longer the moment arm), the greater the moment of inertia, and therefore the greater the angular momentum for a given angular velocity. In many cases the moment of inertia, and hence the angular momentum, can be simplified by, where is the radius of gyration, the distance from the axis at which the entire mass may be considered as concentrated. Similarly, for a point mass the moment of inertia is defined as, where is the radius of the point mass from the center of rotation, and for any collection of particles as the sum, Angular momentum's dependence on position and shape is reflected in its units versus linear momentum: kg⋅m2/s or N⋅m⋅s for angular momentum versus kg⋅m/s or N⋅s for linear momentum. When calculating angular momentum as the product of the moment of inertia times the angular velocity, the angular velocity must be expressed in radians per second, where the radian assumes the dimensionless value of unity. (When performing dimensional analysis, it may be productive to use orientational analysis which treats radians as a base unit, but this is not done in the International system of units). The units if angular momentum can be interpreted as torque⋅time. An object with angular momentum of can be reduced to zero angular velocity by an angular impulse of . The plane perpendicular to the axis of angular momentum and passing through the center of mass is sometimes called the invariable plane, because the direction of the axis remains fixed if only the interactions of the bodies within the system, free from outside influences, are considered. One such plane is the invariable plane of the Solar System. Angular momentum and torque Newton's second law of motion can be expressed mathematically, or force = mass × acceleration. The rotational equivalent for point particles may be derived as follows: which means that the torque (i.e. the time derivative of the angular momentum) is Because the moment of inertia is , it follows that , and which, reduces to This is the rotational analog of Newton's second law. Note that the torque is not necessarily proportional or parallel to the angular acceleration (as one might expect). The reason for this is that the moment of inertia of a particle can change with time, something that cannot occur for ordinary mass. Conservation of angular momentum General considerations A rotational analog of Newton's third law of motion might be written,