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**Idea: **In order to stretch a spring, it is
necessary to do external work on the spring. This work is stored
in the spring
and is called the **elastic potential energy** ( *PE*_{s} ).
*PE*_{s} depends on the spring constant, *k* , and the net
displacement from equilibrium *x* and is given by:

PE_{s} = kx^{ 2}.
| (4) |

The elastic potential energy contributes to the total mechanical
energy of the harmonic oscillator:

E_{ total}
| = |
KE + PE_{s} + PE_{ grav}
| |

= |
mv^{ 2} + kx^{ 2} + mgy
| (5) |

In the absence of friction, the total mechanical energy is
conserved, i.e. it is constant throughout the motion. This is
expressed
graphically (for zero gravitational potential,
*PE*_{ grav} = 0 )
in Fig.13.1.

**Idea: **When a harmonic oscillator reaches its maximum
displacement, *x* = *A* , it must turn around and go back. At this
turning
point, the velocity is zero, and the total mechanical
energy can be written in terms of the amplitude ( *PE*_{grav} = 0 ):

E_{ total} = kA^{ 2}.
| (6) |

v = .
| (7) |

v_{ max} = .
| (8) |

10/9/1997