Dimensional Analysis Flashcards: Equation Checks & Coefficient Units
Practice 48 short physics prompts: check equation dimensions, derive coefficient units, judge function inputs, and recognize what a dimensional pass can establish.
دربارهٔ این دسته
A missing power of time can survive several lines of algebra. These 48 dimensional analysis flashcards give you short scalar physics proposals to check before you trust an equation.
The cards practice six notation and scope decisions, ten complete-term equation checks, fourteen unknown coefficient calculations, eight function-input tasks, six constrained repairs, and four limits of dimensional evidence. Each equation-check prompt maps a fully defined proposal to a dimensional verdict and a short ledger. Each coefficient prompt asks for one coefficient’s mass, length, and time powers together with a coherent SI unit. Function prompts check exp, ln, sine, and cosine inputs under the usual dimensionless-argument and radian convention; logarithm prompts also state the real-domain conditions.
Repairs ask you to change one specified exponent, unit, ledger entry, or function argument. They repair dimensional consistency without claiming to recover a unique physical law. The limit cards ask what dimensions can establish about numerical factors, signs, real models, and undefined symbols.
You should already know the usual mechanics quantities and basic algebra. The deck leaves out unit-name lookup tables, prefixes, numerical conversions, medical calculations, full-course or exam coverage, calculus derivations, Buckingham Pi, and similarity analysis. It does not automatically reverse every expression: deriving a coefficient and explaining a mismatch are the useful retrieval tasks here.
The sequence establishes notation and complete-term checks first, then mixes coefficient and function tasks before constrained repairs and evidence limits. Closely related prompts stay at least three unrelated cards apart. Long-term spacing belongs to the review scheduler.
The underlying facts were checked against NIST SP 330 §2, Durham’s dimensional-analysis guidance, and Oxford’s calculations-with-units guidance. All trial expressions, calculations, wording, and arrangement were independently authored; no source exercises, competitor cards, or examination material were copied.
AI tools drafted and sequenced the cards and generated the decorative cover. Every final front and back was manually read and solved during author review; the cover was checked at full size and across anticipated crops. The original questions, answers, organization, metadata, and generated cover are shared under CC0 1.0 to the extent applicable rights exist. The scientific facts remain common knowledge, and the cited sources retain their own rights. This is independent general physics practice, with no curriculum or examination alignment claimed.
کارتهای این دسته
کارت ۱
پرسش
In mechanics dimensional notation, what do M, L, T and the brackets [q] mean for a quantity q?
پاسخ
M denotes mass, L length, and T time; [q] denotes the dimension of q. These symbols describe dimensions, not chosen measurement units.
کارت ۲
پرسش
A displacement x is measured in metres or centimetres. What stays the same in its dimensional description when the unit changes?
پاسخ
Its dimension stays L (length). Metres and centimetres are different units for the same dimension; the numerical value changes with the unit.
کارت ۳
پرسش
For nonzero scalar quantities q and r, how do you obtain the dimensions of qr and q/r from [q] and [r]?
پاسخ
Multiply dimensions for qr and divide them for q/r. Add base-dimension exponents in a product and subtract them in a quotient.
کارت ۴
پرسش
For a scalar equation y = a + b, where y, a, and b are physical quantities, what dimensional condition must hold for all three complete terms?
پاسخ
[y] = [a] = [b]. Check each complete addend, not just a product formed from the whole right-hand side. Matching dimensions is necessary, but does not establish physical validity.
کارت ۵
پرسش
Does the scalar proposal x = uv/a + 0.3at² pass a dimensional check? Here x is displacement, u and v velocities along one axis, a nonzero acceleration, and t elapsed time.
پاسخ
Passes. [uv/a] = (LT⁻¹)²/(LT⁻²) = L and [at²] = (LT⁻²)T² = L, matching [x]. The numerical factor 0.3 is dimensionless; this does not establish a motion law.
کارت ۶
پرسش
A coefficient k stays constant during an experiment. Does that fact alone tell you whether k is dimensionless?
پاسخ
No. Being constant means its value does not change in that setting. Its dimensions must come from its definition or the equation in which it appears.
کارت ۷
پرسش
Does the scalar proposal F = ρAv² pass a dimensional check? Here F is force magnitude, ρ mass density, A area, and v speed.
پاسخ
Passes. [ρAv²] = (ML⁻³)L²(L²T⁻²) = MLT⁻², matching force. The check does not determine a numerical coefficient or establish a force model.
کارت ۸
پرسش
Two quantities have the same dimensions. Is that enough to establish that adding them is physically meaningful?
پاسخ
No. Their physical meaning and scalar or vector character must also fit. Matching dimensions is a necessary condition for an allowed sum, not permission to add arbitrary quantities.
کارت ۹
پرسش
Does the scalar proposal E = mv + Fx pass a dimensional check? Here E is energy, m mass, v speed, F force magnitude, and x distance.
پاسخ
Fails. [mv] = MLT⁻¹, while [Fx] = ML²T⁻² and [E] = ML²T⁻². The mv addend has the wrong dimensions for this sum.
کارت ۱۰
پرسش
In the scalar proposal F = bv, F is force magnitude and v speed. What dimensions and coherent SI unit must b have? Use M, L, T for mass, length, time.
پاسخ
[b] = MT⁻¹; coherent SI unit kg·s⁻¹. Divide force by speed: (MLT⁻²)/(LT⁻¹) = MT⁻¹. Substitution gives (MT⁻¹)(LT⁻¹) = MLT⁻².
کارت ۱۱
پرسش
Does the scalar proposal P = Fv + E/t pass a dimensional check? Here P is power, F force magnitude, v speed, E energy, and t nonzero elapsed time.
پاسخ
Passes. [Fv] = (MLT⁻²)(LT⁻¹) = ML²T⁻³ and [E/t] = ML²T⁻³, matching power. Each complete addend passes.
کارت ۱۲
پرسش
In the scalar proposal E = cx³, E is energy and x displacement. What dimensions and coherent SI unit must c have? Use M, L, T for mass, length, time.
پاسخ
[c] = ML⁻¹T⁻²; coherent SI unit kg·m⁻¹·s⁻². Divide energy by length cubed: ML²T⁻²/L³ = ML⁻¹T⁻². Substitution gives (ML⁻¹T⁻²)L³ = ML²T⁻².
کارت ۱۳
پرسش
Is the argument in exp(t/τ) dimensionally valid under the usual dimensionless-argument convention? Here t is elapsed time and τ a positive time scale.
پاسخ
Yes. [t/τ] = T/T = 1, so the argument is dimensionless. The exponential itself is also dimensionless; τ > 0 keeps the denominator nonzero.
کارت ۱۴
پرسش
Does the scalar proposal v² = u² + ar² pass a dimensional check? Here v and u are speeds, a acceleration, and r distance.
پاسخ
Fails. [v²] = [u²] = L²T⁻², but [ar²] = (LT⁻²)L² = L³T⁻². The ar² addend has an extra power of length.
کارت ۱۵
پرسش
In the scalar proposal p = dv², p is pressure and v speed. What dimensions and coherent SI unit must d have? Use M, L, T for mass, length, time.
پاسخ
[d] = ML⁻³; coherent SI unit kg·m⁻³. Divide pressure by speed squared: (ML⁻¹T⁻²)/(L²T⁻²) = ML⁻³. Substitution gives (ML⁻³)(L²T⁻²) = ML⁻¹T⁻².
کارت ۱۶
پرسش
In the scalar proposal x = A + Bt + Ct², x is displacement and t elapsed time. What dimensions and coherent SI unit must B have? A and C have the dimensions needed for their terms. Use M, L, T for mass, length, time.
پاسخ
[B] = LT⁻¹; coherent SI unit m·s⁻¹. The complete term Bt must have dimension L, so [B] = L/T. Substitution gives (LT⁻¹)T = L.
کارت ۱۷
پرسش
Is ln(r/r₀) dimensionally valid, and is it defined as a real number? Here r and r₀ are positive lengths; use the usual dimensionless-argument convention.
پاسخ
Yes to both. [r/r₀] = L/L = 1, and r/r₀ > 0. The ratio is dimensionless and meets the real logarithm’s positive-argument requirement.
کارت ۱۸
پرسش
Does the scalar proposal a = (v − u)/t pass a dimensional check? Here a is acceleration, v and u velocities along one axis, and t nonzero elapsed time.
پاسخ
Passes. The difference v − u has dimension LT⁻¹; dividing by t gives LT⁻², matching acceleration. Both terms inside the difference have the same dimensions.
کارت ۱۹
پرسش
In the scalar proposal F = k/r², F is force magnitude and r a positive distance. What dimensions and coherent SI unit must k have? Use M, L, T for mass, length, time.
پاسخ
[k] = ML³T⁻²; coherent SI unit kg·m³·s⁻². Multiply force by r²: (MLT⁻²)L² = ML³T⁻². Substitution gives ML³T⁻²/L² = MLT⁻².
کارت ۲۰
پرسش
What dimensions and coherent SI unit must k have in exp(−kt²)? Here t is elapsed time and k a scalar coefficient; use the usual dimensionless-argument convention and M, L, T for mass, length, time.
پاسخ
[k] = T⁻²; coherent SI unit s⁻². The argument requires [k]T² = 1. The minus sign changes neither dimensions nor the real exponential’s domain.
کارت ۲۱
پرسش
Does the scalar proposal p = F/A + ρv pass a dimensional check? Here p is pressure, F force magnitude, A positive area, ρ mass density, and v speed.
پاسخ
Fails. [F/A] = ML⁻¹T⁻², matching pressure, but [ρv] = (ML⁻³)(LT⁻¹) = ML⁻²T⁻¹. The second addend is incompatible.
کارت ۲۲
پرسش
In the scalar proposal v = h√r, v is speed and r a positive distance. What dimensions and coherent SI unit must h have? Use M, L, T for mass, length, time.
پاسخ
[h] = L¹ᐟ²T⁻¹; coherent SI unit m¹ᐟ²·s⁻¹. Divide speed by √r: LT⁻¹/L¹ᐟ² = L¹ᐟ²T⁻¹. Substitution gives (L¹ᐟ²T⁻¹)L¹ᐟ² = LT⁻¹.
کارت ۲۳
پرسش
Is sin(vt) dimensionally valid under the usual dimensionless-argument and radian convention? Here v is speed and t elapsed time.
پاسخ
No. [vt] = (LT⁻¹)T = L, so the argument is a length rather than dimensionless. A sine output cannot repair an invalid input.
کارت ۲۴
پرسش
In the scalar proposal E = jm², E is energy and m mass. What dimensions and coherent SI unit must j have? Use M, L, T for mass, length, time.
پاسخ
[j] = M⁻¹L²T⁻²; coherent SI unit kg⁻¹·m²·s⁻². Divide energy by mass squared: ML²T⁻²/M² = M⁻¹L²T⁻². Substitution gives (M⁻¹L²T⁻²)M² = ML²T⁻².
کارت ۲۵
پرسش
Does the scalar proposal E = F/t pass a dimensional check? Here E is energy, F force magnitude, and t nonzero elapsed time.
پاسخ
Fails. [F/t] = MLT⁻³, while energy has dimension ML²T⁻². Both the length and time exponents differ.
کارت ۲۶
پرسش
In the scalar proposal F = qat, F is force magnitude, a acceleration, and t elapsed time. What dimensions and coherent SI unit must q have? Use M, L, T for mass, length, time.
پاسخ
[q] = MT⁻¹; coherent SI unit kg·s⁻¹. [at] = LT⁻¹, so [q] = MLT⁻²/(LT⁻¹) = MT⁻¹. Substitution gives (MT⁻¹)(LT⁻¹) = MLT⁻².
کارت ۲۷
پرسش
What dimensions and coherent SI unit must h have in ln(hp)? Here p is positive pressure and h a positive scalar coefficient; use the usual dimensionless-argument convention and M, L, T for mass, length, time.
پاسخ
[h] = M⁻¹LT²; coherent SI unit kg⁻¹·m·s². [h][p] = 1 requires the inverse of ML⁻¹T⁻². The positive values also ensure hp > 0 for a real logarithm.
کارت ۲۸
پرسش
In the scalar proposal p = s/r, p is pressure and r a positive distance. What dimensions and coherent SI unit must s have? Use M, L, T for mass, length, time.
پاسخ
[s] = MT⁻²; coherent SI unit kg·s⁻². Multiply pressure by length: (ML⁻¹T⁻²)L = MT⁻². Substitution gives MT⁻²/L = ML⁻¹T⁻².
کارت ۲۹
پرسش
Does the scalar proposal ρ = m/r² pass a dimensional check? Here ρ is mass density, m mass, and r a positive distance.
پاسخ
Fails. [m/r²] = ML⁻², while mass density has dimension ML⁻³. The denominator supplies area dimensions instead of volume dimensions.
کارت ۳۰
پرسش
In the scalar proposal x = A + Bt + Ct², x is displacement and t elapsed time. What dimensions and coherent SI unit must C have? A and B have the dimensions needed for their terms. Use M, L, T for mass, length, time.
پاسخ
[C] = LT⁻²; coherent SI unit m·s⁻². The complete term Ct² must have dimension L, so [C] = L/T². Substitution gives (LT⁻²)T² = L.
کارت ۳۱
پرسش
Is cos(at/v) dimensionally valid under the usual dimensionless-argument and radian convention? Here a is acceleration, t elapsed time, and v nonzero speed.
پاسخ
Yes. [at/v] = (LT⁻²)T/(LT⁻¹) = 1. The argument is dimensionless, and v ≠ 0 keeps the quotient defined.
کارت ۳۲
پرسش
In the scalar proposal E = wv³, E is energy and v speed. What dimensions and coherent SI unit must w have? Use M, L, T for mass, length, time.
پاسخ
[w] = ML⁻¹T; coherent SI unit kg·m⁻¹·s. Divide energy by speed cubed: (ML²T⁻²)/(L³T⁻³) = ML⁻¹T. Substitution gives (ML⁻¹T)(L³T⁻³) = ML²T⁻².
کارت ۳۳
پرسش
Does the scalar proposal t = r/v + v/a pass a dimensional check? Here t is elapsed time, r distance, v nonzero speed, and a nonzero acceleration.
پاسخ
Passes. [r/v] = L/(LT⁻¹) = T and [v/a] = (LT⁻¹)/(LT⁻²) = T, matching elapsed time. This does not establish the proposal as a motion law.
کارت ۳۴
پرسش
What dimensions and coherent SI unit must q have in sin(qr²)? Here r is distance and q a scalar coefficient; use the usual dimensionless-argument and radian convention and M, L, T for mass, length, time.
پاسخ
[q] = L⁻²; coherent SI unit m⁻². The argument requires [q]L² = 1. With this choice qr² can serve as a dimensionless radian argument.
کارت ۳۵
پرسش
In the scalar proposal P = zr², P is power and r distance. What dimensions and coherent SI unit must z have? Use M, L, T for mass, length, time.
پاسخ
[z] = MT⁻³; coherent SI unit kg·s⁻³. Divide power by r²: ML²T⁻³/L² = MT⁻³. Substitution gives (MT⁻³)L² = ML²T⁻³.
کارت ۳۶
پرسش
The scalar proposal x = ut + a²t/u fails a dimensional check. Here x is displacement, u nonzero velocity, a acceleration, and t elapsed time. Change only the exponent of t in the second addend to make that term match x. What exponent is required?
پاسخ
The exponent must be 3: use a²t³/u. [a²tⁿ/u] = (L²T⁻⁴)Tⁿ/(LT⁻¹) = LTⁿ⁻³; matching L requires n − 3 = 0. This repairs dimensions only, not the motion model.
کارت ۳۷
پرسش
In the scalar proposal v = n/(ρr²), v is speed, ρ positive mass density, and r a positive distance. What dimensions and coherent SI unit must n have? Use M, L, T for mass, length, time.
پاسخ
[n] = MT⁻¹; coherent SI unit kg·s⁻¹. [ρr²] = ML⁻¹, so [n] = (LT⁻¹)(ML⁻¹) = MT⁻¹. Substitution gives MT⁻¹/(ML⁻¹) = LT⁻¹.
کارت ۳۸
پرسش
In the scalar proposal E = αmv², E is energy, m mass, v speed, and α a dimensionless number. Can a dimensional check determine α’s numerical value?
پاسخ
No. [mv²] = ML²T⁻² for every dimensionless α. A numerical value needs additional physical or experimental information.
کارت ۳۹
پرسش
What dimensions and coherent SI unit must j have in cos(jmv)? Here m is mass, v speed, and j a scalar coefficient; use the usual dimensionless-argument and radian convention and M, L, T for mass, length, time.
پاسخ
[j] = M⁻¹L⁻¹T; coherent SI unit kg⁻¹·m⁻¹·s. [mv] = MLT⁻¹, so multiplying by its inverse makes jmv dimensionless.
کارت ۴۰
پرسش
For the scalar proposal F = cv², a unit ledger assigns c the unit kg·s⁻¹. Here F is force magnitude and v speed. Replace only c’s unit with the coherent SI unit required by this proposal.
پاسخ
Use kg·m⁻¹. Force divided by speed squared gives (kg·m·s⁻²)/(m²·s⁻²) = kg·m⁻¹; the stated kg·s⁻¹ would instead give kg·m²·s⁻³ for cv².
کارت ۴۱
پرسش
In the scalar proposal t = g√(m/r), t is elapsed time, m positive mass, and r a positive distance. What dimensions and coherent SI unit must g have? Use M, L, T for mass, length, time.
پاسخ
[g] = M⁻¹ᐟ²L¹ᐟ²T; coherent SI unit kg⁻¹ᐟ²·m¹ᐟ²·s. [√(m/r)] = M¹ᐟ²L⁻¹ᐟ². Multiplying by the stated [g] leaves T.
کارت ۴۲
پرسش
Two scalar proposals differ only by a plus sign versus a minus sign between complete terms that all have the same dimensions. Can dimensional analysis select the physically correct sign?
پاسخ
No. Changing a plus sign to a minus sign does not change the dimensions. The sign needs information about the physical model and the definitions or directions of the quantities.
کارت ۴۳
پرسش
The proposed real logarithm ln(r) uses a positive length r. With a fixed positive reference length r₀, repair only its argument using one quotient of the two lengths. The corrected logarithm must increase as r increases.
پاسخ
Use ln(r/r₀). The argument has dimension L/L = 1 and is positive. The reference length makes the logarithm meaningful without choosing a unit-dependent raw length value. With r₀ fixed, it increases with r; ln(r₀/r) would decrease.
کارت ۴۴
پرسش
A ledger claims [Fr²] = ML²T⁻², with F force magnitude, r distance, and M, L, T mass, length, time. Keep Fr² unchanged and correct only the dimensional ledger.
پاسخ
[Fr²] = ML³T⁻². Force contributes MLT⁻² and r² contributes L², so the length exponents add to 3. The expression therefore does not have energy dimensions.
کارت ۴۵
پرسش
A proposed scalar physics equation passes all term and function-input dimensional checks. What can you conclude about whether it describes a real system?
پاسخ
Only that these dimensional checks found no inconsistency. A pass does not establish the model, assumptions, dependence on variables, numerical factors, or agreement with measurements.
کارت ۴۶
پرسش
In the scalar proposal p = dm/r³, a ledger calls d dimensionless. Here p is pressure, m mass, and r a positive distance. Replace only that entry with the dimensions and coherent SI unit required for d; use M, L, T for mass, length, time.
پاسخ
[d] = L²T⁻²; coherent SI unit m²·s⁻². Since [m/r³] = ML⁻³, [d] = (ML⁻¹T⁻²)/(ML⁻³) = L²T⁻². Substitution gives (L²T⁻²)(ML⁻³) = ML⁻¹T⁻².
کارت ۴۷
پرسش
In the scalar proposal t = r/v + u, t is elapsed time, r distance, and v and u nonzero speeds. A nonzero acceleration a is available. Replace only the addend u with u/aⁿ. Which exponent n makes that addend match time?
پاسخ
The exponent must be 1: use u/a. [u/aⁿ] = L¹⁻ⁿT⁻¹⁺²ⁿ; n = 1 removes length and leaves T. This repairs dimensions only; it does not establish a motion law.
کارت ۴۸
پرسش
You see the proposed equation Q = ab, but the author supplies no definitions or dimensions for Q, a, or b. What information is missing before you can judge dimensional consistency?
پاسخ
The meanings or dimensions of Q, a, and b. Letters alone do not determine dimensions; once supplied, the check compares [Q] with [a][b].
۴۸ کارت
Dimensional Analysis Flashcards: Equation Checks & Coefficient Units
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