Examen de Física I URV 2025 · Exam Physics (1st partial) - January 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]

Exercise 2 (10 punts). Kinematics of surface waves

A swimmer at sea notices that the waves propagate periodically with a sinusoidal profile. He estimates that the vertical distance from trough to crest is 0.45 m, and the distance between consecutive crests is roughly 1.8 m. In addition, he records that 12 full wave crests pass by every 2 minutes.

a)

Determine the harmonic wave equation $y(x,t)$ describing these surface waves, assuming propagation toward the swimmer along the positive direction. [2.5 points]

b)

Find the propagation speed of the wave. [1.5 points]

c)

Determine the oscillation velocity of the water particles at a distance of 10 m from where the waves were generated, at time $t = 5\text{ s}$. [2.5 points]

d)

What is the maximum speed that the oscillating water particles can reach? [2.5 points]