Moment of Inertia Tensor
The moment of inertia for a rigid body moving in space is a tensor formed from the scalars obtained from the moments of inertia and products of inertia about the three coordinate axes. The moment of inertia tensor is constructed from the nine component tensors,
where ei, i=1,2,3 are the three orthogonal unit vectors defining the inertial frame in which the body moves. Using this basis the inertia tensor is given by
This tensor is of degree two because the component tensors are each constructed from two basis vectors. In this form the inertia tensor is also called the inertia binor.
For a rigid system of particles Pk, k=1,...,N each of mass mk with position coordinates rk=(xk, yk, zk), the inertia tensor is given by
where E is the identity tensor
The moment of inertia tensor for a continuous body is given by
where r defines the coordinates of a point in the body and ρ(r) is the mass density at that point. The integral is taken over the volume V of the body. The moment of inertia tensor is symmetric because Iij= Iji.
The inertia tensor defines the moment of inertia about an arbitrary axis defined by the unit vector n as the product,
and can be interpreted as the moment of inertia around the x-axis when the object rotates around the y-axis.
The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,
It is common in rigid body mechanics to use notation that explicitly identifies the x, y, and z axes, such as Ixx and Ixy, for the components of the inertia tensor.
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Other articles related to "moments, inertia, moment of inertia, moment of, moment of inertia tensor, tensor, tensors, inertia tensor":
... is considered to be by some an improvement over the information provided by calculating the area moments of inertia and polar moments of inertia ...
... The following is a list of area moments of inertia ... The area moment of inertia or second moment of area has a unit of dimension length4, and should not be confused with the mass moment of inertia ... If the piece is thin, however, the mass moment of inertia equals the areal density times the area moment of inertia ...
... In a stable state, the largest moments of inertia axis is aligned with the spin axis, with the smaller two moment of inertia axes lying in the plane of the equator ... rotate as a rigid body to realign the largest moment of inertia axis with the spin axis ...
... Another form of inertia is rotational inertia (→ moment of inertia), which refers to the fact that a rotating rigid body maintains its state of uniform rotational motion ... Rotational inertia depends on the object remaining structurally intact as a rigid body, and also has practical consequences For example, a gyroscope uses the property that it resists ...
... The moment of inertia for a rigid body moving in space is a tensor formed from the scalars obtained from the moments of inertia and products of inertia about the three coordinate axes ... The moment of inertia tensor is constructed from the nine component tensors, where ei, i=1,2,3 are the three orthogonal unit vectors defining the inertial frame in which the body moves ... Using this basis the inertia tensor is given by This tensor is of degree two because the component tensors are each constructed from two basis vectors ...
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