For the conical pendulum it is easy to derive the tension, T,
in the string in terms of the length of the string, L, the mass
of the bob, m, and the angular velocity, w. The horizontal
component of the tension produces uniform circular motion with
radius r. If the string moves in a cone of half angle B:
2
m r w = T sin B
From the geometry of the system we see that
r
sin B = ---
L
Leaving a solution:
2
T = m L w
From the vertical components we see that
mg = T cos B
Thus the angle is given by
m g g
cos B = ---- = -----
T 2
L w
and the period is just
2 pi / L cos B \
P = ------ = 2 pi sqrt | --------- |
w \ g /
Now if you solve the problem I posed you will see that you
have been swindled!