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What is the exact defination of moment of inertia and velocity?

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Moment of inertia

INTRODUCTION

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body determines the torque needed for a desired angular acceleration about a rotational axis. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation. It is an extensive (additive) property: the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). One of its definitions is the second moment of mass with respect to distance from an axis r, {\displaystyle I=\int _{Q}r^{2}\mathrm {d} m} {\displaystyle I=\int _{Q}r^{2}\mathrm {d} m}, integrating over the entire mass {\displaystyle Q} Q.

For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about an axis perpendicular to the plane. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix; each body has a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

Applications of multipole moments

The multipole expansion applies to 1/r scalar potentials, examples of which include the electric potential and the gravitational potential. For these potentials, the expression can be used to approximate the strength of a field produced by a localized distribution of charges (or mass) by calculating the first few moments. For sufficiently large r, a reasonable approximation can be obtained from just the monopole and dipole moments. Higher fidelity can be achieved by calculating higher order moments. Extensions of the technique can be used to calculate interaction energies and intermolecular forces.

The technique can also be used to determine the properties of an unknown distribution {\displaystyle \rho } \rho . Measurements pertaining to multipole moments may be taken and used to infer properties of the underlying distribution. This technique applies to small objects such as molecules,[2][3] but has also been applied to the universe itself,[4] being for example the technique employed by the WMAP and Planck experiments to analyze the Cosmic microwave background radiation.

Moment of Inertia: Thin Disk

The moment of inertia of a thin circular disk is the same as that for a solid cylinder of any length, but it deserves special consideration because it is often used as an element for building up the moment of inertia expression for other geometries, such as the sphere or the cylinder about an end diameter. The moment of inertia about a diameter is the classic example of the perpendicular axis theorem For a planar object: