Examen de Física I URV 2023 · (1ª Convocatoria - Enero 2023)

Problem (10 points). Damped oscillator and resonance

A particle of mass $m = 250\text{ g}$ undergoes damped harmonic motion with an oscillation frequency of $5\text{ Hz}$. It is observed that over an interval of $10\text{ s}$, the amplitude of the oscillations decreases by $30\%$.

a)

Determine the elastic spring constant $k$ and the damping parameter $b$. [4 points]

b)

Find the time required for the system's mechanical energy to drop to $10\%$ of its initial value. [3 points]

c)

If a sinusoidal driving force of amplitude $F_0 = 2\text{ N}$ is applied to the oscillator, can resonance in velocity take place? Can resonance in displacement (amplitude) occur? If so, compute the corresponding angular frequency of the driving force in each scenario. [3 points]

A crate of mass $m = 20\text{ kg}$ is pulled up a ramp inclined at an angle of $30^\circ$ with respect to the horizontal to reach an elevation of $h = 3\text{ m}$. The pulling force is applied via a rope with a tension of $F = 200\text{ N}$ forming an angle of $45^\circ$ above the incline surface. The kinetic friction coefficient between the crate and the plane is $\mu_k = 0.2$. (Take $g = 9.8\text{ m/s}^2$).

a)

Calculate the acceleration of the crate up the ramp. [2.5 points]

A crate of mass $m = 20\text{ kg}$ is pulled up a ramp inclined at an angle of $30^\circ$ with respect to the horizontal to reach an elevation of $h = 3\text{ m}$. The pulling force is applied via a rope with a tension of $F = 200\text{ N}$ forming an angle of $45^\circ$ above the incline surface. The kinetic friction coefficient between the crate and the plane is $\mu_k = 0.2$. (Take $g = 9.8\text{ m/s}^2$).

b.

Compute the work carried out by the applied force $F$ during this ascent. Will this work be equal to the work done by friction? Compute the work of friction and compare both values.

Question 2 [5 points]. Energy in gravitational orbits

A celestial body of mass $m_1$ orbits within the gravitational field produced by a body of mass $m_2$.

a)

Assuming the orbit is circular, are the kinetic energy, the gravitational potential energy, and the total mechanical energy constant? Justify your answers thoroughly. [1.5 points]

b)

Assuming the orbit is elliptical, are the kinetic energy, the gravitational potential energy, and the total mechanical energy constant? Justify your answers thoroughly. [1.5 points]

c)

For any quantities that were not constant in the previous sections, specify at which orbital positions they attain their maximum and minimum values. Explain in detail. [2 points]