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 origin: The relation between this linear map and the angular velocity pseudovector formula_74 is the following. Because "W" is the derivative of an orthogonal transformation, the bilinear form is skew-symmetric. Thus we can apply the fact of exterior algebra that there is a unique linear form formula_76 on formula_77 that where formula_79 is the exterior product of formula_80 and formula_81. Taking the sharp "L" of "L" we get Introducing formula_83, as the Hodge dual of "L", and applying the definition of the Hodge dual twice supposing that the preferred unit 3-vector is formula_84 where by definition. Because formula_81 is an