Examen de Física II URV 2024 · Exámenes de : Teoría y Laboratorio (Mayo 2024)

Lab 1. Calculation of errors

Explain how the volume resolution error would change if all length dimensions were determined solely by using a caliper instead of other instruments. Compute the quantitative magnitude of this change using your experimental laboratory data. Would the random error be affected as well? Justify your answer.

Lab 2. Elasticity of a spring and Hooke's law

A force of $20\text{ N}$ is applied to a system of two springs coupled in series, causing the assembly to stretch by $10\text{ cm}$.

a)

Determine the equivalent elastic constant of this spring system.

b)

Assuming the spring constant of the first spring is half that of the second spring, find the individual elastic constant of each spring.

Lab 3. Mechanical resonance in elongation

In the laboratory experiment on mechanical resonance, two contact forces act directly upon the oscillating mass: the elastic restoring force of the spring and the viscous damping force of the immersion fluid. The driving motor does not act directly on the suspended mass. How does the action of the motor turn into a driving resonant force governing the motion of the mass?

Exercise 1 (10 points). Bungee jumper oscillation

An individual of mass $m = 60\text{ kg}$ leaps from a bridge situated at a height $h = 100\text{ m}$ above the ground, secured to an elastic rope of natural length $L_0 = 20\text{ m}$ and spring constant $k$. The jumper experiences free fall until the rope reaches its unstretched length. Beyond this point, the cord stretches and applies an upward force that decelerates the jumper until he momentarily halts and rebounds into oscillatory motion. Assume air resistance and the mass of the rope are negligible ($g = 9.8\text{ m/s}^2$).

a.

Calculate the spring constant $k$ required so that the jumper stops ($v = 0$) just before reaching the ground. [2 points]

b.

Determine the maximum elongation of this same rope if a second jumper of mass $m_2 = 90\text{ kg}$ performs the jump. [2 points]

c.

Compute the speed of this second jumper when reaching the ground level. [2 points]

d.

Establish the differential equation and solve for the equation of motion $x(t)$ of this second jumper once the rope begins to stretch, setting $x(0) = 0$ at the moment the rope starts extending. [4 points]

Exercise 2 (10 points). DC circuit analysis

Consider three identical DC generators, each having an electromotive force $\mathcal{E} = 9\text{ V}$ and an internal resistance $r = 1\ \Omega$.

a.

Calculate the equivalent electromotive force and internal resistance if all three generators are connected: (i) in series, and (ii) in parallel. [3 points]

b.

If the three generators are interconnected in the circuit shown in the diagram, compute the intensity of current flowing through each element in the steady state (taking $R_1, R_2, R_3$ as the branch resistances). [3.5 points]

c.

Assuming the capacitor in the middle branch has a capacitance of $C = 2\ \mu\text{F}$, determine the electric charge accumulated on its plates in terms of the voltage $|V_C|$ across it. [3.5 points]

Question 1 (5 points). Morse potential

A widely used model for describing the vibrational motion of diatomic molecules is the Morse potential. In a specific dimensionless one-dimensional formulation, the potential energy is given by:

$$V(x) = e^{-2x} - 2e^{-x}$$

Analyze and discuss the dynamics of a particle moving in this potential field if its total mechanical energy is $E = -0.5\text{ J}$.

1.

Position $x_{\min}$ and value $V_{\min}$ of the potential energy minimum:

2.

For $E = -0.5\text{ J}$, calculate the inner ($x_1$) and outer ($x_2$) classical turning points, and the maximum kinetic energy $K_{\max}$:

3.

Characterize the nature of the motion for $E = -0.5\text{ J}$ (e.g. bounded vs unbounded, type of motion):

Question 2 (5 points). Electrostatic work

A point charge $Q = +5\text{ nC}$ is located at the origin of coordinates $(0, 0)$.

a.

Determine the energy required to bring a test charge $q = +1\text{ nC}$ from infinity to point $A(3\text{ m}, 4\text{ m})$.

b.

If the charge $q$ is subsequently displaced from point $A(3\text{ m}, 4\text{ m})$ to point $B(4\text{ m}, 3\text{ m})$, what is the amount of work required to perform this second displacement?

Simulation 1. Electric field and potential of a +1 μC charge

In the electrostatic field simulation, clear the workspace and place a single point charge $q = +1\,\mu\text{C}$ at the origin $(0, 0)$. Activate the display of electric field vectors and equipotential lines.

a.

Record the potential $V$, field magnitude $E$, components $E_x, E_y$, and angle $\alpha$ measured at position $(100\text{ m}, 100\text{ m})$.

V

N/C

N/C

N/C

°

b.

Record the same quantities at position $(100\text{ m}, -100\text{ m})$.

V

N/C

N/C

N/C

°

c.

Verify these experimental simulation values against theoretical calculations applying the formulas for a point charge:

$$\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2}\hat{u}_r, \quad V = \frac{1}{4\pi\varepsilon_0} \frac{q}{r}$$

Simulation 2. Electric field and potential of a -1 μC charge

Place a charge $q = -1\,\mu\text{C}$ at the origin $(0, 0)$ and repeat the measurements of point 1 at $(100\text{ m}, 100\text{ m})$ and $(100\text{ m}, -100\text{ m})$. Compare the outcomes with the positive charge case: what differences are observed?

1.

Measurements at position $(100\text{ m}, 100\text{ m})$:

V

N/C

N/C

N/C

°

2.

Measurements at position $(100\text{ m}, -100\text{ m})$:

V

N/C

N/C

N/C

°

3.

Compare the outcomes with the positive charge case: what differences are observed?

Simulation 3. Electric field and potential of a charge dipole (+2 μC and -2 μC)

Clear the workspace and place a charge $q_1 = +2\,\mu\text{C}$ at $(-100\text{ m}, 0)$ and a charge $q_2 = -2\,\mu\text{C}$ at $(100\text{ m}, 0)$. Observe the field vectors, field lines, and equipotential curves. Record the electric potential $V$, field module $E$, vector components $E_x, E_y$, and angle $\alpha$ at the following positions:

a.

$(-100\text{ m}, 0)$ (near the positive charge, e.g. at $(-100\text{ m}, -1\text{ m})$):

V

N/C

N/C

N/C

°

b.

$(0, 0)$:

V

N/C

N/C

N/C

°

c.

$(-150\text{ m}, 0)$:

V

N/C

N/C

N/C

°

d.

$(0, 150\text{ m})$:

V

N/C

N/C

N/C

°

e.

$(-150\text{ m}, 150\text{ m})$:

V

N/C

N/C

N/C

°

Lab 6. Numerical simulation of central forces

A point charge $q_1 = -4\ \mu\text{C}$ is located at the origin $(0, 0)$, and a second point charge $q_2 = +2\ \mu\text{C}$ is located at $(200\text{ m}, 200\text{ m})$.

Simulation 4. Electric field and potential of a three-charge configuration

Reset the simulation screen and configure three point charges: $q_1 = +4\ \mu\text{C}$ at $(0, 0)$, $q_2 = -2\ \mu\text{C}$ at $(0, -100\text{ m})$, and $q_3 = -2\ \mu\text{C}$ at $(0, 100\text{ m})$. Inspect the field lines and equipotential patterns, and record the potential and electric field vector at:

a)

At $(150\text{ m}, 0)$:

V

N/C

b)

At $(0, 150\text{ m})$:

V

N/C

c)

At $(-150\text{ m}, 150\text{ m})$:

V

N/C

d)

At $(200\text{ m}, 0)$:

V

N/C

Lab 6. Numerical simulation of central forces

A point charge $q_1 = -4\ \mu\text{C}$ is located at the origin $(0, 0)$, and a second point charge $q_2 = +2\ \mu\text{C}$ is located at $(200\text{ m}, 200\text{ m})$.

Lab 4. Determination of the center of gravity

Theoretically evaluate the coordinates of the center of gravity of the flat, homogeneous plate shown in the figure below. Take the origin of coordinates at point $A(0, 0)$.

The outer dimensions are $10\text{ cm} \times 10\text{ cm}$. A square central section is cut out from the top edge, leaving two vertical side arms of width $3\text{ cm}$ each, and a bottom horizontal bar of height $3\text{ cm}$.

a)

cm

b)

cm

Lab 5. Measuring the density of liquids

To determine the density of an unknown liquid, a spherical metallic mass of diameter $d = 2.12\text{ cm}$ is utilized. The measured mass of the sphere in air is $m_{\text{air}} = 126.75\text{ g}$, and its apparent mass when fully immersed in the liquid is $m_{\text{app}} = 122.18\text{ g}$.

a)

What is the buoyant force exerted by the liquid on the immersed sphere?

b)

Calculate the density of the liquid.

Lab 6. Numerical simulation of central forces

A point charge $q_1 = -4\ \mu\text{C}$ is located at the origin $(0, 0)$, and a second point charge $q_2 = +2\ \mu\text{C}$ is located at $(200\text{ m}, 200\text{ m})$.

a)

Calculate the magnitude and direction of the net electric field vector at point $A(100\text{ m}, 100\text{ m})$.

N/C

°

b)

Plot or sketch the resulting electric field vector at point $A$.

15. Lab 7. Association of resistances and Kirchhoff's laws

What was the experimental value of the galvanometer current $I_G$ measured in your laboratory session? Under what circumstances can the condition $I_G = 0\text{ A}$ occur? If this condition can be achieved, explain what circuit modifications or parameter settings must be made.