- The equation of motion is a second-order linear differential equation: $m \cdot x''(t) = -c \cdot x'(t) - k \cdot x(t)$.
- Rearranging into standard form gives: $x''(t) + \frac{c}{m} x'(t) + \frac{k}{m} x(t) = 0$.
- This represents a damped harmonic oscillator with mass $m$, damping coefficient $c$, and spring constant $k$.
- The initial conditions are $x(0) = 2$ inches and $x'(0) = 0.5$ ft/s. Note the unit mismatch: convert $x'(0)$ to inches per second by multiplying by 12, giving $x'(0) = 6$ in/s.
- The solver uses `odesolve` to numerically integrate the ODE from $t=0$ to $t=10$ seconds with the given initial conditions.
- The graph displays three solutions labeled $x_{90lb}(t)$, $x_{180lb}(t)$, and $x_{270lb}(t)$, suggesting different masses (90 lb, 180 lb, 270 lb) while keeping $c$ and $k$ constant.
- As mass increases, the system becomes less damped relative to its inertia, leading to slower oscillations and potentially more cycles before settling, as seen in the graph where the red dashed line ($x_{270lb}$) has a longer period and decays more slowly than the orange solid line ($x_{90lb}$).
Parent Tip: Review the logic above to help your child master the concept of mathcad worksheet.
