Learn Extracted exam questions AP Physics 2 2023 Free Response
2023 Free Response
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(10 points, suggested time 20 minutes)
A rectangular tank with a mirrored bottom is filled with water (index of refraction $n_w$). A beam of light passes from air (index of refraction $n_a$) into the water at angle $\theta_i$ from the normal, as shown in Figure 1. Index of refraction $n_w$ is greater than index of refraction $n_a$.
[Figure 1: A rectangular tank of water with a mirrored bottom, labeled "Mirror" along the bottom. A dashed vertical line labeled "Normal" is drawn at the point where a beam enters the top surface of the water at angle $\theta_i$ from the normal. The region above the water surface is labeled $n_a$; the region inside the tank (the water) is labeled $n_w$.]
On the following diagram, sketch the entire path of the beam as the beam enters, travels through, and then exits the water.
[Diagram: A blank rectangular tank with a mirrored bottom, same as Figure 1, showing "Normal" (dashed vertical line) and incident angle $\theta_i$ at the entry point, with $n_a$ labeled above the water surface and $n_w$ labeled inside the tank, for the student to sketch the beam's path.]
Sugar is then added to the water, resulting in a mixture that has a different index of refraction than water. A student considers two models, Model A and Model B, for how the sugar mixes with the water. The models are shown in Figure 2.
Model A: The sugar is uniformly mixed throughout the water, resulting in a mixture with index of refraction $n_m$ such that $n_m > n_w$.
Model B: Layers are formed of varying concentrations of sugar in the water. There are three distinct layers of equal volume. The top layer is only water (index of refraction $n_w$). The middle layer has the same concentration of sugar in the water as in Model A (index of refraction $n_m$). The bottom layer has the highest concentration of sugar (index of refraction $n_b$).
[Figure 2: Two diagrams side by side, each a rectangular tank with a mirrored bottom and a valve/beam entering at angle $\theta_i$ from the "Normal." Model A: a single shaded region labeled $n_m$ inside the tank below the surface labeled $n_a$ (air above), with the beam entering at $\theta_i$. Model B: three horizontal layers inside the tank, from top to bottom labeled $n_w$ (top layer, unshaded), $n_m$ (middle layer, lightly shaded), and $n_b$ (bottom layer, densely shaded/hatched), with air ($n_a$) above and the beam entering at $\theta_i$.]
Consider Model A. Briefly describe how the observed wavelength of light changes, if at all, as the beam travels from air into the mixture.
Relevant angles between the beam and the normal for the various layers present in models A and B are defined in the following table.
| Model A | Model B | ||
|---|---|---|---|
| $\theta_i$ | Incident angle of the beam in air | $\theta_i$ | Incident angle of the beam in air |
| $\theta_1$ | Angle the beam makes with the normal in the mixture in Model A | $\theta_2$ | Angle the beam makes with the normal in the top layer in Model B |
| $\theta_3$ | Angle the beam makes with the normal in the middle layer in Model B | ||
| $\theta_4$ | Angle the beam makes with the normal in the bottom layer in Model B |
Determine an expression for $\theta_4$ in terms of $\theta_i$, $n_a$, and $n_b$.
Rank the angles from greatest to least, with 1 being greatest. If two angles are the same value, give them the same ranking.
____ $\theta_1$ ____ $\theta_2$ ____ $\theta_3$ ____ $\theta_4$
Briefly explain your reasoning using appropriate physics principles and/or mathematical models.
[Figure: Three diagrams. "Original Tank" — a rectangular water tank with a mirrored bottom, beam entering at $\theta_i$ from the Normal, region above labeled $n_a$, water labeled $n_w$. "Model A" — same tank with a single mixture layer labeled $n_m$. "Model B" — same tank with three layers labeled $n_w$ (top), $n_m$ (middle), $n_b$ (bottom).]
For the original tank filled with water, the beam is observed to exit the surface of the water a horizontal distance $d_w$ from the entry point. For models A and B, the horizontal distances are $d_A$ and $d_B$, respectively.
Determine whether $d_A$ and $d_B$ are each greater than, less than, or equal to $d_w$. It is NOT necessary to compare $d_A$ to $d_B$. Briefly justify your answer.
(12 points, suggested time 25 minutes)
Students are given an unknown circuit component that is connected in series to a resistor with known resistance $500\ \Omega$.
The students are asked to experimentally determine whether the component is a resistor or an uncharged capacitor.
Complete the following diagram to show how to use standard circuit equipment to determine whether the component is a resistor or an uncharged capacitor.
[Diagram: A circuit component box (labeled "Circuit Component") connected in series via a wire to a $500\ \Omega$ resistor, drawn with the standard zigzag resistor symbol. The circuit component is drawn as a rectangle with terminals. The student must complete the diagram with standard circuit equipment (e.g., battery, switch, meters).]
Describe an experimental procedure to determine whether the component is a resistor or an uncharged capacitor. Refer to the circuit equipment in the diagram drawn in part (a)(i).
What results would the students expect if the component is an uncharged capacitor? Support your answer in terms of potential difference and charge.
[Figure: A circuit diagram showing a variable resistor $R_{var}$ connected in series with an ammeter (circle with "A"), and a battery with emf $\mathcal{E}$ and internal resistance $r$ (shown as a battery symbol inside a dashed box labeled $\mathcal{E}$, $r$).]
The students conduct a different experiment to determine the emf $\mathcal{E}$ of a battery that is not ideal and has internal resistance $r = 30\ \Omega$. The battery is connected to a variable resistor in a circuit, as shown. The students measure the current $I$ through the circuit for different values of resistance $R_{var}$ of the variable resistor that is connected to the battery. The following table contains the data collected.
| $I$ (A) | $R_{var}$ ($\Omega$) |
|---|---|
| 0.087 | 200 |
| 0.060 | 300 |
| 0.042 | 450 |
| 0.027 | 700 |
| 0.016 | 1200 |
Write an equation describing the circuit in terms of $\mathcal{E}$, $I$, $r$, and $R_{var}$.
Which quantities could be graphed to yield a straight line that could be used to calculate a numerical value for the emf $\mathcal{E}$ of the battery?
Horizontal Axis: ________________ Vertical Axis: ________________
Plot the data points for the quantities indicated in part (b)(ii) on the following graph. Clearly scale and label all axes, including units. Draw a straight line that best represents the data. You may use the blank columns in the table for any quantities you graph other than the given data.
[Graph: A blank grid with light dashed gridlines, with an unlabeled vertical axis (arrow pointing up) and unlabeled horizontal axis (arrow pointing right), for the student to plot and scale the chosen quantities.]
Using the graph from part (b)(iii), determine the emf $\mathcal{E}$ of the battery.
(12 points, suggested time 25 minutes)
[Figure 1: Tank X is a large cylindrical tank, partially filled with water (shaded), with two blocks labeled A and B floating at the bottom-left in front of the tank, and a "Valve" labeled at the bottom-right of the tank. Note: Figure not drawn to scale.]
Tank X is a large cylindrical tank that is partially filled with water, as shown in Figure 1. The bottom of Tank X is connected to a short horizontal pipe. A valve that is initially closed can be opened to allow water to flow through the pipe and exit through the other end of the pipe.
Two blocks, A and B, have identical dimensions and are placed in the tank. Both blocks float at rest and are partially submerged in the water.
The water and air can be modeled as consisting of individual particles that are in continuous random motion. In terms of interactions with both water and air particles, explain why there is an upward buoyant force exerted on each block.
The valve is then opened, and water flows out through the pipe. The surface of the water moves downward. When Block A touches the bottom of Tank X, Block B is still above the bottom of Tank X. Which block has a greater density? Briefly explain your reasoning.
[Figure 2: Tank Y is a large tank shaped like an inverted cone (wide at the top, narrowing toward the bottom), open to the air at the top, filled with water to height $h_0$ above a horizontal pipe at the bottom, which has a closed "Valve." Note: Figure not drawn to scale.]
Tank Y is a large tank with the top open to the air, as shown in Figure 2. The bottom of Tank Y is connected to a short horizontal pipe of radius $r$ with a closed valve. Tank Y is filled with water to height $h_0$ above the horizontal pipe. Tank Y is specially designed so that when the valve is opened, the surface of the water moves downward at constant speed $v_s$.
At time $t = 0$, the valve is opened.
Derive the relationship between the speed $v_p$ at which water exits the pipe and the changing height $h$ of the surface of the water above the pipe to show that
Derive the relationship between $v_p$ and the changing radius $R$ of the top surface of the water to show that
When the radius $R$ of the tank is sufficiently greater than $r$, the speed $v_p$ can be approximated as
Justify this claim.
[Figure 3: Tank Z is a large tank whose top is open to the air, shaped like a cone that is wider at the top and narrows sharply partway down before widening again near a horizontal pipe at the bottom fitted with a closed "Valve." Water fills the tank to height $h_0$ above the horizontal pipe. Note: Figure not drawn to scale.]
Tank Z is a large tank whose top is open to the air and is shaped as shown in Figure 3. The bottom of Tank Z is connected to a short horizontal pipe with a closed valve. Tank Z is filled with water to a height $h_0$ above the horizontal pipe.
At time $t = 0$, the valve of Tank Z is opened.
Does the speed $v_s$ at which the surface of the water moves downward increase, decrease, or remain the same over time as water exits the other end of the pipe? Justify your answer by using or referencing equations from both part (b)(i) and part (b)(ii).
(10 points, suggested time 20 minutes)
Particles A and B each have positive charge $+Q$ and are held fixed at two vertices of an equilateral triangle of side $d$, as shown. Point P is located equidistant from each vertex of the triangle.
[Figure: An equilateral triangle with vertices labeled; Particle A at one vertex and Particle B at another vertex, each carrying charge $+Q$, with side length $d$ between them; Point P is located at the position equidistant from all three vertices (the centroid), including the unoccupied bottom-right vertex referenced in the discussion below.]
Students Y and Z discuss the electric field and the electric potential at Point P after a third charged particle is placed in the bottom-right vertex. The students make the following statements.
Student Y: "If a particle with positive charge $+2Q$ is placed at the bottom-right vertex, the magnitude of the electric field will be zero at Point P."
Student Z: "To make the value of the electric potential zero at Point P, a particle with negative charge $-Q$ should be placed at the bottom-right vertex."
In a coherent, paragraph-length response, evaluate the accuracy of each student's statement. If any aspect of either student's statement is inaccurate, explain how to correct the student's statement. Support your evaluations using appropriate physics principles.
Particles A and B are once again held in place at two vertices of the equilateral triangle. The students want to represent the electric potential energy of a system of particles that is brought very far away to the bottom-right vertex. Scenarios 1 and 2 are considered.
In Scenario 1, a third particle with positive charge $+Q$ is moved from very far away to the bottom-right vertex and then held in place. A bar is shown on the following chart that represents the electric potential energy $U_{i1}$ of the system consisting of all three particles when the third particle with positive charge is very far away from the other particles.
In the grid provided, complete the bar chart.
- Draw a bar to represent the work $W_1$ required to move the third particle with positive charge from very far away to the bottom-right vertex.
- Draw another bar to represent the electric potential energy $U_{f1}$ of the system consisting of all three particles when the third particle with positive charge is held in place at the bottom-right vertex.
The height of each bar should be proportional to the energy represented. If the quantity is zero, write a "0" in that column.
[Chart: A bar chart grid with three labeled columns: $U_{i1}$, $W_1$, $U_{f1}$, and a horizontal "0" baseline. A bar is already shown under column $U_{i1}$ (a small positive bar above the 0 line), representing the initial potential energy of the two-particle system when the third particle is very far away. Scenario 1 title below the chart.]
In Scenario 2, a particle with negative charge $-Q$ is moved from very far away to the bottom-right vertex and then held in place. A bar is shown on the following chart that represents the electric potential energy $U_{i2}$ of the system consisting of all three particles when the particle with negative charge is very far away from the other particles.
In the grid provided, complete the bar chart.
- Draw a bar to represent the work $W_2$ required to move the particle with negative charge from very far away to the bottom-right vertex.
- Draw another bar to represent the electric potential energy $U_{f2}$ of the system consisting of all three particles when the particle with negative charge is held in place at the bottom-right vertex.
The height of each bar should be proportional to the energy represented. If the quantity is zero, write a "0" in that column.
[Chart: A bar chart grid with three labeled columns: $U_{i2}$, $W_2$, $U_{f2}$, and a horizontal "0" baseline. A bar is already shown under column $U_{i2}$ (a small positive bar above the 0 line), representing the initial potential energy of the two-particle system when the negative particle is very far away. Scenario 2 title below the chart.]