Examen de Física I URV 2024 · Physics I - 1st & 2nd Partial Exam (January 2024)

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

Consider a particle moving in one dimension under the action of a conservative force characterized by the potential energy function:

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

where position $x$ is given in centimeters (cm) and the potential energy $U(x)$ in Joules (J).

a)

Does the coefficient of $x^2$ have physical units? If so, state them. [1 point]

b)

Determine the interval of movement allowed for the particle if its total mechanical energy is $E_m = 6\text{ J}$. Sketch the kinetic energy curve $E_k(x)$ strictly within this physically allowed interval. [2 points]

[

,

] cm

c)

Plot the force $F(x)$ acting on the particle as a function of position within this same interval of motion. [2 points]

$F(x) =$

J/cm

d)

Repeat questions b) and c) assuming that the total mechanical energy is instead $E_m = -5\text{ J}$. [4 points]

[

,

] cm

e)

What is the minimum mechanical energy required for the particle to start moving? Justify your answer. [1 point]

$E_m >$

J

Exercise 2 (10 points). Damped and forced harmonic oscillations

A mass of $m = 6\text{ kg}$ hangs from the bottom of a vertical spring having a constant $k = 150\text{ N/m}$. The mass experiences a viscous damping force directly proportional to its instantaneous velocity. At time $t = 0$, the mass is located at $x(0) = 1\text{ cm}$ with an initial velocity of $v(0) = 1\text{ m/s}$. Calculate:

a)

The damping constant $b$ and the period of the damped oscillation, knowing that the damping force has a magnitude of $80\text{ N}$ at a speed of $2\text{ m/s}$. [2 points]

b)

The equation describing the instantaneous position $x(t)$ of the mass fulfilling the initial conditions. [2 points]

c)

A qualitative graphical representation showing the amplitude variation over time, $A(t)$. [2 points]

d)

The angular frequency ($\omega$) of an external periodic force with a static displacement equivalent amplitude of $2\text{ cm}$ that produces amplitude resonance. Find the resulting amplitude and phase of the oscillations at this condition. [2 points]

e)

If the external driving force is given by $F(t) = 5\cos(5t)\text{ N}$, explain whether any type of resonance will be induced. Compute the steady-state amplitude and phase under these specific conditions. [2 points]