Teaching AP Physics 2

Hands-on inquiry A: Does a sealed gas follow an inverse pressure–volume relationship?

Question: For a fixed amount of room-temperature gas, is absolute pressure proportional to 1/V? Equipment: plastic syringe with secure nonpointed connector, pressure sensor rated for the intended range, room thermometer, and data logger or manual recording table. A teacher should check sensor and connector limits. Do not use a needle, compressed-gas cylinder, or heating apparatus.

Measured variables: syringe gas volume V and absolute pressure P; record room temperature T. Controls: sealed gas amount, slow compression, wait for pressure to stabilize after each movement, constant room conditions, same tubing and sensor. Note sensor/tubing dead volume and gauge-versus-absolute reporting. Include dead volume in total V when known and add ambient pressure to gauge readings.

1. Trap approximately 40 mL of air, attach the sensor, check for leaks by monitoring a fixed position, and record an initial equilibrium reading.
2. Move the plunger slowly through at least six teacher-approved volumes, for example 40, 36, 32, 28, 24, and 20 mL when within rated pressure. Wait after each move and record stabilized P. Never force a blocked or overrange device.
3. Repeat on expansion and then repeat selected points. Compare compression and expansion data for heating, friction, or leaks. Stop if a connector loosens or a reading exceeds the approved limit.
4. Convert total volumes to m³. Plot P against V and against 1/V with labelled units and sensible scales; fit the linearized graph. The expected slope is nRT and intercept is near zero in the ideal corrected model. Calculate n from slope/(RT) if calibration permits.

Uncertainty and discussion: use syringe graduation uncertainty and pressure calibration uncertainty; include repeats and uncertainty bars when practical. Look for systematic curvature from dead volume, pressure drift from leaks, and differences between immediately compressed and stabilized readings from temperature changes. An approximately linear P-versus-1/V graph supports the ideal-gas model over the measured range; it does not prove ideal behavior at arbitrary density or temperature. Ask students to connect the rise in P to more frequent normal momentum transfers per area.

Hands-on inquiry B: Can calorimetry identify a metal’s specific heat?

Question: What c best explains the equilibrium temperature of a warm metal and cooler water? Equipment: balance, two temperature probes or thermometer, lidded insulated foam cup, water, small teacher-approved metal samples, warm-water bath at approximately 60–80 °C, tongs and towels. Use eye protection and teacher-approved heat-resistant handling; warm water can scald, so avoid boiling, open flames, sealed heating, and bare-hand contact with the sample.

Measured variables: dry metal mass, water mass, initial metal/bath temperature, initial water temperature, and common final temperature. Controls: same insulated cup and lid, similar transfer time, measured water amount, stirring method, temperature-probe placement away from cup walls, and no phase change. Estimate or separately determine cup/probe heat capacity if equipment allows; otherwise explicitly state the omission.

1. Weigh a dry metal sample and the water. Equilibrate the metal in the warm bath and verify the bath temperature is stable; record the separate water temperature.
2. Transfer the metal quickly with tongs, minimizing carried bath water, into the lidded cup. Gently stir and record temperature versus time until it reaches a uniform maximum and then begins any slow room-cooling drift.
3. Use a common final temperature and write the complete energy balance. With a negligible cup, m_m c_m(T_f−T_m) + m_w c_w(T_f−T_w) = 0. If the cup/probe capacity C_c is known, add C_c(T_f−T_w).
4. Repeat at least three trials with fresh measured water. Calculate each c, plot temperature versus time to diagnose stabilization and leakage, and report mean, scatter, and principal systematic limitations.

Analysis: solve c_m = m_w c_w(T_f−T_w)/[m_m(T_m−T_f)] for the simplified model. Recalculate at plausible temperature endpoints to estimate sensitivity, as in problem 14. A very small water temperature rise makes c sensitive to final-temperature uncertainty. Cup absorption, room losses, and cooling before immersion usually make the uncorrected c too low; carried hot bath water can instead add unaccounted energy. Compare the magnitude with teacher-provided material references without requiring memorized values.

Expected physics: net thermal energy moves from warm metal to cool water until their temperatures match. For the approximately insulated combination, energy is conserved while entropy increases toward equilibrium. Microscopic exchange continues after the net transfer stops. The experiment is a real classroom investigation; the teacher remains responsible for apparatus setup, calibration, supervision, and judging whether the data support the approximations.

Hands-on inquiry: can a voltage map predict force direction? Use a low-voltage DC supply (about 3–6 V), conductive paper, two metal electrodes, insulated leads, a high-input-resistance voltmeter, graph paper, and a ruler. The question is how electrode shape changes equipotential spacing and the inferred electric field. This is a steady conducting-sheet analogue of an electrostatic potential map; the sheet carries current, so it is not a claim that the paper itself is in electrostatic equilibrium.

Fix two parallel bar electrodes and mark their locations. Connect the supply, select one electrode as 0 V, and map measured positions of 1 V, 2 V, 3 V, and so on. Measure each point twice; keep electrode spacing, voltage, paper, probe contact pressure, and meter range controlled. Connect points of equal voltage smoothly. Repeat with a small circular electrode and a bar while holding the voltage scale fixed.

Draw field arrows perpendicular to the equipotentials toward lower voltage. In a central region estimate the normal field magnitude with $E_{\mathrm{avg}}=|\Delta V|/\Delta s$. Plot V against position along a line normal to the bars: a nearly straight line predicts a nearly uniform field; a varying slope predicts a varying field. Compare arrow length estimates with contour spacing. Ask which way a proton and an electron initially accelerate, and whether their paths must follow field lines if they enter with sideways velocity.

Report probe-position uncertainty, voltage resolution, scatter between repeats, and departures near edges. For an estimate E = ΔV/Δs, add relative uncertainties conservatively: δE/E ≈ δ(ΔV)/|ΔV| + δ(Δs)/Δs. Use contour separations large compared with location uncertainty. Expect roughly equally spaced contours between broad central parallel bars and crowded contours near a small electrode. A nonzero fitted intercept may reflect the chosen reference; it does not invalidate the field estimate.

Second station: capacitance versus separation. With the supply disconnected, use a capacitance meter and two broad foil plates separated by measured, uniform insulating spacers. Keep overlap area and spacer material fixed, measure C at several d values, and plot C against 1/d. A slope estimates κε₀A; account for meter/lead capacitance as an offset. Repeat with another dielectric of equal thickness, or vary overlap area at fixed d. Inspect uneven gaps and fringing as systematic errors. Do not infer a dielectric constant from a change in both thickness and material.

Keep voltages low, disconnect before rearranging electrodes, and discharge capacitors through a resistor before handling them. Do not use mains electricity or charged high-voltage capacitors. The teacher verifies equipment and safe procedures.

Hands-on inquiry: how do measurements reveal a resistor’s model? Use a low-voltage adjustable DC supply (0–6 V), a current-limiting resistor, a tested fixed resistor, an incandescent low-voltage bulb, a switch, insulated wires, an ammeter, and a high-resistance voltmeter. Never connect an ammeter directly across a battery. With power off, build a series supply–ammeter–test-element loop and connect the voltmeter across only the element. Ask the teacher to check the meter range and wiring before energizing.

Record several paired current and voltage readings for the fixed resistor, including zero. Change supply voltage gradually; keep the element’s temperature approximately controlled by using low currents, short measurement intervals, and cooling time. Repeat each point and reverse the voltage sweep to look for heating or hysteresis. Plot I vertically against ΔV horizontally. A straight line through zero has slope 1/R; a V-versus-I plot instead has slope R. Compare a best-fit line with plausible steepest/shallowest lines consistent with the measurement uncertainties. Report slope uncertainty and infer resistance; do not force the intercept to zero without checking a zero-reading offset.

Replace the resistor with the bulb while preserving meter arrangement and safe current limits. Curvature and different slopes on the I–V graph can indicate changing resistance as the filament heats. Describe electrical-to-thermal energy conversion and compare brightness with measured P = IΔV. Do not equate visible brightness quantitatively across different bulb designs.

Hands-on RC station: Use a low-voltage supply, a rated capacitor, a known resistor, a charge/discharge switch, and a voltage sensor or high-resistance meter with suitable time resolution. Choose RC of several seconds. Discharge through the resistor first; observe charging from zero and discharge from a known voltage. Record VC against time, using the same polarity convention in both trials. Identify the observed time to about 63% of final charging voltage and to about 37% of initial discharging voltage; compare these landmarks with RC. Repeat with twice the resistance while keeping capacitance and initial/final voltages fixed. Expect slower curves but unchanged endpoint voltage and charge.

Controls include capacitance, source voltage, initial capacitor state, and meter placement. Include resistor and capacitor tolerances, meter resolution, sensor sampling interval, switch timing, and meter loading as uncertainties. Use a measured time range around each landmark rather than inventing precision between samples. Compare doubling-R trials by normalized curve shape or landmark times; no exponential fit or logarithmic time solution is required. At long times check charging branch current approaches zero and discharging voltage approaches zero. Stored energy becomes resistor heating during discharge.

Use only low-voltage classroom equipment, observe capacitor polarity and voltage ratings, switch off before rewiring, and discharge capacitors through a resistor before handling. The teacher verifies safe equipment and procedures; do not use mains circuits.

Hands-on inquiry: isolate the field of a current

Question: Does the magnetic field around a long straight wire scale as I/r? Use a current-limited low-voltage supply, ammeter, long insulated wire, magnetic-field probe (or calibrated magnetometer), meterstick, clamps, and graphing tools. A compass-only version can map direction but cannot establish the quantitative magnitude law.

Measured variables and controls: Measure conventional current I, distance r from the wire axis, and the probe’s tangential field component. Control probe orientation, wire geometry, measurement height, nearby magnetic materials, and the return-current path. Keep the return lead well away and probe near the middle of a straight section long compared with r.

1. With the supply off, map Earth/background B. Choose a probe orientation tangent to the expected wire field and a repeatable zero position.
2. At a fixed r collect at least five current settings. For each magnitude reverse current and compute [B(+I)−B(−I)]/2. Repeat readings three times and record spread.
3. At one safe fixed current repeat at five distances, measuring r from the wire’s center. Include zero-current readings before and after to detect drift.
4. Graph isolated B versus I and B versus 1/r with units and uncertainty bars. Compare fitted slopes with μ₀/(2πr) and μ₀I/(2π), respectively.
5. Reverse current and check the compass or vector direction. Explain why an off-axis probe or nearby return lead would change the result.

Analysis and uncertainty: Record ammeter resolution, probe resolution and repeated spread, distance uncertainty, and angular misalignment. Relative distance error matters most near the wire. Use slope ranges or fit uncertainty; do not force a line through zero until the background subtraction is justified. A nonzero residual intercept or failure at large r can indicate remaining background or finite-wire/return-path effects.

Expected physics: Tangential direction reverses with current; B is proportional to I at fixed r and to 1/r at fixed I. Use q10’s data as an analysis rehearsal, then analyze the class’s own measurements.

Safety and implementation: A teacher selects current limits and wire gauge; start low, prevent shorts, and switch off between trials to limit heating. Use only low-voltage sources. Keep magnets and current-carrying apparatus away from sensitive electronics and follow school guidance for medical devices. This procedure is a feasible classroom plan; the teacher manages apparatus, supervision, and actual implementation.

Hands-on inquiry: estimate a lens focal length from projected images

Question: Does a thin converging lens give image distances consistent with one focal length? Use a low-intensity illuminated arrow or LED object, thin converging lens and holder, white screen, optical bench or metersticks, and ruler. Choose object distances safely beyond the approximate focal length so a real image can be projected.

Variables and controls: Measure d_o and d_i from the lens midline and object/image heights. Change d_o; measure d_i by moving the screen to best focus. Control lens, source color, optical-axis height, object orientation, and distance reference. Keep the screen perpendicular to the axis.

1. Align object, lens center, and screen. Obtain one sharp inverted image. Record the midpoint of the screen positions judged equally sharp, plus the full acceptable-focus range.
2. Choose at least five object distances greater than f, including settings near 2f and farther away. At each setting bracket best focus three times independently and record all values.
3. Plot 1/d_i vertically against 1/d_o horizontally, with reciprocal units. Fit a line with slope and intercept; the model predicts slope −1 and intercept 1/f.
4. Compare f estimates from several pairs and the fitted intercept. Compare measured height ratio with |d_i/d_o| and verify inverted orientation.
5. Place the object inside f. Predict d_i and m, then look through the lens from the transmitted side and describe the virtual image. Confirm that no finite screen position catches that virtual image.

Uncertainty: Record ruler resolution, the range of acceptable focus, alignment differences, and uncertainty in locating the lens midline. Reciprocal-distance errors are largest for small distances; estimate bounds using the reciprocals of d ± uncertainty or use a fit that accounts for them. Do not force a theoretical slope before seeing whether data support it. A constant measurement offset can bias all reciprocal sums.

Expected physics: Real image data should approximate one focal length and slope −1; moving the object toward f from outside sends its real image farther away. Inside f the image is virtual, upright, and enlarged. Failure to project that case supports the ray explanation rather than the claim that the lens has stopped working.

Safety and implementation: Use low-intensity lamps or a classroom ray box. Never focus sunlight into an eye or onto combustible material. Do not point a laser into anyone’s eyes; if a teacher chooses laser equipment, use school-approved beam controls and stops. Secure lens holders and handle glass carefully. The teacher selects apparatus, supervises, and implements the procedure.

Optional second investigation: Use q09’s ray-box semicircular-block setup to plot sin θ_air against sin θ_block, estimate refractive index from the slope, and compare the measured internal critical angle with sin⁻¹(1/n). A model exploration is useful preparation but does not substitute for these measured classroom data.

Inquiry: Do fringe locations support a path-difference model? Use a teacher-approved low-power visible laser, commercial single- and double-slit slides, grating, screen, meterstick, and caliper/slide specifications. Measure slit-to-screen distance L and several dark minima or bright maxima on both sides. Keep wavelength, screen orientation, and the same definition of center fixed. Change only slit width a or separation d for each comparison. Predict which change broadens the pattern before measuring.

1. Secure the laser and slides below eye level; establish a screen that terminates the beam. Never look into the laser or a specular reflection. The teacher manages alignment.
2. Measure L, locate center from symmetric pairs, and record y values with position uncertainty. Use multiple orders rather than only one fringe.
3. For double slits plot bright-fringe y against order m; slope is approximately λL/d when angles are below 10°. For a single slit plot dark-minimum y against nonzero order m; slope is λL/a. Use y/L to check the angular approximation.
4. Repeat measurements and compare slopes for different slides. State which uncertainties come from center choice, finite band width, L, and manufacturing tolerance. Compare the observed slope interval with the prediction.

The expected evidence is a line through the center for small-angle positions, wider spacing for larger λ or L, and narrower spacing for larger a or d. A double-slit photograph should show fine fringes inside a broad diffraction envelope. Do not fit high-angle data with the small-angle formula; use $\sin\theta=y/\sqrt{L^2+y^2}$ instead. Students write a claim, cite two measured comparisons, and explain the path cancellation or addition. The waves model is a supplemental prediction tool; the lab uses actual measured light.

Inquiry: Can measured light spectra distinguish a line source from a thermal source? Use a classroom spectroscope or diffraction grating, a teacher-managed sealed low-current gas discharge lamp, and an enclosed incandescent lamp at two safe settings. Students observe from a fixed distance and through the same aperture. Never open the discharge tube or its supply; avoid hot bulb contact and shield intense glare. No radioactive source is needed.

1. Record which wavelengths are present using a calibrated scale or the grating relation with measured angle. Record angular/scale uncertainty and repeat line locations with another observer.
2. Compare the gas lamp’s separated lines with the incandescent lamp’s broad continuous band. Change only the incandescent setting while keeping geometry and detector/exposure settings fixed. Predict which spectral distribution will favor shorter wavelengths when hotter.
3. Plot detected intensity or a carefully labeled relative-brightness proxy versus wavelength. If the instrument is not spectrally calibrated, do not call the measured peak the true Wien peak or infer an absolute temperature from perceived color alone.
4. Use a supplied single-electron energy-level table to predict two transition wavelengths and test whether candidate line positions agree within uncertainty. Explain why one band cannot identify a whole source uniquely.

The expected physics is discrete bound-state photon energies in the discharge lamp versus a temperature-dependent continuum from the filament. Discuss the spectroscope’s resolution, spectral response, finite line width, calibration bias, and background illumination. A measured missing line may reflect detector sensitivity as well as atomic populations. As a complementary probability activity, shake a large set of two-sided tokens and remove one chosen face each round. Graph survivors and repeat the full trial; compare the average with N₀(1/2)ᵏ and show that the last individual token is unpredictable. Label this a statistical analog, rather than a measurement of nuclear decay or its actual timescale.