A Classical Approach of Newtonian Mechanics

7 CIRCULAR MOTION 7.7 Motion on curved surfaces

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 R of the hoop

playing the role of the tension T 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 v with which the train is sent into the bottom of the loop (i.e.,

the velocity at θ = 0◦). What is the safe range of v? Now, if the train starts at

θ = 0 ◦ with velocity v 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

v^2 > 5 r g. (7.57)

Note, interestingly enough, that this condition is independent of the mass of the

train.

Equation (7.56) yields

vJ^2 =

r R
− r g cos θ. (7.58)
m
Now, the condition for the train to reverse direction without falling off the loop

is vJ^2 = 0 with R > 0. Thus, the train reverses direction when

R = m g cos θ. (7.59)

Note that this equation can only be satisfied for positive R when cos θ > 0. In
other words, the train can only turn around without falling off the loop if the