Damped Oscillator Conceptual Problem and Differential Equation Solution

[ggilvar99]

Hey guys I'm new to the forum and having a little trouble with this conceptual problem.

1. A block of mass m is connected to a spring, the other end of which is fixed. There is also a viscous damping mechanism. The following observations have been made of this system:

i) If the block is pushed horizontally with a force equal to mg, the static compression of the spring is equal to h

ii) The viscous resistive force is equal to mg as the block moves with a speed u.


a) Write the differential equation governing horizontal oscillations of the mass in terms of m, g, h and u.

b) for the particular case of u = 3√gh, what is the angular frequency of the damped oscillations?


2. Homework Equations :

mx'' + λx' + kx = 0


3. The Attempt at a Solution :

F = mg = -kh (x = h)

F = mg = -λu for x' = u

At this point I'm somewhat lost and not sure what they're looking for. If the viscous force = mg at velocity u, how can you translate that into a differential equation that covers all velocities of the mass? Any help would be greatly appreciated

 

[voko]

Can you obtain express $k$ and $\lambda$ from the conditions given?

 

[ggilvar99]

Oh, you're saying to set k = -(mg)/h and lambda = -(mg)/u and plug that into the diffeq? I don't know why that never occurred to me, thanks a lot for the suggestion!