Equation of Motion

The equation of motion is a balance of three forces (click on them for more details): M dx2/d2t = F(v0 - dx/dt, x) - kappa x

The inertia force is the mass of slider M times its acceleration, i.e., the second derivative of the slider position x(t) as a function of time t.

The spring force is caused by stretching or compressing the spring. The force is proportional to the amount of stretching. This is called the Hooke's law. The proportionality factor is the so-called spring constant kappa (the greek letter kappa). In the animation the spring force is shown as a green arrow.

The friction force is caused by the interaction between the slider and the surface of the conveyor belt (moving with the constant velocity v₀). In the animation this force is shown by a red arrow. In addition a red bar shows the dissipated energy per time unit.

In general this is a function of many variables: The relative sliding velocity v0-dx/dt , the position, the force, the history etc. The complexity of the interaction between the surface of two sliding bodies is the central subject of the Friction Lab. For the animation two functions, i.e. friction laws, have been choosen: Coulomb's law and viscous friction. The dynamics in the case of a generalized Coulomb's law is investigated in a paper (PDF 107Kb) I published 1997.

Without any friction we get the equation of motion of a so-called harmonic oscillator. The motion of the slider becomes a sinusoidal oscillation with a constant amplitude. The period T of oscillation is independent of the oscillation amplitude. It is determined by the mass and the spring constant:

In the animation you get this motion when all friction parameters (mu_s and mu_k in the case of Coulomb's law or gamma in the case of viscous friction) are set to zero.