Ejercicio Exercise 1 · Física I URV 2024

a. Does the coefficient multiplying x² carry physical dimensions/units? If yes, identify them.

Solución: J/cm² / J·cm⁻²

b. Determine the interval of motion accessible to the particle if its total mechanical energy is E = 6 J.

Solución: [-√22/3, √22/3] cm / [-1.56, 1.56] cm

c. Graph the conservative force F(x) corresponding to this potential within the allowed domain determined previously.

Solución: F(x) = -18x

d. Repeat questions (b) and (c) assuming that the particle has a mechanical energy of E = -5 J.

Solución: [-√11/3, √11/3] cm / [-1.11, 1.11] cm

e. What is the minimum mechanical energy required for the particle to initiate motion?

Solución: E > -16 J

e. Provide a reasoned explanation for the minimum mechanical energy.

Solución: The minimum of the potential energy is U(0) = -16 J. To initiate motion, kinetic energy must be greater than zero (K > 0), so E = K + U > -16 J.

Sketches and graphs of kinetic energy curves and conservative force (parts b, c, d)

Solución: b. Kinetic energy curve (E = 6 J): $K(x) = E - U(x) = 6 - (9x^2 - 16) = -9x^2 + 22\text{ J}$. This corresponds to an inverted parabola with its maximum at $x = 0$ ($K_{\max} = 22\text{ J}$) and roots at $x = \pm \frac{\sqrt{22}}{3}\text{ cm} \approx \pm 1.56\text{ cm}$. c. Conservative force graph: A straight line passing through the origin $(0, 0)$ with negative slope $-18\text{ J/cm}$, evaluated from $x = -1.56\text{ cm}$ ($F = 28.08\text{ J/cm}$) to $x = 1.56\text{ cm}$ ($F = -28.08\text{ J/cm}$). d. Graphs for E = -5 J: - Kinetic energy: inverted parabola $K(x) = -9x^2 + 11\text{ J}$ with vertex at $(0, 11\text{ J})$ and roots at $x = \pm \frac{\sqrt{11}}{3} \approx \pm 1.11\text{ cm}$. - Conservative force: same linear graph $F(x) = -18x$ plotted over the restricted domain $[-1.11\text{ cm}, 1.11\text{ cm}]$.