Ejercicio Exercise 2 · Física I URV 2024

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]