| φ | = − | U | ( 1 + | a2 | ) = | a2 + b2 | , |
| φ1 | U1 | b2 | a2 − b2 |
(13)
and this, by § 36, is also the ratio of the kinetic energy in the annular interspace between the two cylinders to the kinetic energy of the liquid moving bodily inside r = b.
Consequently the inertia to overcome in moving the cylinder r = b, solid or liquid, is its own inertia, increased by the inertia of liquid (a2 + b2)/(a2 ~ b2) times the volume of the cylinder r = b; this total inertia is called the effective inertia of the cylinder r = b, at the instant the two cylinders are concentric.
With liquid of density ρ, this gives rise to a kinetic reaction to acceleration dU/dt, given by
| πρb2 | a2 + b2 | dU | = | a2 + b2 | M′ | dU | , | |
| a2 − b2 | dt | a2 − b2 | dt |
(14)
if M′ denotes the mass of liquid displaced by unit length of the cylinder r = b. In particular, when a = ∞, the extra inertia is M′.
When the cylinder r = a is moved with velocity U and r = b with velocity U1 along Ox,