Inertia Tensor

Inertia Tensor

In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass (SI units kg·m2, US units lbm ft2), is a property of a distribution of mass in space that measures its resistance to rotational acceleration about an axis. This scalar moment of inertia becomes an element in the inertia matrix when a distribution of mass is measured around three axes in space. This inertia matrix appears in the calculation of the angular momentum, kinetic energy and resultant torque in the dynamics of a rigid body.

Newton's first law, which describes the inertia of a body in linear motion, can be extended to the inertia of a body rotating about an axis using the moment of inertia. That is, an object that is rotating at constant angular velocity will remain rotating unless acted upon by an external torque. In this way, the moment of inertia plays the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration. The symbols I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia.

The moment of the inertia force on a single particle around an axis multiplies the mass of the particle by the square of its distance to the axis, and forms a parameter called the moment of inertia. The moments of inertia of individual particles in a body sum to define the moment of inertia of the body rotating about an axis. For rigid bodies moving in a plane, such as a compound pendulum, the moment of inertia is a scalar, but for movement in three dimensions, such as a spinning top, the moment of inertia becomes a matrix, also called a tensor.

Many systems use a mass with a large moment of inertia to maintain a rotational velocity and resist small variations in applied torque. For example, the long pole held by a tight-rope walker maintains a zero angular velocity resisting the small torque applied by the walker to maintain balance. Another example is the rotating mass of a flywheel which maintains a constant angular velocity resisting the torque variations in a machine.

Read more about Inertia Tensor:  History, Overview, Scalar Moment of Inertia of A Simple Pendulum, Scalar Moment of Inertia of A Rigid Body, Polar Moment of Inertia, Moment of Inertia Matrix, Moment of Inertia Around An Arbitrary Axis, Moment of Inertia Tensor, Moment of Inertia Reference Frames

Other articles related to "inertia tensor, inertia, tensor":

Inertia Tensor - Moment of Inertia Reference Frames - Identities For A Skew-symmetric Matrix
... In order to compare formulations of the inertia matrix in terms of a product of skew-symmetric matrices and in terms of a tensor formulation, the ... vector R=(x, y, z), then the product in the inertia matrix becomes This product can be computed using the matrix formed by the outer product using the identify ...
Inertia Tensor Of Triangle
... The inertia tensor of a triangle (like the inertia tensor of any body) can be expressed in terms of covariance of the body where covariance is defined as area integral over the triangle Covariance for a ...
Parallel Axes Rule - In Classical Mechanics
... theorem) can be generalized to calculate a new inertia tensor Jij from an inertia tensor about a centre of mass Iij when the pivot point is a displacement a from the centre of mass where is the ... 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 ... 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 ...

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