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a.man
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Homework Statement
Two particles, each of mass M, are hung between three identical springs. Each spring is massless and has spring constant k. Neglect gravity. The masses are connected as shown to a dashpot of negligible mass.
The dashpot exerts a force of bv, where v is the relative velocity of its two ends. The force opposes the motion. Let x1 and x2 be the displacement of the two masses from equilibrium.
a. Find the equation of motion for each mass.
b. Show that the equation of motion can be solved in terms of the new dependent variables y1 = x1 + x2 and y1 = x1 - x2.
c. Show that if the masses are initially at rest and mass 1 is given initial velocity v0, the motion of the masses after a sufficiently long time is
x1=x2
= (v0/2ω)* sin(ωt)
Evaluate ω.
Homework Equations
x.. + γx.+ ω2x = 0
x = Ae-γt/2cos(ωt + ∅)
The Attempt at a Solution
I think I managed to get the first two by just using different F = -kx formulae and adding -bv to it.
The second part I just added and subtracted the equations to get:
y..1 + ω2y1 + [itex]\frac{2bv}{m}[/itex] = 0
and
y..2 + 3ω2y2 = 0
I think these are right, as far as I know. The thing is, I'm not sure how to proceed from here. I tried using:
x = Ae-γt/2cos(ωt + ∅)
I can get it very close to the answer, but I don't think that's how you're supposed to do this... using what I got in part 2 would be more relevant, I think.
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