Examen de Física II URV 2024 · Extraordinary Exam - Physics

Exercise 1 (10 points). Mass-spring system and damped oscillations

A mass of $200\text{ g}$ is suspended from a system of two parallel springs as illustrated below. The elastic constants of the two springs are $k_1 = 0.9\text{ N/m}$ and $k_2 = 1.1\text{ N/m}$, respectively.

At time $t = 0$, the mass is released from rest at a position $-2\text{ cm}$ away from equilibrium. Assuming negligible air resistance:

a.

Determine the equivalent elastic constant of the assembly, as well as the pulsation (angular frequency) and the oscillation period. [2 points]

b.

Write the equation of motion $x(t)$ in SI units (express $x$ in meters). [2 points]

c.

Compute the total mechanical energy of the system at any given moment. How is this energy divided between kinetic and potential forms when the mass reaches $+1\text{ cm}$ from the equilibrium position? [2 points]

Exercise 1 (10 points). Mass-spring system and damped oscillations

A mass of $200\text{ g}$ is suspended from a system of two parallel springs as illustrated below. The elastic constants of the two springs are $k_1 = 0.9\text{ N/m}$ and $k_2 = 1.1\text{ N/m}$, respectively.

Assuming now that friction with air produces a resistive force equal in magnitude to one-fourth of the instantaneous velocity:

a)

Find the new angular frequency, the new period, and the corresponding equation of motion. (2 points)

b)

Sketch the total energy as a function of time and calculate the time elapsed until half of the initial energy is dissipated. (2 points)

Exercise 2 (10 punts). Gravitation and satellite orbital mechanics

A satellite with a mass of $500\text{ kg}$ travels in orbit around the Earth at an altitude of $350\text{ km}$.

a.

What kind of orbit is described by the satellite around the Earth? [1 point]

b.

Taking the mass of the Earth as $M = 5.97 \times 10^{24}\text{ kg}$ and its radius as $R = 6371\text{ km}$, determine the satellite's orbital speed and period. What is its angular velocity? [3 points]

c.

What minimum kinetic energy must the satellite have to completely escape from its orbit to infinity? [2.5 points]

d.

A second satellite moves around the Earth in an elliptical orbit characterized by an eccentricity of $e = 0.8$. Calculate its speed at the closest point to the Earth (perigee), given that the distance at the farthest point (apogee) is $10^4\text{ km}$ and the angular momentum at that point equals $8 \times 10^{12}\text{ kg}\cdot\text{m}^2/\text{s}$. [3.5 points]