AP Physics 2 FRQ Practice

🧲AP Physics 2

Practice FRQ 1 of 121/12

1. A sealed cylinder contains a sample of monatomic ideal gas. A frictionless, insulating piston of cross-sectional area $A_p = 2.00× 10^{-3}\ \text{m}^2$ can move vertically without leaking gas. The piston has mass $m_p = 8.00\ \text{kg}$ , and the external pressure above the piston is constant at $P_{\text{atm}} = 1.00× 10^5\ \text{Pa}$ . Initially the gas is in equilibrium at temperature $T_1 = 300\ \text{K}$ and volume $V_1 = 3.00× 10^{-3}\ \text{m}^3$ , as shown in Figure 1. The gas is then heated slowly so that the piston rises while maintaining mechanical equilibrium with the surroundings.

Figure 1. Sealed cylinder containing monatomic ideal gas under a frictionless, insulating piston in mechanical equilibrium with atmospheric pressure.

Black-and-white line diagram of a vertical cylinder-and-piston apparatus, centered on the page.

Overall geometry and layout:
- Draw a tall, upright cylinder with straight vertical sidewalls and a flat bottom. The cylinder is shown as a 2D cross-section (a rectangle with an open top), occupying the central half of the figure width and most of the figure height.
- The top of the cylinder is open to the outside atmosphere.

Piston:
- Inside the cylinder, draw a horizontal piston disk as a thick, straight horizontal line spanning the full inner width of the cylinder.
- Place the piston clearly in the upper half of the cylinder, leaving a gas region below it and an atmosphere region above it.
- On the piston, place the label text exactly: "piston".
- Next to the piston (connected by a short leader line), include BOTH given piston properties as visible text on two separate lines:
- "m_p = 8.00 kg"
- "A_p = 2.00 × 10^−3 m^2"
- Add a note on/near the piston reading exactly: "frictionless, insulating".

Gas region (below piston):
- The region inside the cylinder below the piston is the gas volume. Lightly shade this region (very light gray) to distinguish it from the region above the piston.
- Centered within this shaded region, place a two-line label that includes the initial-state variables exactly as visible text:
- "V_1 = 3.00 × 10^−3 m^3"
- "T_1 = 300 K"
- Add an additional label near the same region reading exactly: "monatomic ideal gas".

Atmosphere region (above piston):
- The region above the piston up to the open top of the cylinder is unshaded and represents the outside air.
- In this upper region (not inside the gas), place the label exactly: "P_atm = 1.00 × 10^5 Pa".
- Also include the word "atmosphere" near that label.

Motion indication:
- Draw a single, bold, vertical arrow immediately to the right of the piston, pointing straight upward.
- Place the text "piston can move upward" next to this arrow.

Sealing statement:
- Near the cylinder sidewall (mid-height), add a short note reading exactly: "sealed (no gas leakage)".

Mechanical equilibrium cue:
- Place a small text note near the piston reading exactly: "mechanical equilibrium".

No axes, no graph grid, no additional numbers beyond those listed above.

Figure 2. Molecular-speed vector representations for two equal-volume gas samples (A and B).

Black-and-white diagram with two identical, side-by-side square containers showing molecular velocity arrows.

Container arrangement:
- Draw two equal-sized squares aligned horizontally with a clear gap between them.
- Center them on the page so they form a left box and a right box.
- Above the left square, center the title text exactly: "Gas A".
- Above the right square, center the title text exactly: "Gas B".
- Under each square, centered, add the text exactly: "equal volume".

Molecules / arrows (must be exactly matched in count and placement pattern):
- In EACH square, draw exactly 8 velocity arrows (representing 8 molecules) located at the same relative positions in both squares, so the only difference is arrow length.
- Use this identical placement pattern inside each square:
1) one arrow near the top-left interior corner,
2) one near the top-right interior corner,
3) one near the bottom-left interior corner,
4) one near the bottom-right interior corner,
5) one near the center,
6) one halfway between center and left wall,
7) one halfway between center and right wall,
8) one halfway between center and bottom wall.
- Arrow directions (identical directional set in both boxes):
- Exactly two arrows point to the right (horizontal),
- exactly two arrows point to the left (horizontal),
- exactly two arrows point upward (vertical),
- exactly two arrows point downward (vertical).
(This ensures an unambiguous symmetric distribution.)

Arrow lengths (the only difference between A and B):
- In Gas A (left box), all 8 arrows have the SAME short length.
- In Gas B (right box), all 8 arrows have the SAME long length.
- Define the long length precisely relative to the short length: each long arrow in Gas B is exactly twice the drawn length of each short arrow in Gas A.

Student prompts (visible text in the figure):
- To the right of both boxes (in a vertical list), include two checkbox-style prompts exactly as text:
- "Higher temperature: A B"
- "Greater pressure in identical rigid container (same number of molecules): A B"
(Leave A and B as selectable options; do not mark an answer.)

Style constraints:
- All arrows are thin black lines with triangular arrowheads.
- No numerical axes, no extraneous labels, and no additional arrows beyond the specified 8 per box.

Figure 3. Rigid-container gas: temperature change from T to 2T and the corresponding change in average molecular speed.

Black-and-white diagram showing one rigid container with two side-by-side states (before and after), so the student modifies arrow lengths.

Overall layout:
- Draw two identical square boxes aligned horizontally with a small gap between them, centered on the page.
- Above the left box, write exactly: "Initial: T".
- Above the right box, write exactly: "Final: 2T".
- Above both boxes (centered at the top of the figure), add the title text exactly: "Rigid container".

Rigid-container indication:
- Draw each box with thick walls.
- Under each box, write exactly: "V = constant".

Molecules/arrows (fixed count and directions):
- In the left box, draw exactly 6 velocity arrows (6 molecules) at fixed positions:
- one near top-left interior,
- one near top-right interior,
- one near bottom-left interior,
- one near bottom-right interior,
- one at the center,
- one midway between center and the right wall.
- Directions in the left box must be exactly:
- two arrows pointing right,
- two arrows pointing left,
- one arrow pointing up,
- one arrow pointing down.
- All 6 arrows in the left box have the SAME length (call this the reference length).

Right box (student modification target):
- In the right box, draw the same 6 arrows in the same positions and directions, but render them as editable/blank targets by drawing them in a lighter gray (or dashed outline) at the SAME initial reference length as the left box.
- Next to the right box, add the instruction text exactly: "Modify arrow lengths to show new average speed".

Numerical precision for the intended change:
- Include a boxed note beneath both boxes stating exactly: "When temperature doubles (T → 2T), average molecular speed increases by a factor of √2".
- The figure should therefore be constructed so that, when completed, every arrow in the right box should be drawn exactly √2 times the reference length of the arrows in the left box (do not pre-draw the longer arrows; leave as the student’s modification).

No extra symbols:
- Do not include any pressure values or variable n.
- No random arrow directions beyond those specified.

Complete the following tasks in Figures 2 and 3.

•

Indicate which gas (A or B) has the higher temperature in Figure 2.

•

Indicate which gas (A or B) would exert the greater pressure if each gas were placed in an identical rigid container with the same number of molecules.

•

In Figure 3, the gas temperature is increased from $T$ to $2T$ while the gas remains in a rigid container. Indicate how the average molecular speed changes by modifying the arrow lengths to represent the new average molecular speed.

ii.

During the slow heating of the gas in Figure 1, the piston rises and the gas expands at constant pressure. Derive an expression for the number of moles $n$ of the gas in terms of $V_1$ , $T_1$ , $P_{\text{atm}}$ , $m_p$ , $A_p$ , and the ideal gas constant $R$ . Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 4. Cylinder-and-piston system heated by a hot reservoir through a conducting plate (thickness L, area A).

Black-and-white apparatus diagram showing heat flow from a hot thermal reservoir into the gas through a flat conducting plate.

Left-to-right arrangement (single horizontal line of components):
1) Hot reservoir (left):
- Draw a large vertical rectangle on the left third of the figure.
- Inside it, centered, write exactly: "Hot reservoir".
- Near the top inside the reservoir, write exactly: "T_h".

2) Conducting plate (middle interface):
- Immediately to the right of the hot reservoir, draw a thin vertical slab (a narrow rectangle) representing the conducting plate, fully contacting the reservoir on its left face.
- Label the slab centered as: "conducting plate".
- Mark the plate thickness with a double-headed horizontal arrow spanning from the left face of the slab to the right face of the slab, and place the label exactly above that arrow: "L".
- Indicate the plate face area by placing the label exactly: "Area = A" directly below the slab.

3) Cylinder with gas and piston (right):
- Immediately to the right of the conducting plate, draw the vertical cylinder (2D cross-section) with a piston inside, similar style to Figure 1, occupying the right third of the figure.
- The left wall (or base contact surface) of the cylinder is drawn flush against the right face of the conducting plate to show thermal contact through the plate.
- Inside the cylinder below the piston, label the gas region exactly: "gas".
- On the piston, label exactly: "piston".

Heat-flow arrow:
- Draw a bold arrow pointing horizontally from left to right, starting inside the hot reservoir, passing across the plate, and pointing into the gas region.
- Above this arrow, write exactly: "Q".

Thermal-contact note:
- Centered beneath the plate-cylinder interface, write exactly: "thermal contact through plate".

No extraneous numbers:
- Do not add numerical values for L, A, or T_h (only symbols as shown). No axes.

Indicate whether the entropy of the gas is increasing, decreasing, or remaining constant at that instant. The cylinder is now placed in thermal contact with a large reservoir at constant temperature $T_H = 600\ \text{K}$ by pressing the bottom of the cylinder against a flat aluminum plate, as shown in Figure 4. The plate has thickness $L = 5.00× 10^{-3}\ \text{m}$ , thermal conductivity $k = 205\ \text{W/(m·K)}$ , and contact area with the cylinder of $A = 2.00× 10^{-3}\ \text{m}^2$ . While the gas warms, the piston continues to move so that the gas pressure remains constant. Consider the instant when the gas temperature is $T = 400\ \text{K}$ .

Assume heat transfer occurs only by conduction through the plate and that the temperature of the plate surface in contact with the gas is equal to the gas temperature at that instant.

Increasing
Decreasing
Remaining constant

Justify your answer.