in the notation of elliptic integrals. The function F1 (sin β) was
tabulated by A. M. Legendre for values of β ranging from 0° to 90°.
The following table gives the period, for various amplitudes α, in
terms of that of oscillation in an infinitely small arc [viz. 2π√(l /g)]
as unit.
| α/π | τ | α/π | τ |
| .1 | 1.0062 | .6 | 1.2817 |
| .2 | 1.0253 | .7 | 1.4283 |
| .3 | 1.0585 | .8 | 1.6551 |
| .4 | 1.1087 | .9 | 2.0724 |
| .5 | 1.1804 | 1.0 | ∞ |
The value of τ can also be obtained as an infinite series, by expanding the integrand in (18) by the binomial theorem, and integrating term by term. Thus
| τ = 2π √ | l | · { 1 + | 12 | sin2 12α + | 12 · 32 | sin4 12α + . . . }. |
| g | 22 | 22 · 42 |
If α be small, an approximation (usually sufficient) is
In the extreme case of α = π, the equation (17) is immediately integrable; thus the time from the lowest position is
This becomes infinite for ψ = π, showing that the pendulum only tends asymptotically to the highest position.
The variation of period with amplitude was at one time a hindrance to the accurate performance of pendulum clocks, since the errors produced are cumulative. It was therefore sought to replace the circular pendulum by some other contrivance free from this defect. The equation of motion of a particle in any smooth path is
| d 2s | = −g sin ψ, |
| dt 2 |
where ψ is the inclination of the tangent to the horizontal. If sin ψ were accurately and not merely approximately proportional to the arc s, say
 |
| Fig. 67. |
the equation (21) would assume the same form as § 12 (5). The motion along the arc would then be accurately simple-harmonic, and the period 2π √(k/g) would be the same for all amplitudes. Now equation (22) is the intrinsic equation of a cycloid; viz. the curve is that traced by a point on the circumference of a circle of radius 14k which rolls on the under side of a horizontal straight line. Since the evolute of a cycloid is an equal cycloid the object is attained by means of two metal cheeks, having the form of the evolute near the cusp, on which the string wraps itself alternately as the pendulum swings. The device has long been abandoned, the difficulty being met in other ways, but the problem, originally investigated by C. Huygens, is important in the history of mathematics.
The component accelerations of a point describing a tortuous curve, in the directions of the tangent, the principal normal, and the binormal, respectively, are found as follows. If OV→, OV′→ be vectors representing the velocities at two consecutive points P, P′ of the path, the plane VOV′ is ultimately parallel to the osculating plane of the path at P; the resultant acceleration is therefore in the osculating plane. Also, the projections of VV′→ on OV and on a perpendicular to OV in the plane VOV′ are δv and vδε, where δε is the angle between the directions of the tangents at P, P′. Since δε = δs/ρ, where δs = PP′ = vδt and ρ is the radius of principal curvature at P, the component accelerations along the tangent and principal normal are dv/dt and vdε/dt, respectively, or vdv/ds and v 2/ρ. For example, if a particle moves on a smooth surface, under no forces except the reaction of the surface, v is constant, and the principal normal to the path will coincide with the normal to the surface. Hence the path is a “geodesic” on the surface.
If we resolve along the tangent to the path (whether plane or tortuous), the equation of motion of a particle may be written
| mv | dv | = T, |
| ds |
where T is the tangential component of the force. Integrating with respect to s we find
i.e. the increase of kinetic energy between any two positions is equal to the work done by the forces. The result follows also from the Cartesian equations (2); viz. we have
whence, on integration with respect to t,
| 12m (ẋ2 + ẏ2 + ż2) | = ∫ (Xẋ + Yẏ + Zż) dt + const. |
| = ∫ (X dx + Y dy + Z dz) + const. |
If the axes be rectangular, this has the same interpretation as (24).
Suppose now that we have a constant field of force; i.e. the force acting on the particle is always the same at the same place. The work which must be done by forces extraneous to the field in order to bring the particle from rest in some standard position A to rest in any other position P will not necessarily be the same for all paths between A and P. If it is different for different paths, then by bringing the particle from A to P by one path, and back again from P to A by another, we might secure a gain of work, and the process could be repeated indefinitely. If the work required is the same for all paths between A and P, and therefore zero for a closed circuit, the field is said to be conservative. In this case the work required to bring the particle from rest at A to rest at P is called the potential energy of the particle in the position P; we denote it by V. If PP′ be a linear element δs drawn in any direction from P, and S be the force due to the field, resolved in the direction PP′, we have δV = −Sδs or
| S = − | ∂V | . |
| ∂s |
In particular, by taking PP′ parallel to each of the (rectangular) co-ordinate axes in succession, we find
| X = − | ∂V | , Y = − | ∂V | , Z = − | ∂V | . |
| ∂x | ∂y | ∂z |
The equation (24) or (26) now gives
i.e. the sum of the kinetic and potential energies is constant when no work is done by extraneous forces. For example, if the field be that due to gravity we have V = ƒmgdy = mgy + const., if the axis of y be drawn vertically upwards; hence
| 12mv 2 + mgy = const. | (30) |
This applies to motion on a smooth curve, as well as to the free motion of a projectile; cf. (7), (14). Again, in the case of a force Kr towards O, where r denotes distance from O we have V = ∫ Krdr = 12Kr 2 + const., whence
| 12mv 2 + 12Kr 2 = const. | (31) |
It has been seen that the orbit is in this case an ellipse; also that if we put μ = K/m the velocity at any point P is v = √μ. OD, where OD is the semi-diameter conjugate to OP. Hence (31) is consistent with the known property of the ellipse that OP2 + OD2 is constant.
The forms assumed by the dynamical equations when the axes of reference are themselves in motion will be considered in § 21. At present we take only the case where the rectangular axes Ox, Oy rotate in their own plane, with angular velocity ω about Oz, which is fixed. In the interval δt the projections of the line joining the origin to any point (x, y, z) on the directions of the co-ordinate axes at time t are changed from x, y, z to (x + δx) cos ω δt − (y + δy) sin ωδt, (x + δx) sin ω δt + (y + δy) cos ω δt, z respectively. Hence the component velocities parallel to the instantaneous positions of the co-ordinate axes at time t are
In the same way we find that the component accelerations are
Hence if ω be constant the equations of motion take the forms
These become identical with the equations of motion relative to fixed axes provided we introduce a fictitious force mω2r acting outwards from the axis of z, where r = √(x2 + y2), and a second fictitious force 2mωv at right angles to the path, where v is the component of the relative velocity parallel to the plane xy. The former force is called by French writers the force centrifuge ordinaire, and the latter the force centrifuge composée, or force de Coriolis. As an application of (34) we may take the case of a symmetrical Blackburn’s pendulum hanging from a horizontal bar which is made to rotate
