Dashpots and the Work - Kinetic Energy Theorem

[Taulant Sholla]

Homework Statement

I need to accommodate a dashpot in an intentionally simple work-kinetic energy analysis method. For example, for a box being dragged up a ramp via a rope while attached to a spring, I can deal with the work done by gravity, rope tension, spring force, and friction via the following method, along with enough known constants and sufficient displacement and velocity boundary conditions?

Homework Equations

K_A+W_NET=K_B
... where A is the position of the box at the bottom of the ramp, and B is the position of the box at the top of the ramp.

The Attempt at a Solution

For gravity, friction, spring, and tension this becomes...
K_A + ∫(mg)ds + ∫Tds + ∫(ks)ds + ∫μ_kF_Nds = K_B

The work terms for the dashpot are:
∫bvds = ∫b(ds/dt)ds = ∫b(ds/dt)(ds/dt)dt = ∫bv²dt

Are any of these terms integrate-able if I'm limited to, again, only knowing displacement and/or velocity boundary conditions?

 

[haruspex]

Are any of these terms integrate-able if I'm limited to, again, only knowing displacement and/or velocity boundary conditions?

The only way I can see is if you can write out and solve the general equation of motion. Observations of the boundary conditions should then allow you to deduce the velocity as a functionof time. That's not entirely satisfactory because it depends on theory, not pure observation.

 

[rude man]

∫bvds = ∫b(ds/dt)ds = ∫b(ds/dt)(ds/dt)dt = ∫bv²dt

If your question is whether the work done on the dashpot is a function of some system variable, e.g. v, what does your first integral above tell you?
(Related: why is it a dumb idea to drive your car fast instead of slowly over a given distance in order to save gas?)