A block of mass $m$ is attached to a spring of constant $k$ on a smooth horizontal surface. The block is given an initial velocity $v_0$. If the spring is also attached to a dashpot providing a damping force $F=-cv$, what is the work done by the damping force until the block comes to rest?

Solution

1. The work-energy theorem states that the net work done on the block is equal to the change in its kinetic energy: $$\begin{array}{rl}{W}_{net}& =\Delta K\\ & =K_f-K_i\end{array}$$.
2. The net work is the sum of the work done by the conservative spring force ($W_s$) and the non-conservative damping force ($W_d$): $W_s+W_d=0-\frac{1}{2}m{v}_0^2$.
3. Since the spring force is conservative, its work is stored as potential energy, and the damping force is the only force that dissipates the initial mechanical energy until the block comes to rest.
4. Because the block starts with kinetic energy $\frac{1}{2}m{v}_0^2$ and ends with zero total mechanical energy, the work done by the damping force must equal the negative of the initial kinetic energy.
5. Hence the answer is (A).