{ "format": "recallweave-deck/1", "title": "Units and dimensions: conversions that preserve meaning", "attribution": "Original AI-assisted questions, fictional quantities and worked explanations by RecallWeave contributors, 2026-10-08. Primary references: BIPM, The International System of Units (SI), 9th edition, version 4.01 (June 2026), sections 2.3.3–2.3.4, 3, 4 and 5.4.1, https://www.bipm.org/en/publications/si-brochure and https://doi.org/10.59161/AUEZ1291 ; NIST, Metric (SI) Prefixes, https://www.nist.gov/pml/owm/metric-si-prefixes ; NIST SP 330, section 4, https://www.nist.gov/pml/special-publication-330/sp-330-section-4 . Reference definitions inform the unit relations; no source exercise text is reproduced. Worked solutions and distractors are original. This introductory practice lesson is not a validated assessment; no endorsement by BIPM or NIST is implied.", "license": "No separate license grant is made by this deck. Exercises, examples and explanations are original RecallWeave course content. The linked BIPM and NIST references retain their own terms; this notice does not relicense those sources or other repository content.", "concepts": [ "Factors and prefixes", "Area and volume", "Rates and density", "Dimensions and models" ], "items": [ { "id": "units-factor-unity", "concept": "Factors and prefixes", "prerequisites": [], "prompt": "A cable is 2.4 m long. Given 1 m = 100 cm, which calculation multiplies by a ratio equal to one AND cancels m to leave the answer directly in cm?", "options": [ "2.4 m × (1 cm / 100 m) = 0.024 cm", "2.4 m × (100 cm / 1 m) = 240 cm", "2.4 m × (1 m / 100 cm) = 0.024 m^2/cm", "2.4 m + 100 cm = 340 cm" ], "answer": 1, "explanation": "100 cm and 1 m are the same length, so their ratio is one. Multiplying 2.4 m by (100 cm / 1 m) cancels m and leaves 240 cm. The physical length has not changed. The inverse ratio in the third option also equals one, but it leaves m^2/cm rather than cm directly; it is an unhelpful orientation here, not a different physical dimension. The first ratio is not one, and adding another length changes the quantity.", "transfer": "Express 0.085 km in m using a written conversion factor. Explain why the numerical value changes while the length stays the same." }, { "id": "units-prefix-case", "concept": "Factors and prefixes", "prerequisites": [], "prompt": "An energy label reads 4 MJ. Using SI prefix symbols with their exact letter case, which value in J represents the same energy?", "options": [ "0.004 J", "4000 J", "4 J", "4000000 J" ], "answer": 3, "explanation": "Capital M is mega, meaning 10^6. Therefore 4 MJ = 4 × 10^6 J = 4000000 J. Lowercase m is milli, meaning 10^−3: 4 mJ would be 0.004 J. The 4000 J option substitutes kilo for mega, and 4 J drops the prefix entirely. Prefix symbols are case-sensitive.", "transfer": "Convert both 7 mJ and 7 MJ to J. How many times larger is the second energy than the first? Explain why changing the letter case changes the quantity." }, { "id": "units-mass-chain", "concept": "Factors and prefixes", "prerequisites": [], "prompt": "A sample has mass 360 mg. Which mass in kg is equal to it? Use 1000 mg = 1 g and 1000 g = 1 kg.", "options": [ "0.00036 kg", "0.36 kg", "360000 kg", "0.00000036 kg" ], "answer": 0, "explanation": "Use both conversions: 360 mg × (1 g / 1000 mg) × (1 kg / 1000 g) = 0.00036 kg. The first step gives 0.36 g, not 0.36 kg. There are 1000000 mg in 1 kg. Multiplying by 1000 reverses the needed change in numerical value; dividing by an extra 1000 goes too far.", "transfer": "Express 0.012 kg in g and in mg. Explain why standard prefixed mass-unit names are formed from gram, even though kilogram is the SI base unit." }, { "id": "units-square-factor", "concept": "Area and volume", "prerequisites": [ "Factors and prefixes" ], "prompt": "A panel has area 36 cm^2. What is the same area in m^2? Use 1 cm = 0.01 m.", "options": [ "0.36 m^2", "360000 m^2", "0.0036 m^2", "0.000036 m^2" ], "answer": 2, "explanation": "Square the entire length conversion: 1 cm^2 = (0.01 m)^2 = 0.0001 m^2. Thus 36 cm^2 = 36 × 0.0001 m^2 = 0.0036 m^2. A linear factor of 0.01 is insufficient for area; using its reciprocal reverses the conversion, and cubing it would be appropriate to volume rather than area.", "transfer": "Express an area of 0.0045 m^2 in mm^2, showing a conversion factor raised to the second power. Explain why multiplying the numerical value by only 1000 is insufficient." }, { "id": "units-cube-edge", "concept": "Area and volume", "prerequisites": [ "Factors and prefixes" ], "prompt": "A cube has edge length 20 mm. Using volume = edge^3 and 1 mm = 0.001 m, what is its volume in m^3?", "options": [ "0.000008 m^3", "0.008 m^3", "0.02 m^3", "8 m^3" ], "answer": 0, "explanation": "First convert the edge: 20 mm = 0.02 m. Then cube it: (0.02 m)^3 = 0.000008 m^3. Alternatively, 8000 mm^3 × (0.001 m / 1 mm)^3 gives the same result. A cubic unit requires the cubed length factor; simply relabelling the edge as a volume or using a linear factor on a volume changes the result.", "transfer": "The cube's edge is increased to 40 mm. Calculate its new volume in m^3 and the ratio of new volume to old volume. Explain why doubling an edge does not merely double the volume." }, { "id": "units-litre-bridge", "concept": "Area and volume", "prerequisites": [ "Factors and prefixes" ], "prompt": "A jug contains 750 mL. Given 1000 mL = 1 L and 1 L = 0.001 m^3, which pair gives that same volume in L and in m^3?", "options": [ "0.75 L and 0.75 m^3", "0.00075 L and 0.00000075 m^3", "750 L and 0.75 m^3", "0.75 L and 0.00075 m^3" ], "answer": 3, "explanation": "750 mL × (1 L / 1000 mL) = 0.75 L. Then 0.75 L × (0.001 m^3 / 1 L) = 0.00075 m^3. Each step changes the numerical representation of the same volume. In particular, 1 mL = 0.000001 m^3 = 1 cm^3; litres and cubic metres do not have the same unit size.", "transfer": "Express 1.8 L in cm^3 and in m^3 using two unit-cancelling chains. Show that the two answers describe the same volume." }, { "id": "units-speed-denominator", "concept": "Rates and density", "prerequisites": [ "Factors and prefixes" ], "prompt": "A cart moves steadily at 54 km/h. Which speed in m/s is equivalent? Use 1 km = 1000 m and 1 h = 3600 s.", "options": [ "194.4 m/s", "15 m/s", "54000 m/s", "0.015 m/s" ], "answer": 1, "explanation": "Convert both the numerator and the denominator: 54 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 15 m/s. The h cancels because the conversion places h in the numerator. The numerical factor is 1000/3600, not 3600/1000. Changing only kilometres or only hours leaves a missing factor.", "transfer": "A belt moves at a constant 1.25 m/s without stopping. How many minutes does a marked point take to travel 750 m, and what is the belt's speed in km/h? Show the unit cancellations." }, { "id": "units-density-two-scales", "concept": "Rates and density", "prerequisites": [ "Factors and prefixes", "Area and volume" ], "prompt": "A uniform fictional fluid has mass density 1.2 g/mL. What is the equivalent density in kg/m^3? Use 1 g = 0.001 kg and 1 mL = 0.000001 m^3.", "options": [ "0.0012 kg/m^3", "1.2 kg/m^3", "1200 kg/m^3", "1200000 kg/m^3" ], "answer": 2, "explanation": "Density is mass divided by volume. Substitute both unit sizes: 1.2 g/mL = 1.2 × 0.001 kg / (0.000001 m^3) = 1200 kg/m^3. The net factor is 1000 because the denominator shrinks by a factor of one million while the numerator shrinks by one thousand. The fluid's density has not physically changed.", "transfer": "A sealed sample occupies 0.25 L and retains the stated density of 1.2 g/mL. Find its mass in kg. Solve once using g/mL and once using kg/m^3, and compare the results." }, { "id": "units-flow-time", "concept": "Rates and density", "prerequisites": [ "Factors and prefixes", "Area and volume" ], "prompt": "A pump delivers a constant 2.4 L/min into an empty tank with no leaks. What is that flow rate in m^3/h? Use 1 L = 0.001 m^3 and 1 h = 60 min.", "options": [ "0.144 m^3/h", "0.00004 m^3/h", "144 m^3/h", "0.0024 m^3/h" ], "answer": 0, "explanation": "2.4 L/min × (0.001 m^3 / 1 L) × (60 min / 1 h) = 0.144 m^3/h. One hour contains 60 delivery intervals of one minute, so the per-hour numerical value grows by 60 before the volume-unit conversion. Dividing by 60 uses the time factor in the wrong direction; ignoring either conversion leaves a factor missing.", "transfer": "At this same constant flow rate, how many mL enter the tank in 25 s? Assume there are still no leaks. Write the calculation so that both L and min cancel." }, { "id": "units-power-dimensions", "concept": "Dimensions and models", "prerequisites": [ "Rates and density" ], "prompt": "On a constant-mass object, a constant net force F acts parallel to the motion. Use F = mass × acceleration, work E = F × displacement, and average power P = E / elapsed time. Acceleration has unit m/s^2. Which base-unit expression represents P?", "options": [ "kg m/s^2", "kg m^2/s^2", "kg m^2/s", "kg m^2/s^3" ], "answer": 3, "explanation": "Force has units kg × m/s^2 = kg m/s^2, called N. Multiplying by distance gives energy units kg m^2/s^2, called J. Dividing that energy by time gives power units kg m^2/s^3, called W. Thus 1 W = 1 J/s. A force unit, an energy unit, or an energy unit multiplied by time cannot replace a power unit.", "transfer": "A constant force of 6 N acts parallel to a displacement of 2.5 m over 3 s. Find the energy transferred in J and the average power in W, and expand both results in kg, m and s." }, { "id": "units-pressure-dimensions", "concept": "Dimensions and models", "prerequisites": [ "Area and volume", "Rates and density" ], "prompt": "For a uniform normal force F over area A, pressure is p = F/A. Given 1 N = 1 kg m/s^2 and 1 Pa = 1 N/m^2, which expression equals 1 Pa?", "options": [ "1 kg m/s^2", "1 kg m^2/s^2", "1 kg/(m s^2)", "1 kg/(m^2 s^2)" ], "answer": 2, "explanation": "Divide the force unit by the area unit: (kg m/s^2)/m^2 = kg m^(1−2)/s^2 = kg/(m s^2). Therefore 1 Pa = 1 kg m^−1 s^−2. The other expressions either keep force units, use energy units, or subtract one too many powers of length.", "transfer": "A uniform normal force of 180 N acts over 300 cm^2. Calculate the pressure in Pa and in kPa. Explain how the area conversion enters the calculation." }, { "id": "units-dimensions-not-proof", "concept": "Dimensions and models", "prerequisites": [ "Rates and density" ], "prompt": "An object moves in one dimension from rest with constant positive acceleration a for elapsed time t > 0. A student proposes displacement x = a t^2. What does dimensional analysis ALONE allow you to conclude?", "options": [ "The dimensions match, so the proposed physical law must be correct.", "Both sides have length dimension, but the check cannot establish the formula or determine its numerical coefficient.", "The expression is inconsistent because a t^2 has velocity dimension.", "Dimensional analysis determines the coefficient 1/2 without any further physical reasoning." ], "answer": 1, "explanation": "Acceleration has dimension length/time^2, so a t^2 has length dimension, just like displacement. This necessary check cannot determine a dimensionless coefficient. For the stated constant-acceleration motion from rest, velocity rises linearly from 0 to a t; its average is a t/2, giving displacement a t^2/2. That additional motion reasoning shows why a dimensionally consistent proposal can still be wrong.", "transfer": "For the same motion, compare x = a t and x = 3 a t^2. Which proposal can be discarded by dimensions alone? Use velocity-versus-time reasoning to assess the coefficient in the other proposal." } ] }