Change in the amplitude of a damped spring block oscillator

[CrazyNeutrino]

Homework Statement

A block is acted on by a spring with spring constant k and a weak friction force of constant magnitude f . The block is pulled distance x0 from equilibrium and released. It oscillates many times and eventually comes to rest.

Show that the decrease of amplitude is the same for each cycle of oscillation.

2. The attempt at a solution
$$\Delta \mathbf{E} = E_f -E_i = \frac{1}{2}kx_0^2 - \frac{1}{2}k(x_0-\Delta x)^2 = f \cdot D$$
where $\Delta x$ is the change in amplitude after one oscillation and D is the distance traveled by the block in one oscillation. I am stuck here. How do I find D? It can't simply be 4x_0 because the distance from the point of rest to the equilibrium decreases on every half oscillation. Where do I go from here?

 

[Orodruin]

The friction force is assumed weak, so you can approximate the leading order effect by assuming harmonic motion and $\Delta x \ll x_0$.