The figure at the right
shows an idealized pendulum, with a "massless" string or rod
of *length* **L** and a *bob *of mass **m**. The open circle
shows the rest position of the bob. When the bob is moved from its rest
position and let go, it swings back and forth. The time it takes the pendulum
to swing from its farthest right position to its farthest left position
and back to its next farthest right position is the *period *of the
pendulum.

The primary forces acting
on the bob are the gravitational force that makes it move in the first
place and the force exerted by the string to keep it moving along a circular
path. In addition, there may be a *damping* force from friction at
the pivot or air resistance or both. We will construct a model to describe
how the angle of the pendulum varies as a function
of time **t**.

Let **s(t)** be the distance
along the arc from the lowest point to the position of the bob at time
**t**, with displacement to the right considered positive. Let **(t)**
be the corresponding angle with respect to the vertical. The figure shows
tangential and radial components of gravitational force on the pendulum
bob. The radial component is exactly balanced by the force exerted by the
string, so the only relevant force producing the motion is the tangential
component of the gravitational force. For the moment, we ignore the damping
force, if any.

The gravitational force
is directed downward and has magnitude **mg** (mass x acceleration),
where **g** is the gravitational acceleration constant, 32.17 feet/sec^{2}
or 9.708 meters/sec^{2} near sea level. Thus, the force acting
in the tangential direction is **-mg sin()**. (The negative sign
is because this force is in the negative direction when is
positive and vice versa.) Since this force is mass x acceleration, it follows
that

Now **s** and
are related as arc length and central angle in a circle of radius **L**:
**s = L **. Thus, the second derivative of **s** is **L**
times the second derivative of . That brings us to our *undamped*
model differential equation with a single dependent variable, the angular
displacement :

Next, we add damping to
the model. We make the simplest possible assumption about the damping force,
that it is proportional to velocity. Since arc length and central angle
are themselves proportional (with proportionality constant **L**), it
makes no difference whether we use linear or angular velocity. Having selected
as our dependent variable, we will represent the damping as
proportional to angular velocity, say, **-b (d/ dt)**. The negative sign is because the damping force has to be opposite the direction of motion.
When we include this term in the model, our equation becomes

When we bring all the terms to the left-hand side, our model equation becomes

This equation is similar to the undamped spring equation

with replacing
**y**, **g** replacing **k**, and **L** replacing one occurrence
of **m**. But there is an important difference between the two equations:
the presence of the sine function in pendulum equation. Recall that for
springs, trigonometric functions turned up only in the *solutions*.
We know the pendulum problem must have solutions, because we see the pendulum
move. But the presence of **sin** in the differential equation makes it impossible
to give a simple formula that describes a solution function.

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