MOTION OF A SPINNING BODY 65
If this torque be not applied, the locomotive tends to overturn, rotating away from the center of the curve about a horizontal axis perpendicular to the plane of the diagram. Substituting in (60) the data of the problem, we find the value of the centripetal torque tending to overturn the locomotive to be
L, = 293,000 slugs (88 ft. per sec.)27 ft. (0.998) = 246,600 lb.-ft. 32.1 2000 ft.
(b) In order that a rotating axle may be turned about a vertical axis, it must be acted upon by a gyroscopic torque toward the center of the curve, about a horizontal axis perpendicular to the axle. The reaction of this torque acts upon the locomotive in the opposite direction (Art. 36).
The value of this torque acting on the locomotive, due to all six pairs of wheels, is
L2 = (2 hs + 3 he" + hs"') w cos o (61)
where hs' represents the angular momentum of each of the pairs of pilot wheels with connecting axle, and h2" and hs'" represent the angular momenta of the drivers and trailers, respectively.
From the data of the problem:
= v 2100 0.7(20) ft.) 188 ft. per sec. _
hs = KS ws - ms k12( r, = 32.1 slugs 12 ) 20 - 4701
Similarly, we find that
hs" = 35858 and h,"' = 7858 per sec W cos 8 = R cos of = 88 ft 2000 ft. 0.998
= 0.044 radian per sec. Hence the magnitude of the gyroscopic torque is
L = [2(4701) + 3(35858) + 7858]0.044 = 5493 lb.-ft.
(c) The gravitational torque about a horizontal axis parallel to the rails is clockwise and has the magnitude
L3 = mgb sin 0
= 293000 lb. wt. (7 ft.)0.07 = 143,570 lb.-ft. (62)