General springs

Applet by Fu-Kwun Hwang
  • An object that is attached to a spring that is slightly stretched or compressed by a displacement x will experience a restoring force given by Hooke's Law:
    Fr(x)= - k x ,where k is a constant
    The solution to this equation is a simple harmonic oscillation (we assume that the mass of the spring is negligible).
  • Consider mass that is attached to a spring that is hung from the ceiling:
    If the spring is stetched, or compressed, by an amount x, then the force exerted on the mass is equal to F(x) = m g - k x
    The equilibrium position is x = m g / k
  • Damping forces:
    A viscous damping force can be modeled as Fb= - b v, where b is a constant and v is the velocity of the load.
  • External, harmonic forces:
    Fext = fosin(cwt)
    1. w2 = wo2 - (b/2m)2 where wo2 = k/m, wo is the nature frequency of the system
    2. if c=0. then fo = 0.
  • The net force acts on the mass is F = m g - k x - b v + fosin(cwt)


  1. You can enter values of m, k, b, and f (m can also be changed by clicking on the +/- buttons)
    • b=0., f=0 - simple harmonic motion(SHM)
    • b!=0. (try 0.1) - damped oscillation
    • f!=0. (try 5.0) - forced oscillation
  2. You can drag the left mouse button to change the initial position of the mass.
  3. The animation starts when the mouse button is released.
  4. If you drag with right mouse button (or press ---> Button), the spring will also move with constant speed in the horizontal direction.
  5. The green arrow measures the displacement x from the unstretched point.
  6. The blue arrow measures the displacement x from the equilibrium point (F=0).
  7. The red arrow represents the velocity v of the mass.
  8. Each time you click the mouse button, the coordinate of the mouse is shown in the text Field. (MKS unit, x/v verses t)
  9. External driving force:
    1. There is no external force if c=0, i.e. fo = 0.
    2. Otherwise Fext = fo * sin(c*w* t), where w2 = k/m - (b/2m)2