Examen de Física I URV 2024 · Primer Parcial (19/01/2024)

Exercise 1 (10 points). Potential energy and conservative force

A particle moves under the action of a one-dimensional conservative force whose potential energy is described by:

$$U(x) = 9x^2 - 16$$

where $x$ is given in centimeters (cm) and $U(x)$ is expressed in Joules (J).

a.

Does the coefficient multiplying $x^2$ carry physical dimensions/units? If yes, identify them. [1 point]

b.

Determine the interval of motion accessible to the particle if its total mechanical energy is $E = 6\text{ J}$. Sketch the kinetic energy curve $E_{\text{kin}}(x)$ restricted to this physically allowed range. [2 points]

c.

Graph the conservative force $F(x)$ corresponding to this potential within the allowed domain determined previously. [2 points]

d.

Repeat questions (b) and (c) assuming that the particle has a mechanical energy of $E = -5\text{ J}$. [4 points]

e.

What is the minimum mechanical energy required for the particle to initiate motion? Provide a reasoned explanation. [1 point]

Exercise 2 (10 points). Damped and driven oscillations

A mass of $m = 6\text{ kg}$ hangs vertically from a spring of stiffness constant $k = 150\text{ N/m}$. The damping force is linear with the instantaneous velocity of the body. At $t = 0$, the mass is observed at a displacement $x(0) = 1\text{ cm} = 0.01\text{ m}$ with a downward/positive velocity $v(0) = 1\text{ m/s}$. Answer the following:

a.

Find the damping parameter $b$ (or $\beta$) and the natural damped period $T$, knowing that the resistive force equals $80\text{ N}$ when the speed reaches $2\text{ m/s}$. [2 points]

b.

Write the explicit expression for the position as a function of time, $x(t)$, incorporating the stated initial values. [2 points]

c.

Plot the damping envelope representing the time-dependent amplitude $A(t)$. [2 points]

d.

Determine the driving angular frequency $\omega_{\text{ext}}$ required to obtain amplitude resonance if an external driving force of amplitude $F_0 = 0.02\text{ N}$ is applied. Calculate the steady-state amplitude and the phase shift under these conditions. [2 points]

e.

If the external force is modified to $F(t) = 5\cos(5t)\text{ N}$, state whether resonance takes place and identify its type. Compute the resulting amplitude and phase. [2 points]