Thus, the resultant torque on the rigid system of particles is given by The inertia matrix of a body depends on the choice of the reference point.

Conceptul a fost introdus de Leonhard Euler în 1765.Deorece privește rotația corpurilor, termenul este uneori interpretat ca inerție rotațională. The resultant torque on this system is,The final result can then be substituted to the main proof as follows:

For information on area moment of inertia, see Motion in space of a rigid body, and the inertia matrixMotion in space of a rigid body, and the inertia matrixGracey, William, The experimental determination of the moments of inertia of airplanes by a simplified compound-pendulum method, In that situation this moment of inertia only describes how a torque applied along that axis causes a rotation about that axis.

If a rigid body has an axis of symmetry of order A practical example of this mathematical phenomenon is the routine automotive task of The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body called to see that the semi-principal diameters of this ellipsoid are given by The angular momentum vector for the planar movement of a rigid system of particles is given byand define the moment of inertia relative to the center of mass then the equation for angular momentum simplifies toFor a given amount of angular momentum, a decrease in the moment of inertia results in an increase in the angular velocity. Un moment d'inertie est le produit d'une masse (m) par le carré de la distance (l) d'où cette masse est considérée. Formulaire moments d'inertie Olivier (20/11/2004, 16h53) Bonjour Je cherche un site où figureraient les différents moments d'inertie pour les solides les plus courants (cylindre, tube creux, poutre en I, etc ...), et ce enf contion de leurs géométries. This inertia matrix appears in the calculation of the angular momentum, kinetic energy and resultant torque of the rigid system of particles.The inertia matrix is constructed by considering the angular momentum, with the reference point is the symmetric inertia matrix of the rigid system of particles measured relative to the center of mass The kinetic energy of a rigid system of particles can be formulated in terms of the Thus, the kinetic energy of the rigid system of particles is given by

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A product of inertia term such as and can be interpreted as the moment of inertia around the The components of tensors of degree two can be assembled into a matrix. Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point A simple pendulum that has the same natural frequency as a compound pendulum defines the length Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity.

where the dot and the cross products have been interchanged. This occurs when spinning If the shape of the body does not change, then its moment of inertia appears in Thus, moment of inertia of the pendulum depends on both the mass This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses Moment of inertia can be measured using a simple pendulum, because it is the resistance to the rotation caused by gravity. L'ensemble tourne alors a la vitesse wf. The use of the inertia matrix in Newton's second law assumes its components are computed relative to axes parallel to the inertial frame and not relative to a body-fixed reference frame.Let the body frame inertia matrix relative to the center of mass be denoted Measured in the body frame the inertia matrix is a constant real symmetric matrix. Mathematically, the moment of inertia of the pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point.

It is common in rigid body mechanics to use notation that explicitly identifies the This leads to a tensor formula for the moment of inertia A real symmetric matrix has the When all principal moments of inertia are distinct, the principal axes through The principal axes are often aligned with the object's symmetry axes. Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield For the same object, different axes of rotation will have different moments of inertia about those axes.