where n = √μ. If P be the initial position of the particle, we may conveniently take OP as axis of x, and draw Oy parallel to the direction of motion at P. If OP = a, and ṡ0 be the velocity at P, we have, initially, x = a, y = 0, ẋ = 0, ẏ = ṡ0 whence
x = a cos nt, y = b sin nt,
(10)
if b = ṡ0/n. The path is therefore an ellipse of which a, b are conjugate semi-diameters, and is described in the period 2π/√μ; moreover, the velocity at any point P is equal to √μ·OD, where OD is the semi-diameter conjugate to OP. This type of motion is called elliptic harmonic. If the co-ordinate axes are the principal axes of the ellipse, the angle nt in (10) is identical with the “excentric angle.” The motion of the bob of a “spherical pendulum,” i.e. a simple pendulum whose oscillations are not confined to one vertical plane, is of this character, provided the extreme inclination of the string to the vertical be small. The acceleration is towards the vertical through the point of suspension, and is equal to gr/l, approximately, if r denote distance from this vertical. Hence the path is approximately an ellipse, and the period is 2π √(l/g).
| Fig. 65. | Fig. 66. |
The above problem is identical with that of the oscillation of a particle in a smooth spherical bowl, in the neighbourhood of the lowest point. If the bowl has any other shape, the axes Ox, Oy may be taken tangential to the lines of curvature at the lowest point O; the equations of small motion then are
| d2x | = −g | x | , | d2y | = −g | y | , |
| dt2 | ρ1 | dt2 | ρ2 |
(11)
where ρ1, ρ2, are the principal radii of curvature at O. The motion is therefore the resultant of two simple vibrations in perpendicular directions, of periods 2π √(ρ1/g), 2π √(ρ2/g). The circumstances are realized in “Blackburn’s pendulum,” which consists of a weight P hanging from a point C of a string ACB whose ends A, B are fixed. If E be the point in which the line of the string meets AB, we have ρ1 = CP, ρ2 = EP. Many contrivances for actually drawing the resulting curves have been devised.
It is sometimes convenient to resolve the accelerations in directions having a more intrinsic relation to the path. Thus, in a plane path, let P, Q be two consecutive positions, corresponding to the times t, t + δt; and let the normals at P, Q meet in C, making an angle δψ. Let v (= ṡ) be the velocity at P, v + δv that at Q. In the time δt the velocity parallel to the tangent at P changes from v to v + δv, ultimately, and the tangential acceleration at P is therefore dv/dt or s̈. Again, the velocity parallel to the normal at P changes from 0 to vδψ, ultimately, so that the normal acceleration is v dψ/dt. Since