University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

156

Pages:

5
Author:

Jabin F.

# Product of Inertia, P.I. (Ixy): Ixy = Σmᵢxᵢyᵢ = ∫dmxy # Inertia Tensor: It is a 3x3 matrix whose elements are M.I. & P.I. terms. Inertia Tensor: I = | Ixx Ixy Ixz | | Iyx Iyy Iyz | | Izx Izy Izz | # Principal Axes: A set of 3 mutually perpendicular axes for which all product of inertia terms are zero. # Principal M.I.: M.I. about principal axes is called Principal M.I. # If axes are symmetric, then those are definitely principal axes. # If axes are not symmetric, then those axes may/may not be principal axes. (Check values of P.I.) # Axes oblique to 1d or 2d systems are never principal axes! # We can define principal axes through every point of a rigid body. So there can be an infinite no. of principal axes for an object. [Diagram of a rigid body with principal axes] # In a given question, if principal axes are not specified, then consider P.A. through origin. Eigen value of inertia tensor gives principal M.I. Eigen value of inertia tensor gives direction of principal axes. Moment of inertia of an object about any axis (in û direction) is given in terms of inertia tensor as: I = û^T * I * û