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## Homework Statement

The block shown in the drawing is acted by a spring with spring constant ##k## and a weak friction force of constant magnitude ##f##. The block is pulled a distance ##x_0## from equilibrium and released. It oscillates many times and eventually comes to rest.

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

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

(Ans. ##n=\frac{1}{4}[(kx_0/f)-1] \approx kx_0/(4f)##)

## Homework Equations

## The Attempt at a Solution

Let x be the displacement of block from the equilibrium position. Applying Newton's second law on the block,

[tex]m\frac{d^2x}{dt^2}=-kx-f[/tex]

Solving the D.E

$$x(t)=A\sin(\omega t)+B\cos(\omega t)-\frac{f}{k}$$

where A and B are constants and ##\omega^2=\frac{k}{m}##.

From the initial conditions, ##A=0## and ##B=x_0+f/k##. Hence, the equation of motion is

$$x(t)=\left(x_0+\frac{f}{k}\right)\cos(\omega t)-\frac{f}{k}$$

I am stuck here. I guess the above equation is wrong. Part a of the question hints me that the amplitude of motion should have a time dependence. The amplitude I get is ##x_0+f/k## which is time independent.

Any help is appreciated. Thanks!