Can a Damped Anti-Spring System Explode Under Certain Conditions?

[jjark24]

Imagine a fictitious universe where springs want to stretch: the spring force is proportional to, and in the same direction as, displacement from equilibrium. We'll call these anti-springs.

(a) Set up a differential equation modeling the motion of a damped anti-spring if the mass is m = 1 kg, the damping coefficient is b = 3 N/(m/s), and the anti-spring constant is k = 4 N/m.

(b) Are there any initial conditions that make the anti-spring 'explode' (i.e., experience arbitrarily large displacements)? If so, what is an example? If not, why not?

(c) Are there any non-0 initial conditions that keep the anti-spring from exploding? If so, what is an example? If not, why not?

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No idea where to even start on this problem. Any help would be appreciated!

I know that (a) is d2y/dt2 + 3dy/dt - 4y = 0

 

[lanedance]

so how about showing a), in thr normal world the spring force with damping is
ma = m d2x/dt2 = F = -kx -b dx/dt

in the ficticious world
m d2x/dt2 = +kx -b dx/dt

or
m d2x/dt2 b dx/dt -kx = 0

which lines up with what you have

now how about trying to solve the DE?

or if you don't want to solve the whole thing or its too difficult, test the intial behaviour for different initial conditions and consider the limiting behaviour as x gets large...

 

[vela]

That's a good start. You're done with part (a). Now solve the equation. You need to analyze the solution to answer parts (b) and (b).