I hate to burst your bubble, but you guys are way off
. Air resistance is absolutely negligible at the speeds of a typical plumb bob.
The so-called "small angle approximation" arises in the derivation of the equation of motion for a planar pendulum, where one assumes that the amplitude of oscillations is small. This means that the formula for the period
T = 2 * Pi * Sqrt [L / g]
is only a limiting case for small theta and breaks down at larger angles
I'll sketch the derivation here for anyone interested:
Resolving the forces on the bob, parallel and perpendicular to the string we obtain
T = m g cos (theta)
parallel to the string, and
m a = - m g sin (theta) --- (*)
perpendicular to the string, where T is the tension in the string, m is the mass of the bob, theta is the acute angle betweent the string and the vertical and a is the tangential acceleration of the bob.
A little mathematics will convince you that
a = s'' = L theta''
where L is the length of the string and the double-prime indicates the second derivative wrt time (recall the arc-length formula, then differentiate twice wrt time). So we can write Equation (*) as
theta'' = - (g/L) sin (theta)
This equation is too complicated to solve as it stands. But you may know that sin (x) is approximately x for small x (where x is measured in radians). Thus, for small angles, we can use the approximation
theta'' = - (g / L) theta
from which it immediately follows that the angular frequency is Sqrt[g / L], and hence the period is 2 * Pi * Sqrt[L/g].
