Note 20 — Chapter XI

To this may be added the extremely useful theorem for such problems, that if any directed quantity L, say, characteristic of the motion of a body, be associated with a line or axis Ol, which is changing in direction, it causes a rate of production of the same quantity for a line or axis instantaneously at right angles to Ol, towards which Ol is turning with angular velocity ω, of amount ωL. If M be the amount of the quantity already existing for this latter line or axis, the total rate of growth of the quantity is there M + ωL. For example, a particle moving with uniform speed v in a circle of radius r, has momentum mv along the tangent. But the tangent is turning round as the particle moves with angular speed v ⁄ r, towards the radius. The rate of growth of momentum towards the centre is therefore

mv × v ⁄ r = mv² ⁄ r.