Mechanical System, Energy dissipated

[hfenton]

Homework Statement

A cart with mass m = 20 kg is initially moving horizontally with constant velocity 0.8 m/s, when it strikes a mechanical “snubber” device that consists of a plate supported by two parallel springs (each with stiffness k = 1000 N/m), and a single damper (dashpot) with viscous friction coefficient b = 80 N-s/m. If the maximum deflection of the snubber’s springs after impact with the cart is 0.0605 m, then how much energy was dissipated by the dashpot during the time from impact to maximum deflection? Neglect friction in the cart’s wheels and neglect the mass of the snubber device. Assume that the cart remains in contact with the snubber from impact to maximum deflection.

Homework Equations

The idea is to set up a mathematical model of this system, but no specific equations were really introduced in class. This is what I know...

F = kx for springs, with relative displacement of F = k(x1-x2)
Dashpot Force = b(v1-v2)
W = Fx where power is the rate of work
Power dissipated by the dashpot = bv^2

The Attempt at a Solution

I came up with a mathematical solution of F = mg = b(v1-v2) + 2k(x1-x2)

where initially v1 = .8 m/s v2 = 0, x1 = 0 and x2 = .0605

I have no idea what to do to then get to an equation that will yield power dissipation for just the dashpot component. Any help would be appreciated! Thanks

 

[Q_Goest]

Hi hfenton,
The cart has some mass and some velocity which represents the energy the cart has because of that motion. That energy is kinetic energy. Can you figure out the amount of kinetic energy the cart has?

When the cart hits the springs/dashpot assembly, it compresses the springs. Do you know how to find the energy required to compress a spring a given amount?

Assuming the cart has more energy than the springs have absorbed, then the remaining energy must have been dissipated by the dashpot. Would you agree?