Dimensional Analysis Flashcards: Equation Checks & Coefficient Units

Question 1

In mechanics dimensional notation, what do M, L, T and the brackets [q] mean for a quantity q?

Accepted Answer

M denotes mass, L length, and T time; [q] denotes the dimension of q. These symbols describe dimensions, not chosen measurement units.

Question 2

A displacement x is measured in metres or centimetres. What stays the same in its dimensional description when the unit changes?

Accepted Answer

Its dimension stays L (length). Metres and centimetres are different units for the same dimension; the numerical value changes with the unit.

Question 3

For nonzero scalar quantities q and r, how do you obtain the dimensions of qr and q/r from [q] and [r]?

Accepted Answer

Multiply dimensions for qr and divide them for q/r. Add base-dimension exponents in a product and subtract them in a quotient.

Question 4

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?

Accepted Answer

[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.

Question 5

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.

Accepted Answer

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.

Question 6

A coefficient k stays constant during an experiment. Does that fact alone tell you whether k is dimensionless?

Accepted Answer

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.

Question 7

Does the scalar proposal F = ρAv² pass a dimensional check? Here F is force magnitude, ρ mass density, A area, and v speed.

Accepted Answer

Passes. [ρAv²] = (ML⁻³)L²(L²T⁻²) = MLT⁻², matching force. The check does not determine a numerical coefficient or establish a force model.

Question 8

Two quantities have the same dimensions. Is that enough to establish that adding them is physically meaningful?

Accepted Answer

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.

Question 9

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.

Accepted Answer

Fails. [mv] = MLT⁻¹, while [Fx] = ML²T⁻² and [E] = ML²T⁻². The mv addend has the wrong dimensions for this sum.

Question 10

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.

Accepted Answer

[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.

Question 11

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.

Accepted Answer

Passes. [Fv] = (MLT⁻²)(LT⁻¹) = ML²T⁻³ and [E/t] = ML²T⁻³, matching power. Each complete addend passes.

Question 12

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.

Accepted Answer

[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.

Question 13

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.

Accepted Answer

Yes. [t/τ] = T/T = 1, so the argument is dimensionless. The exponential itself is also dimensionless; τ > 0 keeps the denominator nonzero.

Question 14

Does the scalar proposal v² = u² + ar² pass a dimensional check? Here v and u are speeds, a acceleration, and r distance.

Accepted Answer

Fails. [v²] = [u²] = L²T⁻², but [ar²] = (LT⁻²)L² = L³T⁻². The ar² addend has an extra power of length.

Question 15

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.

Accepted Answer

[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.

Question 16

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.

Accepted Answer

[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.

Question 17

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.

Accepted Answer

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.

Question 18

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.

Accepted Answer

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.

Question 19

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.

Accepted Answer

[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.

Question 20

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.

Accepted Answer

[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.

Question 21

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.

Accepted Answer

Fails. [F/A] = ML⁻¹T⁻², matching pressure, but [ρv] = (ML⁻³)(LT⁻¹) = ML⁻²T⁻¹. The second addend is incompatible.

Question 22

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.

Accepted Answer

[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.

Question 23

Is sin(vt) dimensionally valid under the usual dimensionless-argument and radian convention? Here v is speed and t elapsed time.

Accepted Answer

No. [vt] = (LT⁻¹)T = L, so the argument is a length rather than dimensionless. A sine output cannot repair an invalid input.

Question 24

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.

Accepted Answer

[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.

Question 25

Does the scalar proposal E = F/t pass a dimensional check? Here E is energy, F force magnitude, and t nonzero elapsed time.

Accepted Answer

Fails. [F/t] = MLT⁻³, while energy has dimension ML²T⁻². Both the length and time exponents differ.

Question 26

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.

Accepted Answer

[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.

Question 27

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.

Accepted Answer

[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.

Question 28

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.

Accepted Answer

[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.

Question 29

Does the scalar proposal ρ = m/r² pass a dimensional check? Here ρ is mass density, m mass, and r a positive distance.

Accepted Answer

Fails. [m/r²] = ML⁻², while mass density has dimension ML⁻³. The denominator supplies area dimensions instead of volume dimensions.

Question 30

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.

Accepted Answer

[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.

Question 31

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.

Accepted Answer

Yes. [at/v] = (LT⁻²)T/(LT⁻¹) = 1. The argument is dimensionless, and v ≠ 0 keeps the quotient defined.

Question 32

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.

Accepted Answer

[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.

Question 33

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.

Accepted Answer

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.

Question 34

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.

Accepted Answer

[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.

Question 35

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.

Accepted Answer

[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.

Question 36

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?

Accepted Answer

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.

Question 37

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.

Accepted Answer

[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.

Question 38

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?

Accepted Answer

No. [mv²] = ML²T⁻² for every dimensionless α. A numerical value needs additional physical or experimental information.

Question 39

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.

Accepted Answer

[j] = M⁻¹L⁻¹T; coherent SI unit kg⁻¹·m⁻¹·s. [mv] = MLT⁻¹, so multiplying by its inverse makes jmv dimensionless.

Question 40

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.

Accepted Answer

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.

Question 41

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.

Accepted Answer

[g] = M⁻¹ᐟ²L¹ᐟ²T; coherent SI unit kg⁻¹ᐟ²·m¹ᐟ²·s. [√(m/r)] = M¹ᐟ²L⁻¹ᐟ². Multiplying by the stated [g] leaves T.

Question 42

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?

Accepted Answer

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.

Question 43

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.

Accepted Answer

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.

Question 44

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.

Accepted Answer

[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.

Question 45

A proposed scalar physics equation passes all term and function-input dimensional checks. What can you conclude about whether it describes a real system?

Accepted Answer

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.

Question 46

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.

Accepted Answer

[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.

Question 47

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?

Accepted Answer

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.

Question 48

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?

Accepted Answer

The meanings or dimensions of Q, a, and b. Letters alone do not determine dimensions; once supplied, the check compares [Q] with [a][b].

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Dimensional Analysis Flashcards: Equation Checks & Coefficient Units