AP Physics 2 FRQ Practice

🧲AP Physics 2

Practice FRQ 1 of 121/12

1. A rigid, sealed, thermally insulated cylinder contains

n = 0.50\ \text{mol}

of an ideal monatomic gas. The gas is initially in thermal equilibrium at temperature

T_1 = 300\ \text{K}

and pressure

P_1 = 1.20\times10^5\ \text{Pa}

. A frictionless piston is then attached to the cylinder so that the gas can expand while remaining sealed. The piston is initially locked. The cross-sectional area of the piston is

A = 2.0\times10^{-3}\ \text{m}^2

. The cylinder is placed in thermal contact with a large thermal reservoir at temperature

T_R = 450\ \text{K}

by inserting a solid copper rod between the cylinder wall and the reservoir, as shown in Figure 1. The copper rod has length

L = 0.25\ \text{m}

, cross-sectional area

A_{rod} = 1.5\times10^{-4}\ \text{m}^2

, and thermal conductivity

k = 400\ \text{W/(m· K)}

. Heat transfer occurs only through the rod.

Figure 1. Thermally insulated gas cylinder with locked frictionless piston and heat transfer only through a copper rod to a 450 K reservoir.

A clean, black-and-white physics apparatus diagram (no perspective; straight-on side view).

Overall layout (left to right):
- On the far left is a large rectangular block labeled exactly "Thermal reservoir" with the text "T_R = 450 K" centered inside the block.
- Immediately to the right of the reservoir block is a straight, horizontal copper rod that extends rightward and touches the left end wall of the gas cylinder.
- To the right of the rod is a rigid cylindrical chamber (drawn as a long horizontal rectangle with thicker end walls), labeled "Rigid, sealed, thermally insulated cylinder" above the chamber. The cylinder’s outer boundary is wrapped with a dashed outline or a surrounding band labeled "Thermal insulation" to indicate insulation everywhere except where the rod contacts.
- On the far right end of the cylinder is a piston drawn as a vertical thick plate fitting the cylinder cross-section. The piston is attached to the cylinder end and can slide horizontally. The piston is explicitly labeled "Frictionless piston".

Gas region:
- The interior of the cylinder (between left wall and piston face) is lightly shaded or filled with small dots to indicate gas and labeled in the center with three separate lines of text:
1) "n = 0.50 mol ideal monatomic gas"
2) "Initial state (locked): T_1 = 300 K"
3) "P_1 = 1.20 × 10^5 Pa"

Locked piston indicator:
- The piston is shown with a locking mechanism: a small pin or latch symbol drawn directly above the piston rod/edge, with an arrowed label "Locked" pointing to the latch. The latch must visually prevent motion (e.g., a bar through a slot).

Heat-transfer-only path:
- The copper rod is the only connection between the cylinder and the reservoir. The cylinder walls elsewhere must show no contact with the reservoir (a visible gap), reinforcing “heat transfer occurs only through the rod.”
- At the rod, place the label "Copper rod" above it.

Exact dimension and parameter labels (must be visible text):
1) Rod length: A double-headed dimension arrow drawn parallel to the rod, spanning from the rod’s left end (at the reservoir face) to the rod’s right end (at the cylinder wall contact). Center the text above that arrow: "L = 0.25 m".
2) Rod cross-sectional area: A label near the rod reading "A_rod = 1.5 × 10^-4 m^2". To avoid ambiguity, include a short leader line pointing to the rod.
3) Rod thermal conductivity: A label near the rod reading "k = 400 W/(m·K)" with a leader line to the rod.
4) Piston cross-sectional area: Next to the piston face (the vertical plate), place the text "A = 2.0 × 10^-3 m^2" with a leader line pointing to the piston face.

Contact points and spatial relationships (explicit):
- The rod must touch the reservoir on its left end and touch the cylinder’s left outer wall on its right end (drawn as direct end-to-end contact, no gaps).
- The piston must be drawn flush with the cylinder’s right opening, with the gas region entirely on the left side of the piston face.
- No other rods, wires, or openings are shown; the cylinder is explicitly sealed.

No axes, no graph grid, no extraneous text beyond the labels listed above.

Figure 2. Microscopic snapshot of gas particles at T1 = 300 K for drawing velocity vectors with different directions and speeds.

A rectangular “microscopic view” box representing a small region inside the gas. The box has a thin black border.

Title/labeling:
- Centered above the box, place the text: "Gas at T_1 = 300 K".

Particles (exact count and placement pattern):
- Inside the box, draw exactly 8 identical small filled circles (gas particles) spread across the box so none overlap and none touch the border.
- Arrange them in two rows of four particles:
- Top row: four particles evenly spaced left to right across the upper half of the box.
- Bottom row: four particles evenly spaced left to right across the lower half of the box.
- The two rows are vertically separated so the top row is clearly above the bottom row.

Velocity vectors to be completed (blank arrows with fixed starting points):
- From the center of each particle, draw a thin arrow shaft with an arrowhead but leave the direction and/or length open for student completion by rendering the arrows as faint gray outlines.
- Each arrow must clearly start at the particle’s center.

Required arrow-length categories (so the student can indicate “faster” unambiguously):
- Provide two clearly distinct arrow-outline lengths:
- Exactly 4 particles have “long-arrow outlines” (faster category): the outline length is visually twice the short-arrow outline length.
- Exactly 4 particles have “short-arrow outlines” (slower category).
- Enforce the pattern:
- Top row: long, short, long, short (left to right).
- Bottom row: short, long, short, long (left to right).

Direction variety requirement (space for many directions):
- The faint arrow outlines must be drawn so they do not already imply a single preferred direction; they should be shown as blank/neutral guides that could be rotated. (For instance, show only a short faint line segment without a fixed arrowhead direction, or show a faint arrow with a removable arrowhead.)

Legend for speed (must be visible):
- Below the box, include a small legend with two sample arrows:
- A short arrow labeled "slower"
- A long arrow labeled "faster"

No numbers besides "T_1 = 300 K". No axes. No additional labels.

Figure 3. Single-particle collision with a wall segment showing the momentum-change direction normal to the wall.

A simplified collision diagram with one wall segment and one gas particle, drawn in black-and-white.

Wall segment:
- Draw a thick straight vertical wall segment on the right side of the figure, spanning most of the height.
- Label the wall segment with text placed just to the right of it: "Cylinder wall".

Wall normal direction (must be explicit):
- At the midpoint of the wall segment, draw a short dashed line perpendicular to the wall pointing leftward into the gas region.
- Label this dashed line "normal".

Particle and motion:
- Draw one small filled circle (particle) to the left of the wall, positioned so there is clear empty space between the particle and the wall.
- Draw an incoming velocity arrow (solid) from the particle pointing diagonally toward the wall, so that it clearly intersects the wall at a point slightly above the wall’s midpoint. Label this arrow "v_in".
- Draw an outgoing velocity arrow (solid) starting at the collision point on the wall and pointing diagonally away from the wall back into the gas region (leftward), symmetric to the incoming arrow with respect to the wall normal (mirror reflection). Label this arrow "v_out".

Momentum-change vector:
- At the collision point on the wall, draw a bold arrow labeled exactly "Δp".
- The Δp arrow must point purely leftward, exactly along the wall-normal dashed line direction (perpendicular to the wall), not angled.

Pressure connection (brief text in-figure):
- Below the collision drawing, include a single sentence in small text: "Many collisions per unit time on the wall produce the gas pressure."

No other particles, no container outline besides the one wall segment, and no axes.

i. Complete the following tasks in Figures 2 and 3.

• Indicate in Figure 2 the directions of typical velocity vectors for several gas particles at

T_1 = 300\ \text{K}

, and indicate which particles would be moving faster by drawing longer arrows.

• Indicate in Figure 3 the direction of the momentum change

\Delta \vec{p}

of a particle during a collision with the wall segment shown, and explain briefly how many such collisions relate to the gas pressure.

ii. While the piston remains locked, the gas is warmed by the reservoir until it reaches

T_2 = 450\ \text{K}

. Assume the volume remains constant during this heating.

Derive an expression for the final pressure

P_2

in terms of

P_1

T_1

, and

T_2

. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Then calculate the numerical value of

P_2

Figure 4. P–V diagram for the initial locked heating (constant volume) and the subsequent piston motion to a final resting state.

A blank-but-scaled pressure–volume graph with clearly numbered axes (no grid).

Axes:
- Horizontal axis labeled "V (m^3)" with an arrow at the positive (right) end.
- Vertical axis labeled "P (Pa)" with an arrow at the positive (up) end.

Exact axis ranges and tick labels (must be printed as numbers):
- V-axis: start at 0 at the origin. End at 0.012 m^3 at the far right. Tick marks and labels at: 0, 0.003, 0.006, 0.009, 0.012.
- P-axis: start at 0 at the origin. End at 2.4 × 10^5 Pa at the top. Tick marks and labels at: 0, 0.6 × 10^5, 1.2 × 10^5, 1.8 × 10^5, 2.4 × 10^5.

State points and labels:
- Place a solid filled point labeled "1" at the tick intersection corresponding to P_1 = 1.20 × 10^5 Pa and V_1 = 0.003 m^3.
- Place a second solid filled point labeled "2" directly above point 1 (same volume), at the tick intersection corresponding to P_2 = 1.80 × 10^5 Pa and V = 0.003 m^3.
- Next to point 2, add the label "T_2 = 450 K (locked, V constant)".

Process indication (student-identification region):
- Draw a thin vertical line segment connecting point 1 to point 2 to indicate the constant-volume heating while locked.
- From point 2, draw a dashed placeholder curve extending rightward to a third, unlabeled open circle located exactly on the horizontal level P = 1.20 × 10^5 Pa at volume V = 0.0045 m^3.
- Label the open circle "f" (final) and place the text "piston unlocked → comes to rest" near the dashed curve.

Style requirements:
- Solid line for the known constant-volume segment (1→2).
- Dashed line for the unknown/qualitative path (2→f) that students classify as isothermal/isobaric/adiabatic.
- No additional curves, no shaded regions, no grid.

After the gas reaches

T_2 = 450\ \text{K}

, the piston is unlocked. The outside of the piston is exposed to a constant atmospheric pressure

P_{atm} = 1.00\times10^5\ \text{Pa}

. The piston has mass

m_p = 6.0\ \text{kg}

and remains frictionless. The gas stays in thermal contact with the reservoir through the rod, so during the piston motion the gas temperature is maintained at

T = 450\ \text{K}

. Use

g = 9.8\ \text{m/s}^2

. Given values:

$n = 0.50\ \text{mol}$
$T = 450\ \text{K}$
$A = 2.0\times10^{-3}\ \text{m}^2$
$P_{atm} = 1.00\times10^5\ \text{Pa}$
$m_p = 6.0\ \text{kg}$
$g = 9.8\ \text{m/s}^2$
$R = 8.31\ \text{J/(mol· K)}$

B. Indicate whether the process from the moment the piston is unlocked until the piston comes to rest is best modeled as isothermal, isobaric, or adiabatic (see Figure 4).
______ Isothermal
______ Isobaric
______ Adiabatic

Justify your answer.

Then calculate the final gas pressure

P_f

and the final gas volume

V_f

after the piston comes to rest.