Ejercicio Exercise 1 · Física I URV 2024

a. Does the coefficient of x² have physical units? If so, state them.

Solución: $\text{J/cm}^2$ / $\text{J/m}^2$ / $\text{N/cm}$ / $\text{N/m}$

b. Determine the lower turning point x_min (cm) for Em = 6 J

Solución: $-\frac{\sqrt{22}}{3}$ cm / $-1{,}563$ cm

b. Determine the upper turning point x_max (cm) for Em = 6 J

Solución: $\frac{\sqrt{22}}{3}$ cm / $1{,}563$ cm

c. Expression of the force F(x) acting on the particle

Solución: $-18x$

d. Lower turning point x_min (cm) for Em = -5 J

Solución: $-\frac{\sqrt{11}}{3}$ cm / $-1{,}106$ cm

d. Upper turning point x_max (cm) for Em = -5 J

Solución: $\frac{\sqrt{11}}{3}$ cm / $1{,}106$ cm

e. Minimum mechanical energy required for the particle to start moving

Solución: $-16$

e. Justification of the minimum mechanical energy required

Solución: At $x = 0$, $U(x)$ reaches its minimum $U(0) = -16\text{ J}$. At $E_m = -16\text{ J}$, $E_k = 0$ and the particle is in static equilibrium; for motion to occur, $E_k > 0$, so $E_m > -16\text{ J}$.

Sketches and Plots: b) Ek(x) curve for Em = 6 J; c) F(x) plot for Em = 6 J; d) Ek(x) and F(x) plots for Em = -5 J

Solución: b. Sketch of $E_k(x)$ for $E_m = 6\text{ J}$: Inverted parabola symmetric around $x = 0$ with a maximum of $E_k(0) = 22\text{ J}$, dropping to $0\text{ J}$ at the turning points $x = \pm\frac{\sqrt{22}}{3}\text{ cm} \approx \pm 1{,}563\text{ cm}$. Strictly plotted inside this interval. c. Plot of $F(x)$: Straight line passing through the origin with negative slope $-18\text{ J/cm}^2$ (or $-1800\text{ N/cm}$). At $x = -1{,}563\text{ cm}$, $F = +28{,}14\text{ J/cm} = +2814\text{ N}$; at $x = 0$, $F = 0$; at $x = +1{,}563\text{ cm}$, $F = -28{,}14\text{ J/cm} = -2814\text{ N}$. d. Plots for $E_m = -5\text{ J}$: $E_k(x) = 11 - 9x^2$ is an inverted parabola reaching peak $E_k(0) = 11\text{ J}$ and vanishing at $x = \pm\frac{\sqrt{11}}{3}\text{ cm} \approx \pm 1{,}106\text{ cm}$. The force $F(x) = -18x$ is identical in functional form but plotted over the narrower domain $[-1{,}106\text{ cm}, 1{,}106\text{ cm}]$, spanning from $+19{,}9\text{ J/cm}$ to $-19{,}9\text{ J/cm}$.