The Pendulum

Part 1: Background: Modeling the Pendulum

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 theta 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 theta(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/sec² or 9.807 meters/sec² near sea level. Thus, the force acting in the tangential direction is -mg sin(theta). (The negative sign is because this force is in the negative direction when theta is positive and vice versa.) Since this force is mass x acceleration, it follows that

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

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 theta as our dependent variable, we will represent the damping as proportional to angular velocity, say, -b (d theta / 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 damped, unforced spring equation

with theta 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. Indeed, the Existence-Uniqueness Theorem for second-order equations assures that there will be a unique solution for any given initial conditions. But the presence of sin in the differential equation makes it impossible to give a simple formula that describes a solution function.

Last modified: March 20, 1998