10 Turnkey Virtual STEM Laboratories & AP Physics Problem Sets

PHYS 211 / AP Physics C

Lab #1: Harmonic Oscillator Dynamics & Hairspring Escapement Isochronism

Horology & Mechanics

Learning Objectives: Investigate the relationship between balance wheel moment of inertia (I), hairspring restoring torque (k), and oscillation frequency (BPH). Quantify rate drift caused by lift angle variations.

Governing Equations: T = 2\pi \sqrt{\frac{I}{k}}, \quad \text{Beat Period } \Delta t = \frac{3600}{\text{BPH}} \text{ sec}

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. If a balance wheel caliber operates at 28,800 BPH, what is its fundamental harmonic frequency in Hertz (Hz)?

One full oscillation cycle (Hertz) comprises two alternations (beats). Thus 28,800 BPH / 7,200 = 4.0 Hz.

2. How does a decrease in balance wheel moment of inertia I affect the watch movement rate?

Since T = 2π√(I/k), reducing I decreases period T, causing the escapement to complete more beats per hour (running fast).

3. At 28,800 BPH and a 52° lift angle, what is the contact duration of the impulse pin within the pallet fork?

With 4 Hz oscillation and average amplitude 280°, the angular velocity at equilibrium yields approximately 9 milliseconds of impulse engagement.

CHEM 141 / Materials Science 201

Lab #2: Archimedes Hydrostatic Density & Sovereign Gold Karat Assay

Precious Metals & Metallurgy

Learning Objectives: Measure dry mass and buoyant immersion displacement of precious metal specimens to determine specific gravity and resolve alloy purity non-destructively.

Governing Equations: SG = \frac{W_{\text{air}}}{W_{\text{air}} - W_{\text{water}}}, \quad \text{Karat} = 24 \times \frac{\rho_{\text{alloy}} - \rho_{\text{base}}}{\rho_{\text{pure}} - \rho_{\text{base}}}

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. A suspected 1 troy oz bullion coin weighs 31.10 g in air and 29.49 g when suspended in distilled water at 20°C. What is its specific gravity?

SG = 31.10 / (31.10 - 29.49) = 31.10 / 1.61 = 19.32, matching pure 24K gold exactly.

2. Why does a tungsten core (density ~19.25 g/cm³) pass hydrostatic tests but fail acoustic ultrasound tests?

Tungsten closely mimics gold density (19.25 vs 19.32 g/cm³), but sound travels at ~5,200 m/s in tungsten vs ~3,240 m/s in gold.

3. What is the theoretical specific gravity of 14K yellow gold (58.3% Au, 20.8% Ag, 20.9% Cu)?

Weighted inverse density calculation yields SG approximately 13.07 to 13.4 depending on alloy ratio.

CIVL 310 / Structural Mechanics

Lab #3: Elastic Deflection & AISC L/360 Limit States in Structural I-Beams

Structural Engineering

Learning Objectives: Calculate centerline elastic deflection for ASTM A992 wide-flange beams under concentrated loads, checking conformance against the AISC L/360 architectural serviceability limit.

Governing Equations: \delta_{\text{max}} = \frac{P L^3}{48 E I_x} \le \frac{L}{360}

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. For a 6.0 meter span beam, what is the maximum allowable deflection under the standard AISC L/360 live-load limit?

Allowable deflection = L / 360 = 6000 mm / 360 = 16.67 mm.

2. If beam span length L is doubled while keeping load P and section Ix constant, how does centerline deflection change?

Deflection is proportional to L³. Doubling L increases deflection by 2³ = 8 times.

3. What is the elastic modulus (E) of ASTM A992 structural steel in SI units?

Structural steel has an elastic modulus of approximately 200,000 MPa (200 GPa or 29,000 ksi).

AERO 402 / Orbital Mechanics

Lab #4: Hohmann Interplanetary Transfer & Trajectory Δv Optimization

Planetary Science & Astrodynamics

Learning Objectives: Derive the semi-major axis and velocity change requirements (Δv1 and Δv2) to transition a spacecraft from Earth circular orbit to Mars orbit using an elliptical Hohmann transfer.

Governing Equations: \Delta v_1 = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1 + r_2}} - 1 \right), \quad a = \frac{r_1 + r_2}{2}

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. What is the semi-major axis of a Hohmann transfer ellipse from Earth (1.0 AU) to Mars (1.524 AU)?

Semi-major axis a = (r1 + r2) / 2 = (1.0 + 1.524) / 2 = 1.262 AU.

2. How long does a one-way Hohmann transfer from Earth to Mars take, using Kepler’s Third Law (T = a^(3/2))?

Period of full ellipse = (1.262)^1.5 = 1.417 years. The transfer is half the orbit: 1.417 / 2 = 0.708 years = ~8.5 months (258 days).

3. Where must the initial propulsion burn Δv1 be applied along the Earth parking orbit?

To raise aphelion to Mars orbit, a prograde tangential burn must be performed at transfer orbit perihelion.

GEOL 204 / Paleontology Lab

Lab #5: Trilobite Biostratigraphic Biozones & Radiometric Correlation

Paleontology & Geochronology

Learning Objectives: Correlate Cambrian stratigraphic strata using index fossil ranges (Paradoxides davidis, Olenellus) calibrated with potassium-argon dating of intercalated volcanic bentonite beds.

Governing Equations: t = \frac{1}{\lambda} \ln \left( 1 + \frac{^{40}\text{Ar}}{0.105 \cdot {}^{40}\text{K}} \right), \quad \text{Duration } \Delta t = \text{FAD} - \text{LAD}

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. What defining characteristic makes a fossil taxon an optimal index fossil (biozone marker)?

Optimal index fossils evolved rapidly (narrow temporal stratigraphic range) and spread globally across multiple marine basins.

2. In which geologic period did Paradoxides trilobites serve as the dominant biostratigraphic marker?

Paradoxides trilobites are hallmark Middle Cambrian (Miaolingian) index fossils spanning ~509 to ~497 Ma.

3. If a volcanic ash layer above a fossil bed contains 12.5% of its original radioactive parent isotope, how many half-lives have elapsed?

100% → 50% (1) → 25% (2) → 12.5% (3). Exactly three half-lives have elapsed.

PHYS 152 / Acoustics Lab

Lab #6: Standing Wave Harmonics & 12-Tone Equal Temperament Tuning

Acoustics & Harmonics

Learning Objectives: Synthesize open acoustic pipe standing waves, measuring node/antinode spacing and computing the cent discrepancy between pure harmonic overtones and 12-TET intervals.

Governing Equations: f_n = n \frac{v}{2 L}, \quad \text{Cents } c = 1200 \log_2 \left( \frac{f_2}{f_1} \right)

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. What is the theoretical frequency of the pure harmonic fifth (3:2 ratio) above concert pitch A4 (440 Hz)?

Pure fifth = 440 × 1.5 = 660.00 Hz (in 12-TET it is tempered to 440 × 2^(7/12) ≈ 659.25 Hz, flat by ~1.95 cents).

2. In an open-open cylindrical organ pipe of length L = 1.0 m, what is the fundamental frequency (v = 343 m/s)?

f1 = v / (2L) = 343 / (2 × 1.0) = 171.5 Hz.

3. How many cents are in a standard equal-tempered chromatic semitone?

An octave is divided into 1,200 cents across 12 semitones, making each semitone exactly 100 cents.

MECH 304 / Precision Mechanics

Lab #7: Swiss Lever vs Co-Axial Escapement Friction & Lift Angles

Horology & Mechanics

Learning Objectives: Compare the sliding friction coefficients and tangential impulse mechanisms between the traditional Swiss lever escapement and the Daniels co-axial escapement.

Governing Equations: W_{\text{loss}} = \int \mu N \, ds, \quad \eta_{\text{escapement}} = \frac{E_{\text{balance}}}{E_{\text{barrel}}}

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. Why does the Daniels co-axial escapement require significantly less lubrication on the pallet jewels than the Swiss lever?

The co-axial escapement replaces sliding friction with rolling/tangential push, virtually eliminating lubricant deterioration.

2. What is the typical lift angle range for modern Swiss lever escapement calibers (ETA 2824-2, Sellita SW200-1)?

Standard Swiss lever calibers operate with lift angles between 50° and 54° (e.g., ETA 2824 is 50°, Sellita SW200 is 50°, Seiko NH35 is 53°).

3. How does an incorrect lift angle setting on a timing machine (timegrapher) affect rate accuracy vs amplitude reading?

Daily rate is calculated purely from interval between consecutive beats, so it remains exact; only amplitude calculation relies on lift angle.

MATL 305 / Metallurgy Lab

Lab #8: Ternary Au-Ag-Cu Phase Equilibrium & Karat Specific Gravity

Precious Metals & Metallurgy

Learning Objectives: Analyze the density and color variation across the gold-silver-copper ternary system to determine exact scrap melt value and alloy composition.

Governing Equations: \rho_{\text{alloy}} = \frac{100}{\frac{\%\text{Au}}{\rho_{\text{Au}}} + \frac{\%\text{Ag}}{\rho_{\text{Ag}}} + \frac{\%\text{Cu}}{\rho_{\text{Cu}}}}

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. What metal is added in high proportions to gold and copper to formulate Rose Gold?

Rose gold achieves its distinctive red/pink hue by increasing copper content (e.g. 75% Au, 22.25% Cu, 2.75% Ag for 18K Red Gold).

2. Calculate the pure gold content in 50 grams of 18-karat gold jewelry:

Purity = 18 / 24 = 0.75. Fine weight = 50 g × 0.75 = 37.50 grams.

3. At a spot gold price of $2,700/troy oz, what is the gross melt value of 37.50 grams of pure gold (31.1035 g/oz)?

Troy ounces = 37.50 / 31.1035 = 1.2056 oz. Gross melt value = 1.2056 × $2,700 = $3,255.26.

CIVL 320 / Advanced Structures

Lab #9: Euler Column Buckling & Effective Length Factors (K)

Structural Engineering

Learning Objectives: Calculate critical buckling compressive load (Pcr) for wide-flange structural columns across pinned, fixed, and free boundary conditions.

Governing Equations: P_{\text{cr}} = \frac{\pi^2 E I}{(K L)^2}, \quad \lambda = \frac{K L}{r}

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. What is the theoretical effective length factor K for a column with both ends fixed against rotation and translation?

For fixed-fixed boundaries, inflection points occur at 0.25L and 0.75L, yielding theoretical K = 0.5 (AISC recommended design value is 0.65).

2. How does doubling the column length L affect the critical Euler buckling capacity Pcr?

Pcr is inversely proportional to L². Doubling length reduces buckling capacity to (1/2)² = 1/4 (a 75% reduction).

3. About which principal axis (strong axis Ix or weak axis Iy) will an unbraced wide-flange column buckle first?

Buckling occurs about the axis with the minimum radius of gyration / moment of inertia, which is the weak axis (Iy).

ASTR 302 / Celestial Mechanics

Lab #10: Lagrange Libration Equilibria & Roche Tidal Disruption Limits

Planetary Science & Astrodynamics

Learning Objectives: Evaluate gravitational equipotential wells for three-body orbital mechanics, identifying Trojan stability regions (L4/L5) and satellite breakup limits.

Governing Equations: d_{\text{Roche}} = R_M \left( 2 \frac{\rho_M}{\rho_m} \right)^{1/3}, \quad \nabla \Phi_{\text{eff}} = 0

Interactive Laboratory Environment:

Pre-calibrated parameter state ready for observation and empirical data collection.

Launch Virtual Lab Simulator ↗

Experimental Data Recording Table:

Trial #	Controlled Variable	Observed Metric	Theoretical Value	% Relative Error
Trial 1 (Baseline)
Trial 2 (Elevated)

📝 Concept Check: Auto-Graded Exercises

1. Where are the James Webb Space Telescope (JWST) and Gaia observatories stationed?

Sun-Earth L2 sits approximately 1.5 million kilometers behind Earth, shielded from solar radiation and offering an unobstructed deep-space view.

2. What prevents satellites from surviving inside a planet’s Roche limit?

Inside the Roche limit, the gravitational gradient across the body exceeds its own gravitational binding force, tearing fluid/loose bodies apart.

3. Why are Lagrange points L4 and L5 dynamically stable while L1, L2, and L3 require active station-keeping?

Although L4 and L5 are potential maxima, the Coriolis force deflects perturbed objects into closed stable orbits around the libration points.