Suppose that the object moves from point to
point in Fig. 65. In doing so, it gains potential energy
,
where is the angular coordinate of the object measured with
respect to the downward vertical. This gain
in potential energy must be offset by a corresponding loss in kinetic energy. Thus,

(298) |

Here, is the velocity at point ( ), and is the velocity at point ( ).

Let us now examine the radial acceleration of the object at point . The radial forces
acting on the object are the reaction of the vertical hoop, which acts towards
the centre of the hoop, and the component
of the object's weight,
which acts away from the centre of the hoop. Since the object is executing
circular motion with instantaneous tangential velocity , it must experience
an instantaneous acceleration towards the centre of the hoop. Hence,
Newton's second law of motion yields

Note, however, that there is a constraint on the reaction that the hoop
can exert on the object. This reaction must always be *positive*. In other
words, the hoop can push the object away from itself, but it can never pull
it towards itself. Another way of putting this is that if the reaction ever becomes
negative then the object will fly off the surface of the hoop, since it is no longer
being pressed into this surface. It should be clear, by now, that the problem
we are considering is exactly analogous to the earlier problem of an object attached to the end
of a piece of string which is executing a vertical circle, with the reaction of the
hoop playing the role of the tension in the string.

Let us imagine that the hoop under consideration is a ``loop the loop'' segment in a
fairground roller-coaster. The object sliding around the inside of the loop then becomes
the roller-coaster train. Suppose that the fairground operator can vary the
velocity with which the train is sent into the bottom of the loop
(*i.e.*, the velocity at
). What is the safe
range of ? Now, if the train starts at
with velocity then
there are only *three* possible outcomes. Firstly, the train can execute a complete
circuit of the loop. Secondly, the train can slide part way up the loop, come to a halt,
reverse direction, and then slide back down again. Thirdly, the train can slide part way up the loop,
but then fall off the loop. Obviously, it is the third possibility that the fairground
operator would wish to guard against.

Using the analogy between this problem and the problem of a mass on the end of a piece
of string executing a vertical circle, the condition for the roller-coaster train to
execute a complete circuit is

Equation (300) yields

(302) |

(303) |

(304) |

(305) |

Consider a skier of mass skiing down a hemispherical mountain of radius , as shown in Fig. 66. Let be the angular coordinate of the skier, measured with respect to the upward vertical. Suppose that the skier starts at rest () on top of the mountain ( ), and slides down the mountain without friction. At what point does the skier fly off the surface of the mountain?

Suppose that the skier has reached angular coordinate . At this stage, the
skier has fallen though a height
. Thus, the tangential
velocity of the skier is given by energy conservation:

Equations (306) and (307) can be combined to give

(308) |