Ejercicio Exercise 2 · Física I URV 2024

a) The damping constant b

Solución: $40$ N·s/m / $40$ kg/s

a) The period of the damped oscillation

Solución: $1{,}686$ s

b) The equation describing the instantaneous position x(t) of the mass fulfilling the initial conditions

Solución: $e^{-\frac{10}{3}t} [0{,}01\cos(3{,}727t) + 0{,}2773\sin(3{,}727t)]$ m / $0{,}2775 e^{-3{,}333t}\cos(3{,}727t - 1{,}535)$ m

c) A qualitative graphical representation showing the amplitude variation over time, A(t)

Solución: The envelope curve representing the oscillation amplitude decreases exponentially over time according to $A(t) = A_0 e^{-\gamma t} = 0.2775 e^{-3.333t}\text{ m}$. The curve starts at $A(0) \approx 0.278\text{ m}$ on the vertical axis and decays asymptotically toward zero as $t \to \infty$, with characteristic decay time $\tau = 1/\gamma = 0.3\text{ s}$.

d) The angular frequency (ω) that produces amplitude resonance

Solución: $1{,}667$ rad/s / $\frac{5}{3}$ rad/s

d) Resulting amplitude at amplitude resonance

Solución: $0{,}0201$ m / $2{,}01$ cm

d) Phase of the oscillations at amplitude resonance

Solución: $0{,}464$ rad / $26{,}57^\circ$

e) Explain whether any type of resonance will be induced

Solución: Velocity resonance (phase resonance or frequency of maximum power absorption), because $\omega = \omega_0 = 5\text{ rad/s}$.

e) Steady-state amplitude under these conditions

Solución: $0{,}025$ m / $2{,}5$ cm

e) Steady-state phase under these conditions

Solución: $\pi/2$ rad / $1{,}571$ rad / $90^\circ$