# Damped Harmonic Oscillator Scenario

Suppose we have the following setup, a mass on a spring attached to a wall, as shown in the diagram:

We are completely ignoring friction between the mass and the ground here, and we have an ideal spring too. The mass starts in the equilibrium position. Let's define a coordinate system such that the origin is the equilibrium position of the spring and the positive x axis to the right of the equilibrium point.

Let's now submerge this system in a fluid A, such that the motion of the mass is heavily over-damped (e.g molasses). We give the mass a kick to the right such that it has some initial velocity:

$$v = v_0$$ away from the wall.

Let's now submerge the system in a fluid B, so now the motion of the mass is critically damped. We give it the same initial velocity velocity away from the wall.

My question is, will the mass achieve a maximum displacement away from the equilibrium position (x=0) in fluid A or fluid B, or will the maximum displacements in each case be equal?

• What do you think will happen and why? – sammy gerbil Jun 22 '17 at 3:42
• My intuition tells me that the critically damped oscillator will achieve a higher maximum displacement, since the over damped oscillator is losing energy at a higher rate, and they start off with the same kinetic energy. The thing that was bugging me in my mind was that the critically damped mass will move more quickly through the fluid, though the overdamped mass will move more slowly, though over a longer period of time. – user154080 Jun 22 '17 at 4:44
• As a result it's unclear to me which one will go further. – user154080 Jun 22 '17 at 4:49