of the masses, over all the three-dimensional space gives the moment of inertia I:

<math>I = \int_V r^2\,dm = \iiint_V {r^2\,\rho\,dv} = \iiint_V {r^2\,\rho\,dx\,dy\,dz} \!</math>

where

V is the volume of the object

r² dm is the infinitesimal moment of inertia

r is the distance from the axis of rotation

dm is an infinitesimal mass

ρ is the mass density of the object

dv is an infinitesimal volume

and dx, dy, and dz are infinitesimal lengths.

Types of moment of inertia

There are an infinite number of moments of inertia for any object, one for every possible axis of rotation through the object's centroid. For convenience, the three moments of inertia typically used for an object are about axes parallel to the three Cartesian axes (X, Y, and Z):

<math>I = \;</math> moment of inertia about the current axis of rotation

<math>I_{xx} = \;</math> moment of inertia about the line through the centroid, parallel to the X-axis

<math>I_{yy} = \;</math> moment of inertia about the line through the centroid, parallel to the Y-axis

<math>I_{zz} = \;</math> moment of inertia about the line through the centroid, parallel to the Z-axis

If the origin of the axis system is positioned at the object's centroid, we can simplify the notation further:

<math>I_{x} = \;</math> moment of inertia about the X-axis

<math>I_{y} = \;</math> moment of inertia about the Y-axis

<math>I_{z} = \;</math> moment of inertia about the Z-axis

Applications of moment of inertia

Torque

A common equation which describes the relationship between the linear force applied to an object, the object's mass, and the object's linear acceleration, in a frictionless setting, is:

<math>F = ma\,</math>

A similar equation can be used to describe the relationship between the rotational force (torque) applied to an object, the object's rotational mass (moment of inertia), and the object's rotational (angular) acceleration, in a frictionless setting, assuming all motion is in a plane and the mass of the object remains constant:

<math>{\tau} = I{\alpha}\,</math>

Where:

<math>{\tau} = \,</math> torque

<math>I = \,</math> moment of inertia

<math>{\alpha} = \,</math> rotational (angular) acceleration

Kinetic Energy

Moment of inertia is also used to allow physicists to rewrite formulas originally written in terms of mass (m) and velocity (v) in terms of angular velocity (ω) and moment of inertia (I).

For example, the kinetic energy (K) of an object is given as:

<math>K = \frac{1}{2} m v^2</math> (1)

where m is the object's mass and v is its velocity.

To find an analogous formula for an object in rotation, we substitute for v and m based upon the definitions of angular velocity and moment of inertia:

<math>v = \omega r \,</math>

<math>m = \frac{I}{r^2}</math>

Giving us

<math>K = \frac{1}{2} \left(\frac{I}{r^2}\right) (\omega r)^2</math>,

which simplifies to

<math>K = \frac{1}{2} I \omega^2</math> (2).

Notice how formulas (1) and (2) appear similar. Although this is not always the case, here we were able to obtain the rotational kinetic energy formula simply by substituting I for m and ω for v in the linear kinetic energy formula, even though the symbols represent different physical quantities with different physical dimensions.

Inertia tensor

The moment of inertia can be used to describe the amount of angular momentum a rigid body possesses, via the relation:

<math>\vec{L} = I \vec{\omega}\,</math>

For the case where the angular momentum is parallel to the angular velocity, the moment of inertia is simply a scalar.

However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity. The definition of the moment of inertia tensor is very similar to that above, except that it is now expressed as a matrix:

<math>I = \sum_i m_i r_i^2 (E-P_i)</math>

where

E is the identity matrix

P is the projection operator.

Alternatively the elements of the inertia tensor can be expressed as:

<math>I_{xx} = \sum_i m_i (y_i^2+z_i^2)</math>

<math>I_{yy} = \sum_i m_i (x_i^2+z_i^2)</math>

<math>I_{zz} = \sum_i m_i (x_i^2+y_i^2)</math>

<math>I_{xy} = I_{yx} = -\sum_i m_i x_i y_i\;</math>

<math>I_{xz} = I_{zx} = -\sum_i m_i x_i z_i\;</math>

<math>I_{yz} = I_{zy} = -\sum_i m_i y_i z_i\;</math>

It is notable that (because it is symmetric), it is always possible to diagonalize the inertia tensor to find the principal axes of the rigid body, those which satisfy the eigenvalue problem:

<math>I\vec{\omega}=\lambda \vec{\omega}.\,</math>

In the case of rotation about a principal axis with constant angular velocity, the angular momentum is, like the angular velocity, along this axis, so it remains constant. Thus no torque is required to maintain this rotation.

See also

• List of moments of inertia
• Torque
• Rotational energy

External links

• http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.inertia.html,
• http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html,
• http://kwon3d.com/theory/moi/iten.htmlda:Inertimoment

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