**Period of the Simple Pendulum using Elementary Mathematics**

Randall D. Peters

Department of Physics

1400 Coleman Ave.

Mercer University

Macon, Georgia 31207

*Copyright Oct 2002*

ABSTRACT-- The period of the simple pendulum is obtained from considerations of kinematics without using calculus.

Referring to the figure, consider the circle on which p moves at constant speed, centered on the equilibrium point of the pendulum. Four cardinal points of the particle's motion are indicated by the integers 0, 1, 2, and 3. Zero corresponds to t = 0, followed by points separated in time by one-quarter period. At any given time, p is located by drawing a vertical line through the center of the bob and finding its intersection with the circle. For the position shown in the figure, p is about 1/8 of a cycle from the starting point 0.

Whereas a similar construct is commonly used to describe only position, here
we will describe all three kinematic variables x, v, and a--corresponding
respectively to position, velocity, and acceleration of the pendulum bob.
These variables are one-dimensional vectors, in which direction is determined
by algebraic sign relative to the horizontal x-axis onto which all projections
are made. In the figure, only velocity vectors are shown, one for the circular
motion (constant magnitude v_{o}), and the other for the pendulum
motion (variable magnitude v).

Insofar as p is concerned, there are two invariant quantitities, the magnitude
of the particle's velocity v_{o} and the magnitude of its acceleration
a_{o}. To obtain the period of the pendulum in terms of its length
L and the acceleration of gravity g, we must specify v_{o }and
a_{o} in terms of the period T, which is readily done. The speed
is simply the circumference of the circle divided by the period of the motion;
i.e., v_{o} = 2px_{o}/T =
w x_{o }. The acceleration is centripetal
and the magnitude is given by a_{o} = v_{o}^{2}/
x_{o} = w^{2} x_{o }.
(It is worth noting that many introductory physics textbooks develop this
expression for centripetal acceleration without employing calculus.)

Looking at the figure, one can see that the pendulum's kinematics is obtained from the particle's kinematics through the expressions

which are valid only if x_{o} << L.

To get the period of the pendulum in terms of L and g, we must now (finally)
employ Newton's 2nd law S F = m a. The net force
is the restoring one in the -x direction ; i.e., S
F = - mg x / L = m a = -mw^{2}
x ._{ } Thus we obtain
w^{2} = g / L, yielding the desired result

without having to do any calculus whatsoever.