Spring-Mass system with friction

[Pi-Bond]

Homework Statement

A block shown in the drawing 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 before coming to a halt.

(a) Show that the amplitude decreases by same amount in each cycle of oscillation.

(b) Find the number of cycles n, the mass oscillates before coming to rest.

Homework Equations

Conservation of energy

The Attempt at a Solution

For (a) I used conservation of energy. Let x_i be the amplitude of a cycle (with energy E_i) and x_f be the amplitude (with energy E_f)

$E_{i}=\frac{1}{2}kx_{i}^{2}$
$E_{f}=\frac{1}{2}kx_{f}^{2}+f(x_{o}+x_{f})$

Yielding

$x_{f}=x_{i}-\frac{2f}{k}$

Showing that the amplitude decreases by 2f/k between subsequent cycles.

For (b) I calculated the distance x for the mass to come to rest at:

$\frac{1}{2}kx_{o}^{2}=fx \implies x=\frac{kx_{o}^{2}}{2f}$

Meanwhile, the total distance covered by the block using the result from (a),

$x=x_{0}+x_{0}-\frac{2f}{k}+x_{0}-\frac{4f}{k}+(...)+x_{0}-\frac{2(n-1)f}{k}$

for n cycles. Simplifying,

$x=nx_{0}-\frac{2f}{k}(1+2+(...)+n-1)$

If I equate those expressions, I don't get the right answer. Any hints?

 

[Modulated]

It looks like your "initial" and "final" energies are not a whole cycle but only half a cycle apart. Can you see why?

 

[Pi-Bond]

So a cycle is when the mass returns to the same side? If that is so, I can't see how talking about amplitude makes any sense, as the mass does not even go x₀ on the other side of the equilibrium point, but rather goes $x_{0}-\frac{2f}{k}$ as I showed.