a true simple-harmonic vibration may be regarded as the orthogonal projection of uniform circular motion; it was pointed out by P. G. Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant angular velocity about the pole, in the direction of diminishing radius vector. When k2 > 4μ, the solution of (29) is, in real form,
where
The body now passes once (at most) through its equilibrium position, and the vibration is therefore styled aperiodic.
To find the forced oscillation due to a periodic force we have
The solution is
| x = | ƒ | cos (σ1t + ε − ε1), |
| R |
provided
| R = { (μ − σ12)2 + k2σ12}12, tan ε1 = | kσ1 | . |
| μ − σ12 |
Hence the phase of the vibration lags behind that of the force by
the amount ε1, which lies between 0 and 12π or between 12π and π,
according as σ12 ≶ μ. If the friction be comparatively slight the
amplitude is greatest when the imposed period coincides with the
free period, being then equal to f /kσ1, and therefore very great
compared with that due to a slowly varying force of the same average
intensity. We have here, in principle, the explanation of the
phenomenon of “resonance” in acoustics. The abnormal amplitude
is greater, and is restricted to a narrower range of frequency, the
smaller the friction. For a complete solution of (34) we must of
course superpose the free vibration (30); but owing to the factor e−t/τ
the influence of the initial conditions gradually disappears.
For purposes of mathematical treatment a force which produces a finite change of velocity in a time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely short. The whole effect is summed up in the value of the instantaneous impulse, which is the time-integral of the force. Thus if an instantaneous impulse ξ changes the velocity of a mass m from u to u′ we have
The effect of ordinary finite forces during the infinitely short duration of this impulse is of course ignored.
We may apply this to the theory of impact. If two masses m1, m2 moving in the same straight line impinge, with the result that the velocities are changed from u1, u2, to u1′, u2′, then, since the impulses on the two bodies must be equal and opposite, the total momentum is unchanged, i.e.
The complete determination of the result of a collision under given circumstances is not a matter of abstract dynamics alone, but requires some auxiliary assumption. If we assume that there is no loss of apparent kinetic energy we have also
Hence, and from (38),
i.e. the relative velocity of the two bodies is reversed in direction, but unaltered in magnitude. This appears to be the case very approximately with steel or glass balls; generally, however, there is some appreciable loss of apparent energy; this is accounted for by vibrations produced in the balls and imperfect elasticity of the materials. The usual empirical assumption is that
where e is a proper fraction which is constant for the same two bodies. It follows from the formula § 15 (10) for the internal kinetic energy of a system of particles that as a result of the impact this energy is diminished by the amount
| 12 (1 − e2) | m1m2 | (u1 − u2)2. |
| m1 + m2 |
The further theoretical discussion of the subject belongs to Elasticity.
This is perhaps the most suitable place for a few remarks on the theory of “dimensions.” (See also Units, Dimensions of.) In any absolute system of dynamical measurement the fundamental units are those of mass, length and time; we may denote them by the symbols M, L, T, respectively. They may be chosen quite arbitrarily, e.g. on the C.G.S. system they are the gramme, centimetre and second. All other units are derived from these. Thus the unit of velocity is that of a point describing the unit of length in the unit of time; it may be denoted by LT−1, this symbol indicating that the magnitude of the unit in question varies directly as the unit of length and inversely as the unit of time. The unit of acceleration is the acceleration of a point which gains unit velocity in unit time; it is accordingly denoted by LT−2. The unit of momentum is MLT−1; the unit force generates unit momentum in unit time and is therefore denoted by MLT−2. The unit of work on the same principles is ML2T−2, and it is to be noticed that this is identical with the unit of kinetic energy. Some of these derivative units have special names assigned to them; thus on the C.G.S. system the unit of force is called the dyne, and the unit of work or energy the erg. The number which expresses a physical quantity of any particular kind will of course vary inversely as the magnitude of the corresponding unit. In any general dynamical equation the dimensions of each term in the fundamental units must be the same, for a change of units would otherwise alter the various terms in different ratios. This principle is often useful as a check on the accuracy of an equation.
The theory of dimensions often enables us to forecast, to some extent, the manner in which the magnitudes involved in any particular problem will enter into the result. Thus, assuming that the period of a small oscillation of a given pendulum at a given place is a definite quantity, we see that it must vary as √(l/g). For it can only depend on the mass m of the bob, the length l of the string, and the value of g at the place in question; and the above expression is the only combination of these symbols whose dimensions are those of a time, simply. Again, the time of falling from a distance a into a given centre of force varying inversely as the square of the distance will depend only on a and on the constant μ of equation (15). The dimensions of μ/x2 are those of an acceleration; hence the dimensions of μ are L3T−2. Assuming that the time in question varies as axμy, whose dimensions are Lx+3yT−2y, we must have x + 3y = 0, −2y = 1, so that the time of falling will vary as a32/√μ, in agreement with (19).
The argument appears in a more demonstrative form in the theory of “similar” systems, or (more precisely) of the similar motion of similar systems. Thus, considering the equations
| d 2x | = − | μ | , | d 2x′ | = − | μ′ | , |
| dt 2 | x2 | dt ′2 | x′2 |
which refer to two particles falling independently into two distinct
centres of force, it is obvious that it is possible to have x in a constant
ratio to x′, and t in a constant ratio to t ′, provided that
| x | : | x′ | = | μ | : | μ′ | , |
| t 2 | t ′2 | x2 | x′2 |
and that there is a suitable correspondence between the initial
conditions. The relation (44) is equivalent to
| t : t ′ = | x32 | : | x′32 | , |
| μ12 | μ′12 |
where x, x′ are any two corresponding distances; e.g. they may be
the initial distances, both particles being supposed to start from rest.
The consideration of dimensions was introduced by J. B. Fourier
(1822) in connexion with the conduction of heat.
 |
| Fig. 64. |
§ 13. General Motion of a Particle.—Let P, Q be the positions of a moving point at times t, t + δt respectively. A vector OU→ drawn parallel to PQ, of length proportional to PQ/δt on any convenient scale, will represent the mean velocity in the interval δt, i.e. a point moving with a constant velocity having the magnitude and direction indicated by this vector would experience the same resultant displacement PQ→ in the same time. As δt is indefinitely diminished, the vector OU→ will tend to a definite limit OV→; this is adopted as the definition
