(34)
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 about a vertical axis half-way between the points of attachment of the upper string. The equations of small motion are then of the type
ẍ − 2ωẏ − ω2x = −p2x, ÿ + 2ωẋ − ω2y = −q2y.
(35)
This is satisfied by
ẍ = A cos (σt + ε), y = B sin (σt + ε),
(36)
provided
| (σ2 + ω2 − p2) A + 2σωB = 0, |
| 2σωA + (σ2 + ω2 − q2) B = 0. |
(37)