MOMENTUM, or MOMENT OF INERTIA. Let us conceive a system of bodies possessing weight, and Immovably attached to a fixed axis, round which the whole system can turn. It Is known from experience, as well as deducible from the laws of motion, that the nearer the bodies are placed to the axis, the more rotatory motion may be communicated by a given force. The moment of inertia is a name given to a mathematical function of the masses in the system and of their positions with respect to the axis, on the magnitude of which the rotatory motion produced by a given pressure, acting for a given time, depends. This function is the sum of the products made by multi plying the number of unite in each mass by the number of units in the square of its distance from the axis. Thum, if m, en", &c.. be the masses of material points situated at the distances r, r', r", kc., from the axis, the moment of inertia is en al" r• 2 + &c. If the body be a continuous solid, and if dm be one of the elements of the mass, at n distance r from the axis, the moment of inertia is the integration being made throughout the whole extent of the solid. Let e n be the axis, and lot a pressure be communicated to the system at the point r, and such as would, were a mass r placed there, cause the system which consists of that single mass only to revolve with a velocity v, being at the distance a from the axis. The momentum of this velocity is rr. Let the system of ea, ne,and es", in consequence of this pressure, begin to revolve with an angular velocity 0 (measured in theoretical units. [ANGLE.] The consequence is, that en, en, and m" begin to revolve with velocities r0, r'e, and r" 8, and momenta m re), na"r"0. Now, if pressures, which would just prevent this motion in the game time as the applied pressure generated it, were applied in the opposite direction, the three pressures so applied would counterbalance the pressure at P. But [Momprrcm] the pressures which in the same time produce motions are to one another as the momenta produced, so that if c.rr represent the pressure at r, those applied in the contrary direction at an, and in", arc care, e m'r 0, and ctie"r"0. But the first acts perpendicularly at the extremity of the arm a, the others at the arms r, r', and r". Hence cm re.r + c r'ap' +cers"r"0.r"=c re.a or 0=c rra.
the denominator of which is what has been called the moment of inertia of the system. Hence it follows that the communication of a given pressure at a given distance from the axis of rotation will cause an angular velocity which is inversely as the moment of inertia : if the masses or their distances were increased in each a way as to double the moment of inertia, the angular velocity produced by a given pressure would be only the half of what it would have been before the change.
The moment of inertia may be represented by Imrtl (sum of all the terms of the form and the whole mass by .7.,»1. Let k be such a distance that, if the whole mass were concentrated at that distance, the moment of inertia would not be altered : that is, let Int x be = Then k is what was called the radius of gyration. [GYRATION.] The property which in most important in the actual determination of momenta of inertia by the integral calculus is one in virtue of which the moment may be found with respect to any axis when it is known with respect to a parallel axis passing through the centre of gravity. Let P Q be an axis passing through e. the centre of gravity, and let be another axis parallel to r Q and distant from it by o It or h, Then, whatever the moment of inertia may be when r q is the axis, That with respect to A LI is found by adding the moment of inertia of the whole system concentrated in o, or Zm x That is Enir== Imp= + Ien x where Imp=ssmornent of inertia of the system with respect to P Q.
Hence it appears that of all axes parallel to a given axis the moment of inertia is least for that axis which passes through the centre of gravity ; so that, ceteris paribus, the greatest motion is produced by a given force when the axis passes through the centre of gravity. Of all the axes which pass through the centre of gravity there are three, each at right angles to the other two, which possess remarkable properties, and are called principal axes. [ROTATION.] From what has been said it may easily be supposed that the moment of inertia is as important in the consideration of rotatory motions as the rectangle in mensuration. We shalt see a further use of this function in OSCILLATION, and also a practical mode of finding the moment of inertia.