7 CIRCULAR MOTION 7.7 Motion on curved surfaces
Figure 66: A skier on a hemispherical mountain.turning point lies in the lower half of the loop (i.e., − 90 ◦ < θ < 90◦). The
condition for the train to fall off the loop is
vJ^2 = −r g cos θ. (7.60)Note that this equation can only be satisfied for positive vJ^2 when cos θ < 0. In
other words, the train can only fall off the loop when it is situated in the upper
half of the loop. It is fairly clear that if the train’s initial velocity is not sufficiently
large for it to execute a complete circuit of the loop, and not sufficiently small
for it to turn around before entering the upper half of the loop, then it must
inevitably fall off the loop somewhere in the loop’s upper half. The critical value
of v^2 above which the train executes a complete circuit is 5 r g [see Eq. (7.57)].
The critical value of v^2 at which the train just turns around before entering the
upper half of the loop is 2 r g [this is obtained from Eq. (7.55) by setting vJ = 0
and θ = 90 ◦]. Hence, the dangerous range of v^2 is
2 r g < v^2 < 5 r g. (7.61)For v^2 < 2 r g, the train turns around in the lower half of the loop. For v^2 > 5 r g,
the train executes a complete circuit around the loop. However, for 2 r g < v^2 <
5 r g, the train falls off the loop somewhere in its upper half.
Consider a skier of mass m skiing down a hemispherical mountain of radius r,as shown in Fig. 66. Let θ be the angular coordinate of the skier, measured with
respect to the upward vertical. Suppose that the skier starts at rest (v = 0 ) on
top of the mountain (θ = 0◦), and slides down the mountain without friction. At
what point does the skier fly off the surface of the mountain?
m
mg cos (^)
R
r cos (^) r^
mg v^