3.2 Problems
107
Eliminating v between the two equations and noting that
cos θ = 1 − h/R
we find h = R/3.
(iv) A motorcyclist goes around in a vertical circle inside a spherical cage. Find the
minimum speed at the top so that he may successfully complete the circular
ride.
Here we equate the reaction on the cage to the total weight of the rider plus
motorcycle
mg = mv
2
/R
or v =
g R
(v) Loop-the-Loop is a track which consists of a frictionless slide connected to a
vertical loop of radius R, Fig. 3.1. Let a particle start at a height h on the slide
and acquire a velocity v at the bottom of the loop.
If v <
√
2g R, the particle will not be able to climb up beyond the point B. It
will oscillate in the lower semicircle about the point D.
If
2g R < v <
5g R
the particle will be able to climb up the arc BC and leave at some point E and
describe a parabolic path. If v >
√
5g R, the particle will be able to execute a
complete circle. This corresponds to a height h = 2.5R.
Fig. 3.1 Loop-the-loop
3.2 Problems
3.2.1 Motion in a Horizontal Plane
3.1 Show that a particle with coordinates x = a cos t, y = a sin t and z = t traces
a path in time which is a helix.
[Adapted from Hyderabad Central University 1988]