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Question

The block shown in the figure, is acted on by a spring with spring constant k and a weak frictional force of constant magnitude f. The block is pulled a distance x0 from equilibrium position and then released. It oscillates many times and ultimately comes to rest. Show that the decrease of amplitude is the same for each cycle of oscillation.
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Solution

Let us assume the that block compress spring up to x,
where xo>x because of loss in energy due to frictional force
Now work done against friction will be loss of spring energy.
So f(xo+x)=k(x2ox2)2
So xox=2fk, is loss of amplitude which is constant.

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