/C \ Q N
(A**) ( -- + a^2) -- - ac sin [alpha]--
\M / L L
- [mu]a(a sin [alpha] + 2c cos [alpha] - [rho] sin^2 [alpha])
+ Ra[rho] cos [alpha] = 0,
Q /A \ N CR + K /A \
(B**) -ac -- + ( -- + c^2) sin [alpha]-- - ------ + 2[mu] ( -- + c^2) cos [alpha]
L \M / L M \M /
+ [mu]c sin [alpha](a - [rho] sin [alpha])
- Rc[rho] cos [alpha] = 0.
The elimination of L, Q, N will lead to an equation for the
determination of n^2, and n^2 must be positive for the motion to be
stable.
If b is the radius of the horizontal circle described by G in steady
motion round the centre B,
(10) b = v/[mu] = (cP - aR)/[mu] = c sin [alpha] - aR/[mu],
and drawing GL vertically upward of length [lambda] = g/[mu]^2, the
height of the equivalent conical pendulum, the steady motion condition
may be written
(11) (CR + K)[mu] sin[alpha] - [mu]^2 sin[alpha] cos[alpha]
= -gM(a cos[alpha] - c sin[alpha])
+ M([mu]^2c sin[alpha] - [mu]Ra) (a sin[alpha] + c cos[alpha])
= gM[b[lambda]^(-1) (a sin [alpha] + c cos [alpha])
- a cos [alpha] + c sin [alpha]]
= gM.PT,
LG produced cuts the plane in T.
Interpreted dynamically, the left-hand side of this equation
represents the velocity of the vector of angular momentum about G, so
that the right-hand side represents the moment of the applied force
about G, in this case the reaction of the plane, which is parallel to
GA, and equal to gM.GA/GL; and so the angle AGL must be less than the
angle of friction, or slipping will take place.