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 $x_0$ 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.

Answer 1

Let us assume the that block compress spring up to $x$,

where $x_o>x$ because of loss in energy due to frictional force

Now work done against friction will be loss of spring energy.

So $f(x_o+x)=\frac{k(x_o^2-x^2)}{2}$

So $x_o-x=\frac{2f}{k}$, is loss of amplitude which is constant.