Moment of inertia is diagonal and torques around the axes act independently of each other. When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram meter squared (kg·m) in SI units and pound-foot-second squared (lbf·ft·s) in imperial or US units. Moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics
Angular velocity of bases. For example, changing to the mobile frame: where formula_46 are unit vectors for the frame fixed in the moving body. This example has been made using the Z-X-Z convention for Euler angles. The angular velocity vector formula_47 defined above may be equivalently expressed as an angular velocity tensor, the matrix (or linear mapping) "W" = "W"("t") defined by: This is an infinitesimal rotation matrix. The linear mapping "W" acts as formula_49: A vector formula_51 undergoing uniform circular motion around an axis satisfies: Given the orientation matrix "A"("t") of a frame, whose columns are the moving orthonormal coordinate vectors