Page 567 : DYNAMICS — DYNAMICS
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II. The change in the linear momentum of a body is proportional to the force acting upon the body; and the direction of the change is the same as the direction of the force.
III. Action is always equal and opposite to reaction; by which is meant that the mutual forces of any two bodies or of any two parts of a body are always equal and oppositely directed.
Lavoisier showed that the amount of matter in any isolated system cannot be increased or diminished by any known means: and therefore, presumably, that “the amount of matter in the universe is a constant.”
6. The next great step in dynamics is the introduction of the idea of work and energy. Work, in dynamics, is employed to denote one definite quantity, viz.: the product of a force multiplied by the distance through which it is exerted, both distance and force being measured in the same direction; while energy is defined simply as the ability to do work, and is, therefore, measured in the same units as work.
By 1847 Helmholtz, together with a number of his contemporaries, had proved that though the energy which a system of bodies possesses may assume a great variety of forms, yet the amount of that energy is a constant quantity. The energy of the universe is continually undergoing transformation; but there is no evidence for thinking that the slightest bit of energy has ever been annihiliated or created by man. This summary is known as the principle of the conservation of energy. Nearly all the problems of dynamics are solved by applying to the particular case in question either Newton’s laws of motion or the law of the conservation of energy.
DYNAMICS OF ROTATION
7. A very great simplification in the treatment of dynamical problems is secured by the fact that all cases in rotation may be solved not only by the same principles, but by the same formulæ, as in the case of translation.
We have only to replace linear displacement, x, by angular displacement, θ; and linear inertia, m, by rotational inertia, I. The following table gives a summary of the principal quantities involved in translation and rotation, and shows that they are identical in form in the two cases:
| TRANSLATION | ROTATION | ||
| Inertia ( = Mass) | m | Rotational inertia ( = moment of inertia) | I |
| Linear Displacement ( = change of position) | x | Angular Displacement | θ |
| Time | t | Time | t |
| Linear Velocity | v = x/t | Angular Velocity | ω = θ/t |
| Linear Acceleration | α = v/t | Angular Acceleration | γ = ω/t |
| Linear Momentum | M = mv | Angular Momentum | G = Iω |
| Force | F = ma | Moment of Force | L = Iγ |
| Energy of Translation | E = ½mv² | Energy of Rotation | E = ½Iω² |
8. Up to this point we have no principle which will determine the direction in which any dynamical process occurs. The law of the conservation of energy would be equally well-satisfied whether a clock-weight ran down and delivered energy to the clock-train and the air, or whether the clock-weight ran up, deriving the necessary energy from the heat in the train and in the air. It has been found convenient to divide all kinds of energy into two groups—energy of position and energy of motion—called potential and kinetic energy respectively. And it has been discovered by Clausius, Kelvin, Helmholtz and others that, in general, “the potential energy of any system tends to become a minimum.” Armed with this theorem, which is, perhaps, the most general principle of dynamics, we are prepared for the solution of all the ordinary problems of mechanics. The method in general consists in writing, in the form of equations, the six conditions of equilibrium, namely, that all the forces shall be zero and that all the moments of force shall be zero, and then solving for the quantity desired.
Upon the eight general principles enunciated above are constructed the entire
