# 1The Damped Spring and Pendulum Problems

## 1.1 Derivation of the Damped Spring and Pendulum Equations

In this chapter we present some simple ordinary differential equation problems to give students a chance to become familiar with PDE2D before proceeding to more difficult problems in later chapters.

Like many second‐order differential equations, the equations used to model the damped spring and pendulum are derived using Newton's second law: Mass times acceleration equals the force acting on the mass.

Suppose a weight of mass hangs from the ceiling on a spring. We will let be the height of this weight, with taken as its height when stationary. Then we will consider three forces acting on this mass: The force of the spring itself will be approximately proportional to the displacement from equilibrium and in the opposite direction, ; a force of friction (perhaps due to the surrounding air or liquid, or the spring itself) approximately proportional to the velocity of the mass and in the opposite direction, ; and an additional external force, which may be caused by some outside agent ...

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