Elastic Potential Energy

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:

$${1\over 2}$$ (4)

This potential energy can be changed to kinetic energy by releasing the spring and allowing it to pull (or push) back towards its equilibrium position.

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

E _total = KE + PE_s + PE _grav

$${1\over 2} {1\over 2}$$ = (5)

where y is the height of the mass as measured from some arbitrary reference point.

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.

  

Figure 13.1: Potential Energy as a Function of Position

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 ):

$${1\over 2}$$ (6)

Combining Eqs(13.5) and (13.6) gives an expression for the velocity as a function of the displacement:

 

$$\sqrt{{k\over m} (A^2 - x^2)}$$ (7)

The maximum velocity is reached at the equilibrium position x = 0 . At this point all of the energy of the system is in the form of kinetic energy:

$$\sqrt{kA^2\over m }$$ (8)