AP Physics C Virtual Labs & Experimental Investigations
Hands-on apparatus · student measurements · evidence-based analysis. All 18 Mechanics and Electricity & Magnetism experiments now use one visible physical apparatus per lab, with simple controls modeled on the AP Physics 1 lab experience. The hidden numerical engine drives the apparatus, recorded data, fits, uncertainty, results, discussion, conclusion, and feedback without exposing a second simulation panel.
A useful Physics C lab starts with a testable question and a physics model, identifies independent/dependent variables and controls, records repeatable measurements, chooses a graph or transformation from the model, treats uncertainty honestly, and finishes with a claim supported by evidence. Calculus matters when the model depends on slopes, areas, rates, integrals, or differential-equation behavior.
Course requirement
AP Physics C courses require a minimum of 25% of instructional time in supervised hands-on laboratory investigations representative of course topics. Students should maintain a lab notebook or portfolio of their work.
Simulation limitation
Virtual labs and simulations can support the laboratory requirement when they are part of supervised course work with genuine student data collection and analysis. Using this independent page by itself does not certify that a school course has met the 25% requirement.
Each Physics C investigation now presents one believable physical setup and only the controls needed to operate it: choose the study or object when required, drag the apparatus, arm the instrument, run the experiment, and record the reading. The numerical engine remains hidden so students never see a second competing simulation.
Simple direct control
The visible interface is reduced to the physical apparatus, a compact study/object selector when needed, instrument controls, and the recorded-data workflow.
One visual experiment
Legacy simulation stages and precision-control panels are hidden from the student view; they remain internal only to preserve the verified Physics C calculations.
Evidence-first graphing
Plot recorded scatter first, add the best-fit/model overlay second, then open the quantitative analysis.
Separate automatic mode
Automatic datasets remain a clearly separate analysis-practice option and are not presented as student-run physical evidence.
Standard Physics C investigation workflow
Research question
State a measurable relationship or model test. Avoid a vague goal such as “study motion.”
Experiment plan
Identify apparatus, independent/dependent variables, controls, repeated trials, and a procedure another student could reproduce.
Measure
Record raw observations with units. Preserve sensor resolution and do not replace measured values with theoretical values.
Analyze
Select the graph, fit, transformation, derivative, integral, or residual test demanded by the physical model.
Uncertainty
Estimate measurement limits, scatter, slope/intercept uncertainty, and whether discrepancies are significant relative to uncertainty.
Results
Report fitted parameters or derived quantities with units and enough precision to match the measurements.
Discussion
Compare data with the model, explain deviations physically, and identify limitations that could bias the result.
Conclusion
Answer the research question directly and support the claim with quantitative evidence from the experiment.
Choose one of 18 investigation pathways
All 18 pathways are fully interactive investigations. Use the course filters to move between the 10 Mechanics and 8 Electricity & Magnetism labs.
18 pathways shown
How to choose the graph from the physics model
| Model pattern | Useful analysis | What slope / area means | Typical mistake |
|---|---|---|---|
| Direct proportionality y=kx | Plot y vs x and inspect intercept/residuals | Slope estimates k | Forcing the fit through the origin without evidence |
| Inverse dependence y=k/x | Plot y vs 1/x | Slope estimates k | Fitting y vs x with a straight line |
| Power law y=kxⁿ | Transform using the predicted power or use log-log analysis | Transformed slope/intercept test the model | Choosing a transformation only because it looks linear |
| Rate relation y=dq/dt | Use local slope / derivative or integrate the rate | Slope is rate; area under rate-time graph is accumulated quantity | Using noisy numerical derivatives without discussing uncertainty |
| Exponential transient | Nonlinear fit or justified logarithmic linearization | Fit parameter gives the time constant | Taking logs of signed/zero values without a valid model domain |
| Conservation law | Compare before/after totals or residual from zero | Residual size tests agreement within uncertainty | Declaring “conserved” because values are merely close |
Uncertainty: the minimum useful treatment
Instrument resolution
Record the smallest meaningful sensor/scale increment and distinguish resolution from trial-to-trial scatter.
Repeated measurements
Use repeat trials to estimate variability. A single measurement cannot reveal random scatter by itself.
Fit uncertainty
When a model uses a slope/intercept, report the fitted quantity and evaluate whether the intercept is physically consistent with the model within uncertainty.
Percent difference is not uncertainty
A percent difference from theory is a comparison metric. It does not replace an estimate of measurement uncertainty or scatter.
Systematic effects
Friction, alignment, sensor zero, finite electrode size, internal resistance, drag, and timing offsets can shift every trial in the same direction.
Evidence threshold
Decide whether a disagreement is larger than the experimental uncertainty before calling the model unsupported.
Full lab + guided investigation pathways
Motion Sensor: Position, Velocity, and Acceleration
Test whether measured motion obeys the calculus links v = dx/dt and a = dv/dt, and whether graph areas recover the corresponding changes in position and velocity. Use a real motion-sensor procedure plus a student-controlled virtual cart, manual and automatic data collection, three synchronized motion graphs, regression, numerical integration, uncertainty, results, discussion, and conclusion.
Physical apparatus and procedure
- Level a dynamics track and place an ultrasonic or optical motion sensor at one end. Aim the sensor so it sees the cart throughout the run without reflections from nearby objects.
- Choose a motion model. For nearly constant velocity, give the cart a gentle push on a low-friction level track. For approximately constant acceleration, use a slight incline or a cart-pulley arrangement with a small net driving force.
- Start data acquisition before releasing the cart. Collect position as a function of time at a fixed sampling rate; if the software also reports velocity and acceleration, retain those channels for comparison.
- Select a clean interval after release and before the cart reaches an end stop. Record at least five well-spaced points or use the full sensor trace.
- Graph x versus t, v versus t, and a versus t. Test derivative consistency, then compare the trapezoid area under v(t) with measured Δx and the area under a(t) with measured Δv.
- Repeat the run or analyze a second clean interval to estimate repeatability. Do not replace noisy measured values with theoretical values.
Keep hands clear of moving carts and pulley masses, use an end stop, and prevent the cart from striking the sensor. Motion sensors can give bad data close to their minimum range or when the target surface is poorly aligned.
Prediction and validity conditions
For constant acceleration, x(t) should be quadratic, v(t) approximately linear, and a(t) approximately constant. The fitted coefficient of t² in x(t) should be about a/2, while the slope of v(t) should be about a.
- Use one coordinate direction and keep its sign convention fixed.
- Analyze one uninterrupted motion interval.
- The cart should remain within the useful sensor range.
- Acceleration should be approximately constant for the quadratic/linear model test.
- Numerical derivatives amplify measurement noise; numerical integrals are usually less noise-sensitive.
Evidence to report
Report the quadratic x(t) fit, linear v(t) fit, acceleration estimate, R² values, Δx compared with ∫vdt, Δv compared with ∫adt, and the dominant random/systematic limitations.
Student controls
| Sample | t (s) | x (m) | v (m/s) | a (m/s²) | x model (m) |
|---|---|---|---|---|---|
| No accepted sensor samples yet. | |||||
Use a quadratic fit for x(t), a linear fit for v(t), and a mean/trend check for a(t). Trapezoid integration tests whether graph areas recover Δx and Δv.
Record at least four distinct times to draw and fit the motion graphs; five or more are recommended.
No analysis yet.
After analysis, compare derivative-based acceleration estimates, integral areas, residuals, and the likely effect of sensor noise/alignment.
State whether the measured data support the derivative and integral links over the tested interval, with quantitative evidence.
Uncertainty and model limitations
- Position resolution: estimate from sensor specifications or repeated stationary readings.
- Timing: account for sample interval and any trigger/release delay.
- Derivative noise: acceleration estimated from velocity changes is especially sensitive to noisy measurements.
- Alignment/systematics: a tilted track, sensor angle, wheel friction, pulley inertia, or drag can make acceleration differ from the intended constant value.
- Endpoint choice: include only the interval where the model assumptions are defensible.
Lab-report prompts
- Which graph gives the most direct evidence for constant acceleration? Explain.
- Does 2×(quadratic t² coefficient) agree with the slope of v(t) within the scatter?
- How closely does ∫vdt agree with measured displacement? Which is more sensitive to noisy data: the derivative or integral test?
- Identify one systematic effect and predict the direction it would shift a fitted parameter.
- Write a one-sentence evidence-based conclusion that includes at least one numerical comparison.
Projectile Motion from Video Data
Test whether horizontal and vertical projectile motion can be modeled independently: constant horizontal velocity with approximately constant downward acceleration. Use a real video-analysis procedure plus a student-controlled virtual launch, manual frame tracking, a separate automatic nine-frame dataset, component fits, a trajectory graph, uncertainty, results, discussion, and conclusion.
Physical apparatus and video procedure
- Use a ball or other compact projectile, a safe launcher/ramp, a meter stick or calibration grid, and a camera fixed so its optical axis is approximately perpendicular to the plane of motion.
- Place the calibration scale in the same plane as the projectile. Define a coordinate origin and record the launch point height. Keep the camera stationary for the entire run.
- Record the launch at a known frame rate. Use a conservative trajectory with a clear landing region and no people in the projectile path.
- Step through the video frame by frame. For each accepted frame, record time and the projectile center coordinates. Use at least six frames spanning most of the flight; more frames improve the fit.
- Fit x versus t with a line and y versus t with a quadratic. Interpret the horizontal slope as v₀ₓ and the vertical quadratic coefficient as approximately −g/2.
- Compare fitted component motion with the measured x–y trajectory. Repeat or use a second launch setting to test whether the same component model remains valid.
Use a low-energy launcher, secure the camera/tripod, establish a clear landing zone, and use eye protection when appropriate. Camera parallax, an incorrect scale plane, motion blur, rolling shutter, uncertain projectile-center tracking, and frame-rate metadata can create systematic bias.
Prediction and validity conditions
For ideal projectile motion after release, horizontal acceleration is approximately zero and vertical acceleration is approximately −g. Therefore x(t) should be linear and y(t) quadratic. The two component descriptions share the same time coordinate but otherwise evolve independently.
- Analyze only the free-flight interval after the projectile leaves the launcher.
- Use one fixed coordinate system; positive y is upward in this lab.
- The camera scale must lie in the plane of motion.
- Air resistance should be small enough that a constant-acceleration model is defensible.
- Do not force the horizontal fit through the origin unless the coordinate origin actually makes x₀=0.
Evidence to report
Report the horizontal linear fit and R², vertical quadratic fit and R², fitted v₀ₓ and v₀ᵧ, fitted g, reconstructed launch speed and angle, trajectory consistency, residual/scatter size, and the dominant random/systematic limitations.
Student controls
| Frame | t (s) | x measured (m) | y measured (m) | x model (m) | y model (m) |
|---|---|---|---|---|---|
| No accepted video frames yet. | |||||
Use a linear fit for x(t), a quadratic fit for y(t), and an x–y trajectory graph. The fitted vertical t² coefficient estimates −g/2.
Record at least five distinct frame times to fit the component models; six or more are recommended.
No analysis yet.
After analysis, compare the horizontal linearity, vertical curvature, fitted acceleration, residuals, and likely effects of calibration/parallax.
State whether the tracked data support independent horizontal and vertical component models, using fitted values and uncertainty-aware evidence.
Uncertainty and model limitations
- Scale calibration: a meter stick outside the plane of motion creates a systematic position scale error.
- Camera perspective: a non-perpendicular camera produces parallax and can make the measured scale position-dependent.
- Frame timing: incorrect frame-rate metadata biases fitted velocities and acceleration.
- Tracking: choosing different points on a blurred projectile increases coordinate scatter.
- Air drag: substantial drag can make horizontal speed decrease and vertical acceleration differ from the simple constant-g model.
Lab-report prompts
- Explain why the x and y equations may be fitted separately even though they describe one trajectory.
- Compare fitted g with the accepted local/model value and decide whether the discrepancy is meaningful relative to experimental limitations.
- Use the fit coefficients to reconstruct launch speed and angle without simply reading the simulation controls.
- Identify one systematic error and predict the direction in which it would bias a fitted parameter.
Newton’s Second Law with an Atwood/Cart System
Test both parts of the model ΣF_ext = Ma: acceleration should increase linearly with net external force at fixed inertial mass and linearly with 1/M at fixed net force. Use a cart, pulley, hanging mass, and acceleration measurement; compare two controlled-variable studies, fit the correct linearized graph, quantify friction/pulley effects, and finish with uncertainty, results, discussion, and conclusion.
Physical apparatus and procedure
- Level a low-friction dynamics track. Connect a cart by a light string over a pulley to a small hanging mass. Use a motion sensor, photogates, or video analysis to measure the cart acceleration.
- Measure the cart, hanger, and slotted masses. Estimate rolling/friction force with a slow pull or force sensor if possible. Record the pulley radius and note that pulley rotational inertia can act like an additional inertial mass I_p/R².
- Study A — force dependence: keep the total moving mass constant by transferring small masses from the cart to the hanger. This changes the driving force while leaving the nominal total mass fixed.
- Study B — mass dependence: keep the hanging/driving mass fixed and add masses to the cart. This changes total inertial mass while keeping the driving force approximately fixed.
- For every setting, release from rest without pushing. Record acceleration over a clean interval after string tension is established and before the hanger reaches the floor or cart reaches the stop.
- For Study A plot a versus F_net. For Study B plot a versus 1/M_eff. Interpret the fitted slope, intercept, residuals, and whether the expected proportionality is supported within uncertainty.
Use a track stop, prevent the hanging mass from striking feet or the floor, keep fingers clear of the pulley, and use small driving masses. A tilted track, string stretch, pulley friction, wheel friction, and pulley rotational inertia can bias the acceleration systematically.
Prediction and validity conditions
For the cart + hanging mass system, the internal string tensions cancel when the entire moving system is modeled. The net external drive is approximately F_net = m_h g − f. If pulley inertia matters, use M_eff = M + I_p/R². Then a = F_net/M_eff.
- Choose the cart + hanger as one system before writing the system equation.
- In Study A, total moving mass must stay fixed as hanging mass changes.
- In Study B, hanging mass and friction conditions should stay fixed as total mass changes.
- Use measured or estimated friction consistently; do not silently set it to zero after observing a nonzero intercept.
- A massive pulley changes the effective inertia even when its axle friction is small.
Evidence to report
Report the fitted slope, intercept, R², residual behavior, inferred mass or force from the slope, percent difference from the independently measured value, and the dominant random/systematic limitations.
Student controls
| Trial | Study | M nominal (kg) | mₕ (kg) | F_net (N) | 1/M_eff (kg⁻¹) | a measured (m/s²) | a model (m/s²) |
|---|---|---|---|---|---|---|---|
| No accepted acceleration trials yet. | |||||||
Study A: fit a versus F_net. Study B: fit a versus 1/M_eff. The fitted intercept is a useful diagnostic for unmodeled systematic effects.
Record at least four distinct settings to fit the proportionality; five or more are recommended.
No analysis yet.
After analysis, compare slope and intercept with the Newton’s-second-law prediction and discuss friction, track level, pulley inertia, and acceleration-measurement scatter.
State whether acceleration is proportional to net force or inverse inertial mass for the selected controlled-variable study, supported by quantitative fit evidence.
Uncertainty and limitations
- Mass: scale resolution affects both M and mₕ.
- Friction: rolling/axle resistance may change with load and direction rather than remain perfectly constant.
- Pulley inertia: ignoring I_p/R² makes the observed acceleration smaller than a = F_net/M.
- Track level: a small slope adds or subtracts a component of gravity from the intended driving force.
- Acceleration extraction: differentiation of position data amplifies sensor/video noise.
- Release: a push, string slack, or oscillating hanger contaminates the clean constant-acceleration interval.
Report prompts
- Define the system and explain why string tension is internal to the cart + hanger system.
- Explain how your procedure changes only the intended independent variable.
- Interpret the fitted slope physically and compare it with 1/M_eff or F_net as appropriate.
- Is the intercept statistically/experimentally consistent with zero? What physical effect could create a nonzero intercept?
- Which limitation is most likely systematic rather than random?
- State a quantitative conclusion without claiming exact agreement.
Static and Kinetic Friction
How do maximum static friction and kinetic friction depend on normal force? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Use a block, force sensor, and adjustable normal load. Increase the pull slowly to identify the peak static force, then measure the approximately steady kinetic force while the block slides. Repeat at controlled normal-force settings without changing the contacting surfaces.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
How do maximum static friction and kinetic friction depend on normal force?
Model-appropriate analysis
Fit f_s,max versus F_N and f_k versus F_N. Slopes estimate μ_s and μ_k; intercepts and residuals expose offsets or changing contact conditions.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | F_N (N) | f_s,max (N) | f_k (N) | μ_s,trial | μ_k,trial | ||
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Fit f_s,max versus F_N and f_k versus F_N. Slopes estimate μ_s and μ_k; intercepts and residuals expose offsets or changing contact conditions.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Spring Force, Work, and Energy
Does an ideal spring obey Hooke’s law, and does measured force–displacement work agree with the change in spring potential energy? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Zero a force sensor, measure displacement from the relaxed spring length, and collect force at controlled extensions. Use the force–position graph to infer k and integrate the measured force magnitude to compare with ½kx². Stay below the elastic limit.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
Does an ideal spring obey Hooke’s law, and does measured force–displacement work agree with the change in spring potential energy?
Model-appropriate analysis
Fit −F versus x for k, compare measured work/energy with ½kx², and inspect residual curvature for departures from Hooke’s law.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | x (m) | F measured (N) | −F/x (N/m) | U from data (J) | U model (J) | ||
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Fit −F versus x for k, compare measured work/energy with ½kx², and inspect residual curvature for departures from Hooke’s law.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Impulse and Collisions
Does measured impulse equal momentum change, and when is kinetic energy conserved during a collision? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Use two carts with force probes or a force sensor, measure velocities immediately before and after impact, and record a high-rate force–time pulse. Compare numerical impulse with Δp, test system momentum, then classify kinetic-energy change for elastic, inelastic, and sticking collisions.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
Does measured impulse equal momentum change, and when is kinetic energy conserved during a collision?
Model-appropriate analysis
Integrate a 41-point force pulse, fit impulse versus momentum change, test p_f versus p_i, and compare K_f/K_i across collision types.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | v_i (m/s) | Impulse J (N·s) | Δp (kg·m/s) | p_i | p_f | K_f/K_i | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Integrate a 41-point force pulse, fit impulse versus momentum change, test p_f versus p_i, and compare K_f/K_i across collision types.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Torque and Rotational Inertia
How do net torque and rotational inertia determine angular acceleration? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Use a rotating platform or disk with known mass distribution. Apply tangential force at a measured radius, correct for friction torque, and measure angular acceleration. Run one controlled study varying net torque at fixed I and another varying I at fixed torque.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
How do net torque and rotational inertia determine angular acceleration?
Model-appropriate analysis
Study A fits α versus τ_net; Study B fits α versus 1/I. Use a free intercept and residual graph, then infer I or τ from the slope.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Study | τ_net (N·m) | I (kg·m²) | 1/I | α (rad/s²) | ||
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Study A fits α versus τ_net; Study B fits α versus 1/I. Use a free intercept and residual graph, then infer I or τ from the slope.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Rolling Energy on an Incline
How does rotational inertia change translational speed for objects rolling without slipping? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Release a rigid body from rest at several vertical drops on the same incline. Measure center-of-mass speed near the bottom and test v² versus h. Compare the inferred β with the object model and deliberately examine a partial-slip condition as a model-limit case.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
How does rotational inertia change translational speed for objects rolling without slipping?
Model-appropriate analysis
Fit v² versus h, infer β from slope 2g/(1+β), inspect residuals, and compare total translational+rotational energy with Mgh.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Object | h (m) | v² measured | β expected | K account ratio | ||
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Fit v² versus h, infer β from slope 2g/(1+β), inspect residuals, and compare total translational+rotational energy with Mgh.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Conservation of Angular Momentum
When external torque is negligible, does angular momentum remain constant as rotational inertia changes? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Spin a low-friction rotating platform with two equal movable masses. Measure I_i and ω_i, move the masses radially, then measure I_f and ω_f. Repeat for several final radii and compare conservation residuals and kinetic-energy change.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
When external torque is negligible, does angular momentum remain constant as rotational inertia changes?
Model-appropriate analysis
Fit ω_f versus 1/I_f so the slope estimates angular momentum. Plot angular-momentum residual and K_f/K_i versus final radius to separate L conservation from energy conservation.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | r_f (m) | I_f (kg·m²) | 1/I_f | ω_f (rad/s) | ΔL/L (%) | K_f/K_i | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Fit ω_f versus 1/I_f so the slope estimates angular momentum. Plot angular-momentum residual and K_f/K_i versus final radius to separate L conservation from energy conservation.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Oscillation Period and Model Limits
Which variables control oscillator period, and where do ideal-spring or small-angle models break down? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Run three controlled studies: spring mass (T² vs m), pendulum length (T² vs L), and pendulum amplitude (T/T₀ vs amplitude). Time multiple cycles for each setting, divide by the cycle count, fit with a free intercept, and use residuals to identify model limits.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
Which variables control oscillator period, and where do ideal-spring or small-angle models break down?
Model-appropriate analysis
Spring: T² vs m with expected slope 4π²/k. Pendulum: T² vs L with expected slope 4π²/g. Amplitude: T/T₀ reveals large-angle departure from the small-angle model.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Study | Setting | T measured (s) | T² (s²) | Model value | Residual / ratio | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Spring: T² vs m with expected slope 4π²/k. Pendulum: T² vs L with expected slope 4π²/g. Amplitude: T/T₀ reveals large-angle departure from the small-angle model.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Electric Field and Equipotential Mapping
How can measured potential reveal electric-field magnitude, direction, and departures from ideal symmetry? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Use conductive paper or a shallow conducting tray with finite electrodes and a high-impedance voltmeter probe. Record x,y,V points, sketch equipotentials, and perform a centerline scan. Infer E_x from −dV/dx and compare with ΔV/d in the central region.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
How can measured potential reveal electric-field magnitude, direction, and departures from ideal symmetry?
Model-appropriate analysis
Fit V(x) on the centerline and infer E_x from the negative slope. Use residuals and off-axis measurements to identify finite-electrode edge effects instead of claiming a perfectly uniform field everywhere.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Probe | x (m) | y (m) | V (V) | ΔV/d ideal E (V/m) | Local model note | ||
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Fit V(x) on the centerline and infer E_x from the negative slope. Use residuals and off-axis measurements to identify finite-electrode edge effects instead of claiming a perfectly uniform field everywhere.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Potential Energy of an Accelerated Charge
Does electric-potential-energy change account for the measured kinetic-energy change of an accelerated charged particle? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Select a charged-particle model, accelerate it through a controlled potential difference, and record final speed. Test ΔK against |q|V_acc and independently test v_f²−v_i² against V_acc. Include a deliberately lossy pathway to diagnose non-ideal transfer.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
Does electric-potential-energy change account for the measured kinetic-energy change of an accelerated charged particle?
Model-appropriate analysis
Fit ΔK versus |q|V_acc (ideal slope 1) and v_f²−v_i² versus V_acc (slope 2|q|/m). Residuals distinguish measurement scatter from systematic energy loss.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Particle | V_acc (V) | |q|V (J) | ΔK measured (J) | v_f²−v_i² | Energy ratio | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Fit ΔK versus |q|V_acc (ideal slope 1) and v_f²−v_i² versus V_acc (slope 2|q|/m). Residuals distinguish measurement scatter from systematic energy loss.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Capacitance, Geometry, and Dielectric
How does a parallel-plate capacitor’s measured capacitance depend on plate area, separation, and dielectric constant, and where do fringing effects limit the ideal model? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Use two parallel conducting plates connected to a capacitance meter or a low-voltage RC measurement system. Run one-variable studies of overlap area A, plate separation d, or dielectric constant κ while holding the other quantities fixed. Keep plates parallel, prevent contact, and include a fringing-sensitive condition so students can identify the limits of C=κε₀A/d rather than forcing ideal agreement.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
How does a parallel-plate capacitor’s measured capacitance depend on plate area, separation, and dielectric constant, and where do fringing effects limit the ideal model?
Model-appropriate analysis
Area study: fit C versus A. Separation study: fit C versus 1/d. Dielectric study: fit C versus κ. Compare the fitted slope with the appropriate κε₀/d, κε₀A, or ε₀A/d prediction, inspect the intercept and residuals, and use the fringing condition to diagnose the ideal parallel-plate model limit.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Study | Setting | C measured (pF) | C ideal (pF) | Transformed x | Residual (%) | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Area study: fit C versus A. Separation study: fit C versus 1/d. Dielectric study: fit C versus κ. Compare the fitted slope with the appropriate κε₀/d, κε₀A, or ε₀A/d prediction, inspect the intercept and residuals, and use the fringing condition to diagnose the ideal parallel-plate model limit.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Ohm, Kirchhoff, and Internal Resistance
Do measured I–V relationships and multi-loop network data satisfy Ohm’s law, junction/loop conservation, and a real source’s internal-resistance model? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Build low-voltage resistor circuits with meters of appropriate resistance. Run three separate studies: a fixed resistor I–V sweep, a branched Kirchhoff network with fixed resistors and changing source voltage, and a source/internal-resistance load sweep. Record polarity and current direction before measurement; keep one topology per dataset.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
Do measured I–V relationships and multi-loop network data satisfy Ohm’s law, junction/loop conservation, and a real source’s internal-resistance model?
Model-appropriate analysis
Ohm study: fit I versus V so slope gives 1/R. Kirchhoff study: compare I_source with I_branch1+I_branch2 and graph loop residuals around zero. Internal-resistance study: fit terminal voltage V_t versus current I; intercept estimates emf and slope −r estimates internal resistance.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Study | Setting | I source (mA) | V terminal/load (V) | Junction residual | Loop / model residual | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Ohm study: fit I versus V so slope gives 1/R. Kirchhoff study: compare I_source with I_branch1+I_branch2 and graph loop residuals around zero. Internal-resistance study: fit terminal voltage V_t versus current I; intercept estimates emf and slope −r estimates internal resistance.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
RC Transient and Time Constant
Does a measured charging or discharging transient follow the exponential RC model, and does the fitted time constant scale with R and C as τ=RC? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Use a low-voltage source, switch, resistor, capacitor, and voltage sensor/data logger. Start every transient from a defined initial condition, sample capacitor voltage through several time constants, and fit either ln|V_C−V_f| versus time or the equivalent exponential model. Run separate R and C scaling studies and a leakage pathway that deliberately violates the ideal single-time-constant assumption.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
Does a measured charging or discharging transient follow the exponential RC model, and does the fitted time constant scale with R and C as τ=RC?
Model-appropriate analysis
For each transient, fit the logarithmic form to obtain τ from slope −1/τ. Then fit measured τ versus R at fixed C or τ versus C at fixed R. The leakage mode adds a second loss path so students can identify systematic departure from a single ideal RC time constant.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Study | Setting | τ measured (s) | τ model (s) | Fit slope (s⁻¹) | τ residual (%) | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
For each transient, fit the logarithmic form to obtain τ from slope −1/τ. Then fit measured τ versus R at fixed C or τ versus C at fixed R. The leakage mode adds a second loss path so students can identify systematic departure from a single ideal RC time constant.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Magnetic Field from Current
How does measured magnetic field depend on current and geometry for a current-carrying wire or solenoid, and when do finite-length effects limit the ideal field models? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Use a low-voltage current supply with a long straight conductor or solenoid and a calibrated magnetic-field probe. Zero the probe away from the apparatus, define the positive field direction, and run separate one-variable studies of wire current, wire distance, or solenoid turns density. Include a finite-geometry condition so students can distinguish model limits from random sensor scatter.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
How does measured magnetic field depend on current and geometry for a current-carrying wire or solenoid, and when do finite-length effects limit the ideal field models?
Model-appropriate analysis
Wire-current study: fit B versus I at fixed r. Wire-distance study: fit B versus 1/r at fixed I. Solenoid study: fit B versus turns density n at fixed current. Compare slope with the ideal long-wire or long-solenoid prediction and use residual structure to diagnose finite-length, alignment, ambient-field, or probe-zero effects.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Study | Setting | B measured (mT) | B ideal (mT) | Transformed x | Residual (%) | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Wire-current study: fit B versus I at fixed r. Wire-distance study: fit B versus 1/r at fixed I. Solenoid study: fit B versus turns density n at fixed current. Compare slope with the ideal long-wire or long-solenoid prediction and use residual structure to diagnose finite-length, alignment, ambient-field, or probe-zero effects.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Faraday Induction and Lenz’s Law
Does measured induced emf scale with the rate of magnetic-flux change, and does its sign consistently oppose the imposed flux change? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Use a search coil connected to a high-impedance voltage sensor and a controllable magnetic-field source or rotating-coil apparatus. Define the coil area-vector orientation before recording signs. Run separate studies varying field-change rate, coil turns, or rotation rate, and include a loaded-coil condition to show why terminal voltage can differ from open-circuit induced emf.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
Does measured induced emf scale with the rate of magnetic-flux change, and does its sign consistently oppose the imposed flux change?
Model-appropriate analysis
Field-rate study: fit signed emf versus −dB/dt so the slope estimates NA and the sign directly tests Lenz’s law. Turns study: fit emf versus N at fixed signed flux-change rate. Rotation study: fit peak |emf| versus angular speed using ε_peak=NABω. Compare open-circuit and loaded-coil behavior without confusing induced emf with current.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Study | Setting | ε measured (mV) | ε Faraday model (mV) | Transformed x | Sign / residual | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
Field-rate study: fit signed emf versus −dB/dt so the slope estimates NA and the sign directly tests Lenz’s law. Turns study: fit emf versus N at fixed signed flux-change rate. Rotation study: fit peak |emf| versus angular speed using ε_peak=NABω. Compare open-circuit and loaded-coil behavior without confusing induced emf with current.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
LR and LC Transient Dynamics
Do LR transients and LC oscillations follow the time constants, periods, and differential-equation behavior predicted by inductive circuit models? This investigation includes manual measurement, a separate automatic dataset, model-selected graphs, residual/model-limit analysis, uncertainty, results, discussion, conclusion, and Print / Save.
Physical apparatus and procedure
Use low-voltage resistor–inductor and inductor–capacitor circuits with voltage/current sensing. For LR studies, begin from a defined switch state and record current through several time constants. For LC studies, charge the capacitor safely, isolate the idealized LC branch, and record oscillation period. Include coil resistance/damping as a model-limit condition rather than treating every inductor as ideal.
- Define one independent variable and hold the stated controls fixed.
- Record raw measurements with units and instrument resolution before calculating derived quantities.
- Repeat or span enough settings to reveal scatter, intercept bias, curvature, or a model-limit transition.
- Choose the graph from the physics model, not because a transformation merely looks linear.
Use supervised, classroom-appropriate apparatus and conservative operating ranges. Secure moving/rotating equipment, use low-voltage electrical setups, and document sensor zero, calibration, alignment, friction, edge effects, or other systematic limitations relevant to this experiment.
Research question
Do LR transients and LC oscillations follow the time constants, periods, and differential-equation behavior predicted by inductive circuit models?
Model-appropriate analysis
LR-R study: fit measured τ versus 1/R at fixed L, with ideal slope L. LR-L study: fit τ versus L at fixed R, with ideal slope 1/R. LC study: fit measured period T versus √(LC), with ideal slope 2π. Use representative transient/oscillation traces and the damping condition to distinguish ideal differential-equation behavior from real coil resistance.
Evidence standard
Use fitted slope/intercept, R², residual behavior, comparison with the independently specified model value, and uncertainty/model-limit evidence. Do not describe approximate agreement as exact conservation or exact proportionality.
Student controls
| Trial | Study | Setting | Measured time (s) | Ideal model (s) | Transformed x | Residual (%) | |
|---|---|---|---|---|---|---|---|
| No accepted measurements yet. | |||||||
LR-R study: fit measured τ versus 1/R at fixed L, with ideal slope L. LR-L study: fit τ versus L at fixed R, with ideal slope 1/R. LC study: fit measured period T versus √(LC), with ideal slope 2π. Use representative transient/oscillation traces and the damping condition to distinguish ideal differential-equation behavior from real coil resistance.
Record enough controlled measurements or run the automatic dataset to generate the model-selected graphs.
No analysis yet.
Interpret slope/intercept, residuals, uncertainty, and the most important physical model limitation.
Answer the research question with quantitative evidence rather than a qualitative “looks right” statement.
Uncertainty checklist
- Instrument resolution and calibration/zero.
- Trial-to-trial scatter and repeatability.
- Whether control variables remained fixed.
- Fit slope/intercept and residual pattern.
- One plausible systematic effect with its likely direction.
- Whether disagreement is larger than the experimental uncertainty.
Report prompts
- State the model and the graph it predicts.
- Explain why the chosen independent variable isolates the intended relationship.
- Interpret the fitted parameter physically.
- Use residuals to distinguish random scatter from model failure.
- Identify the strongest systematic limitation.
- Write a quantitative conclusion and one defensible improvement.
Lab notebook / report template
Experimental FRQ transfer
Procedure writing
Name what to vary, what to measure, what to hold constant, and how repeated measurements make the comparison defensible.
Graph construction
Axes need quantity labels, units, sensible scales, plotted data, and a fit/curve that matches the intended model.
Data → claim
A conclusion should point to a numerical feature—slope, intercept, area, time constant, conservation residual, or uncertainty comparison—not merely say the graph “looks right.”
Model limitations
State a physical mechanism and its directional effect when possible. “Human error” by itself is rarely a useful scientific explanation.
Use appropriate low-voltage electrical sources, secure rotating/moving apparatus, eye protection where required, conservative projectile setups, and instructor-approved procedures. High-voltage electrostatics, mains electricity, strong magnets, hot equipment, and fast rotating systems require qualified supervision and suitable institutional safety controls.
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