Exercici 1 · Física I URV 2025

Exercici 1 (10 punts). Damped and forced oscillations

A body of mass $m = 6\text{ kg}$ is attached to a vertical spring with an elastic constant of $k = 150\text{ N/m}$. During the oscillations, it is noted that the amplitude diminishes gradually over time. Assuming that at $t = 0$ the mass is located at $x_0 = 1\text{ cm}$ from the origin with an initial velocity $v(0) = 1\text{ m/s}$, determine:

a)

The equation for the position as a function of time, $x(t)$, knowing that a damping force of $80\text{ N}$ opposes motion when the mass moves at $2\text{ m/s}$. [2.5 points]

b)

The value of the damping constant $b$ and the oscillation period $T$ of this underdamped motion under the given initial conditions. [2.5 points]

c)

The angular frequency of a sinusoidal external driving force with amplitude $F_0 = 2\text{ N}$ that causes amplitude resonance. Calculate the steady-state amplitude and phase of the resulting oscillation. [2.5 points]

d)

If a driving force $F(t) = 2\cos(5t)\text{ N}$ is applied to the mass, state whether any kind of resonance takes place. Determine the amplitude and phase of the permanent motion driven by this force. [2.5 points]