A block of mass $m$ is attached to a spring of constant $k$ and is placed on a rough horizontal surface with coefficient of friction $\mu$. If the block is released from rest at a distance $x_0$ from the equilibrium position, what is the total distance traveled by the block before it comes to rest permanently?

Solution

1. The initial potential energy stored in the spring when the block is at distance $x_0$ is $U=\frac{1}{2}k{x}_0^2$.
2. The work done by the force of friction over the total distance $d$ is given by $$\begin{array}{rl}{W}_f& =f_{k}d\\ & =\mu mgd\end{array}$$.
3. By the work-energy theorem, the initial energy is entirely dissipated by friction when the block comes to rest, so $\frac{1}{2}k{x}_0^2=\mu mgd$.
4. Solving for the total distance $d$, we obtain $d=\frac{k{x}_0^2}{2\mu mg}$.
5. Hence the answer is (A).