1. **Stating the problem:** We want to analyze the motion of a mass attached to a spring with friction (damping). 2. **Formula used:** The motion is modeled by the second-order differential equation $$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$$ where $m$ is the mass, $c$ is the damping coefficient (friction), $k$ is the spring constant, and $x(t)$ is the displacement. 3. **Important rules:** This is a damped harmonic oscillator. The solution depends on the discriminant $$\Delta = c^2 - 4mk$$. 4. **Cases:** - If $\Delta > 0$, overdamped: two real distinct roots. - If $\Delta = 0$, critically damped: one real repeated root. - If $\Delta < 0$, underdamped: complex conjugate roots leading to oscillations with exponential decay. 5. **Solving:** Assume solution $x(t) = e^{rt}$, substitute into the equation to get characteristic equation $$mr^2 + cr + k = 0$$. 6. **Find roots:** $$r = \frac{-c \pm \sqrt{c^2 - 4mk}}{2m}$$. 7. **General solution:** - Overdamped: $$x(t) = C_1 e^{r_1 t} + C_2 e^{r_2 t}$$ - Critically damped: $$x(t) = (C_1 + C_2 t) e^{rt}$$ - Underdamped: $$x(t) = e^{-\frac{c}{2m} t} (C_1 \cos(\omega t) + C_2 \sin(\omega t))$$ where $$\omega = \frac{\sqrt{4mk - c^2}}{2m}$$. 8. **Interpretation:** Constants $C_1$ and $C_2$ are determined by initial conditions (initial displacement and velocity). This method allows solving the motion of a spring-mass system with friction.