Dimensional Analysis Dimensional Analysis Dimensional Analysis Traps Trap Scenario Student Misconception Correct Concept Example Case Like adds to like, angles have no dimension, but constants and powers need your full attention. Different physical quantities can be added or subtracted if they appear in the same equation. Principle of Homogeneity states only quantities with identical dimensions can be added or subtracted. In the equation v = a + bt , both [a] and [bt] must have the dimensions of velocity [LT -1 ] . Arguments of trigonometric, logarithmic, and exponential functions can have dimensions. Arguments of , , , , and e must be dimensionless ( [M 0 L 0 T 0] ). In y = A (kx - t) , both kx and t must be dimensionless, so [k] = [L -1 ] and [ ] = [T -1 ] . Every constant in physics is a dimensionless numerical value. Many physical constants ( G , h , k B , ) possess specific dimensions and units. The Universal Gravitational Constant G has dimensions [M -1 L 3 T -2 ] , not just a number. A dimensionally correct equation is always physically correct. Dimensional consistency is a necessary but not sufficient condition for physical correctness. The equation s = ut + at 2 is dimensionally consistent ( [L] = [L] ) but physically incorrect because the factor 1/2 is missing. Quantities that are dimensionless are also unitless. Quantities like plane angle and solid angle are dimensionless but possess units ( rad and sr ). The angle = arc / radius has dimensions [L 0 L 0 T 0] but is measured in radians. Numerical constants like 1/2 , , or e can be derived using dimensional analysis. Dimensional analysis cannot determine dimensionless scaling factors or numerical constants. The 2 in the period of a pendulum T = 2 l/g cannot be found using only M, L, T analysis. The power (exponent) in an expression can have dimensions. Powers must always be dimensionless, as they represent the number of times a base is multiplied. In the expression N = N 0 e - t , the product t must be [M 0 L 0 T 0] , making [ ] = [T -1 ] . Dimensional analysis can derive formulas involving more than three independent variables. If a quantity depends on more than three independent variables, unique powers cannot be solved using only M, L, T . A formula involving M, L, T and a fourth variable like Charge Q requires four independent equations. Specific Heat and Latent Heat have the same dimensions as Energy. Specific Heat involves temperature change ( ) and Latent Heat involves phase change without temperature change. Specific Heat s is [L 2 T -2 K -1 ] , whereas Latent Heat L is [L 2 T -2 ] . Resistance, Inductance, and Capacitance have simple, unrelated dimensions. These quantities are interlinked through time constants; specific ratios yield the dimension of time [T] . The combinations RC , L/R , and LC all have the dimension of time [T] . The ratio of any two similar-looking variables is dimensionless. Ratios are only dimensionless if the numerator and denominator have identical units. Velocity gradient (dv/dx) is [T -1 ] , while Pressure gradient (dP/dx) is [ML -2 T -2 ] ; they are not dimensionless ratios. Thermal Conductivity (K) depends only on Energy and Length. Thermal conductivity involves Energy, Length, Time, and Temperature difference. The dimensions of Thermal Conductivity K are [MLT -3 K -1 ] derived from dQ/dt = KA(dT/dx) . Permittivity ( 0) and Permeability ( 0) are dimensionless constants of vacuum. They have complex dimensions that relate to the speed of light and impedance. The product [μ 0 ε 0] has dimensions of [L -2 T 2] , which is the reciprocal of velocity squared (1/c 2) . Force and Surface Tension share the same dimensions because they both involve 'Tension'. Surface Tension is force per unit length, whereas Force is mass times acceleration. Force is [MLT -2 ] but Surface Tension is [MT -2 ] . Frequency and Angular Velocity are dimensionally different because one is rotations and other is radians. Both represent the 'per second' rate and share the same dimensions. Frequency ν and Angular Velocity ω both have dimensions of [T -1 ] . dimensional analysis traps Dimensional analysis is a language that tells you how a physical quantity is built from the base ingredients of nature: mass (M), length (L), time (T), electric current (I), temperature (Θ), amount of substance (N), and luminous intensity (J). Every measured quantity, from speed to energy, can be expressed as a product of powers of these base dimensions. Why care? Because equations in physics are not just algebraic games; they must make sense in the units world too. If left-hand side and right-hand side measure different kinds of things, the equation is meaningless. This is the principle of homogeneity: all additive terms in a correct physical equation share the same dimensions. Once you learn to see dimensions, you gain a fast way to check formulas, convert between unit systems, and even predict the shape of new relations. For example, before knowing the full theory of pendulum motion, dimensional analysis already reveals that the time period T is proportional to the square root of length L and inversely proportional to the square root of g. Dimensions also protect you from common traps: arguments of trigonometric, exponential, and logarithmic functions must be dimensionless; angles are measured in radians, which are dimensionless; and a mean of measurements has the same dimensions as the measurements themselves. The method is not magic—pure numbers like 2π or dimensionless functions cannot be found by it—but as a first-pass filter, it is quick, rigorous, and incredibly helpful in exams where time is short. remember Think of dimensions as the recipe of a quantity. Two different dishes can weigh the same (same units) yet be made from different ingredients; but if two bowls of curry are to be added together, they must both be curries (same kind, i.e., same dimensions). A property that can be measured and expressed as a number with a unit (e.g., length, time, force). Physical quantity Dimension The nature of a physical quantity in terms of base quantities, represented by exponents of M, L, T, I, Θ, N, J. Dimensional formula An expression showing how a derived quantity depends on base dimensions, e.g., velocity has [ M 0 L 1 T -1 ]. In any valid physical equation, all additive terms must have the same dimensions. Principle of homogeneity A quantity with overall dimensions [ M 0 L 0 T 0 I 0 Θ 0 N 0 J 0 ], e.g., angle in radians, refractive index, strain. Dimensionless quantity The SI base quantities (and symbols for their dimensions) are: mass (M), length (L), time (T), electric current (I), thermodynamic temperature (Θ), amount of substance (N), and luminous intensity (J). Every derived quantity can be reduced to a power product of these. For example, acceleration is change in velocity per unit time, so [a] = [v]/[t] = (L T -1 ) T -1 = L T -2 . Energy combines force and distance: [E] = [F][L] = (M L T -2 ) L = M L 2 T -2 . This dimensional view is independent of whether you use SI or cgs units; it captures the essence of the quantity rather than the label of the unit. Base quantities and dimension symbols Mass → M Length → L Time → T Electric current → I Thermodynamic temperature → Θ Amount of substance → N Luminous intensity → J To build a dimensional formula, rewrite the quantity as products or ratios of simpler quantities whose dimensions are known, then collect exponents of M, L, T, I, Θ, N, J. Always cross-check by tracking units alongside dimensions: if you know the SI unit of a quantity (for instance, pressure is N m -2 ), you can also reach its dimensions by substituting the base units (N = kg m s -2 ) and then mapping kg → M, m → L, s → T. SI Units and Formula Physical Quantity Symbol SI Unit Dimensional Formula [M a L b T c] Powers of M (mass), L (length), T (time) only. Decompose from F = ma, W = F·d, P = W/t, p = mv, [η] from Stokes/Poiseuille, Y = stress / strain, G from F = Gm₁m₂/r², h from E = hν. Length metre ( m ) [M 0 L 1 T 0] Mass kilogram ( kg ) [M 1 L 0 T 0] Time second ( s ) [M 0 L 0 T 1] Volume cubic metre ( m 3 ) [M 0 L 3 T 0] Density kg m -3 [M 1 L -3 T 0] Velocity m s -1 [M 0 L 1 T -1 ] Acceleration m s -2 [M 0 L 1 T -2 ] Linear Momentum kg m s -1 [M 1 L 1 T -1 ] Force newton ( N ) [M 1 L 1 T -2 ] Impulse newton-second ( N s ) [M 1 L 1 T -1 ] Work joule ( J ) [M 1 L 2 T -2 ] Energy joule ( J ) [M 1 L 2 T -2 ] Torque newton-metre ( N m ) [M 1 L 2 T -2 ] Power watt ( W ) [M 1 L 2 T -3 ] Pressure pascal ( Pa ) [M 1 L -1 T -2 ] Stress pascal ( Pa ) [M 1 L -1 T -2 ] Strain dimensionless [M 0 L 0 T 0] Young's Modulus pascal ( Pa ) [M 1 L -1 T -2 ] Spring / Force Constant N m -1 [M 1 L 0 T -2 ] Frequency , f hertz ( Hz ) [M 0 L 0 T -1 ] Wavelength metre ( m ) [M 0 L 1 T 0] Angular Velocity rad s -1 [M 0 L 0 T -1 ] Angular Acceleration rad s -2 [M 0 L 0 T -2 ] Moment of Inertia kg m 2 [M 1 L 2 T 0] Angular Momentum kg m 2 s -1 [M 1 L 2 T -1 ] Surface Tension N m -1 [M 1 L 0 T -2 ] Coefficient of Viscosity pascal-second ( Pa s ) [M 1 L -1 T -1 ] Universal Gravitational Constant N m 2 kg -2 [M -1 L 3 T -2 ] Planck's Constant joule-second ( J s ) [M 1 L 2 T -1 ] top 20 si units dimensional formulae Quantity SI unit Dimensional formula Velocity v m s -1 M 0 L 1 T -1 Acceleration a m s -2 M 0 L 1 T -2 Force F N = kg m s -2 M 1 L 1 T -2 Energy E, Work W J = N m M 1 L 2 T -2 Power P W = J s -1 M 1 L 2 T -3 Pressure p Pa = N m -2 M 1 L -1 T -2 Homogeneity check is the fastest way to catch impossible equations. For a valid equation involving addition or subtraction, each term must carry the same dimensions. For example, displacement s, initial velocity u, acceleration a, and time t in uniformly accelerated motion: s = ut + (1/2) a t 2 is dimensionally consistent because [s] = L, [ut] = (L T -1 ) T = L, and [a t 2 ] = (L T -2 ) T 2 = L. If any term turns out with a different dimension (say, L T -1 ), the equation cannot be correct in any unit system. All additive terms have dimension of length. Check of s = ut + (1/2) a t 2 Contrast that with wrong forms like s = u + a t 2 , which adds L T -1 (velocity) to L (displacement): the terms do not match. Or v = u + a x, which wrongly adds velocity and acceleration times displacement: [a x] = (L T -2 ) L = L 2 T -2 . Homogeneity instantly flags such mistakes before any numeric substitution. Also note: multiplicative constants (like 1/2 or 2π) are dimensionless and do not affect homogeneity. Boundary of homogeneity: it applies to additive terms. It does not validate correctness of dimensionless constants or functional forms. For example, T ∝ L/g is dimensionally right, but the actual constant is 2π, which dimensional analysis alone cannot find. tip Units can be the same while dimensions differ in another system, but the deeper rule is: terms must have identical dimensions. Homogeneity is unit-system independent. If two terms have the same units in SI, they are always addable. Dimensional analysis is also a reliable way to convert between unit systems by tracking how the base units scale. For example, 1 erg in cgs is 1 g cm 2 s -2 . Since 1 g = 10 -3 kg, 1 cm = 10 -2 m, and 1 s = 1 s, you can rewrite erg in SI base units and then reassemble to joules. The same method works for pressure (dyn cm -2 to pascal), viscosity (poise to Pa s), or energy density (erg cm -3 to J m -3 ). 1 erg = 1 g cm 2 s -2 1 g = 10 -3 kg 1 cm = 10 -2 m 1 s = 1 s 1 J = 1 kg m 2 s -2 Convert 1 erg to joule using dimensional analysis of base units. easy Value of 1 erg in joules A powerful use of dimensions is to predict how a quantity depends on others when you do not know the exact law. Assume a power-law relation, write the dimensions of each variable, and equate exponents. The method fixes the exponents but not the dimensionless constant. This is especially helpful for oscillations and waves, where characteristic times and speeds emerge from just a few parameters. Using dimensional analysis, find how the time period T of a simple pendulum depends on length L and gravitational acceleration g. medium Assume T = k L a g b [T] = T 1 [L] = L 1 [g] = L 1 T -2 Equate the dimensions on both sides: [T] = [L] a [g] b = L a + b T -2b $. Exponents a and b (and the form of T up to a dimensionless constant) Dimensional analysis cannot give numerical constants (like 2, π, 1/2) or the specific functional form if exponentials/trigonometric functions are involved. Also, arguments of sin, cos, exp, and log must be dimensionless; writing sin(ωt) is fine only if ω has dimensions T -1 . neet-alert You can determine the dimensions of physical constants by embedding them in known laws. For example, the gravitational constant G appears in F = G m 1 m 2 / r 2 . Since [F] = M L T -2 and [m] = M, [r] = L, we get [G] = [F] r 2 / m 2 = M -1 L 3 T -2 . Similarly, the Planck constant h has unit joule second, so [h] = M L 2 T -1 ; the gas constant R has [R] = energy per mole per kelvin = M L 2 T -2 N -1 Θ -1. Gravitational constant has dimensions M -1 L 3 T -2 . Dimensions of G Arguments of trigonometric or exponential functions must be pure numbers, so any angular frequency ω in sin(ωt) must carry T -1 to make ωt dimensionless. Likewise, exp(-x/λ) implies x/λ is dimensionless, so λ must have the same dimension as x. This rule is frequently tested: always check the insides of sin, cos, tan, exp, and log for dimensionlessness. Exponents a, b, c (and the form of F up to a constant) Assume F = k η a r b v c [F] = M L T -2 [η] = M L -1 T -1 [r] = L [v] = L T -1 Equate powers of M, L, T on both sides. hard Find the dependence of viscous drag force F on a small sphere moving slowly through a fluid with dynamic viscosity η. Assume F depends only on η, the sphere’s radius r, and its speed v. The arithmetic mean of repeated measurements has the same dimensions as the measured quantity and is the best estimate of the true value under random errors. Arithmetic mean ( a = a n ) Random errors are unbiased and symmetrically distributed about the true value. All measurements refer to the same physical quantity and share the same dimensions. Best estimate minimizes the sum of squared deviations. a = a n For Z formed by product or quotient of A and B, the worst-case relative error in Z is the sum of relative errors in A and B. Z Z = A A + B B Relative error for product/quotient: ( Z Z = A A + B B ) Errors are small compared to measured values (linearization valid). Worst-case (maximum probable) error adds magnitudes of relative errors. A, B, Z are nonzero. If Z = A n, the relative error in Z equals |n| times the relative error in A. Relative error for a power: ( Z Z = |n| , A A ) Small errors permit linear approximation. Worst-case error uses absolute value of exponent n. Z Z = |n| , A A For Z = A ± B, the maximum absolute error is the sum of absolute errors: ΔZ = ΔA + ΔB. Absolute errors are considered in worst-case (extreme) combination. A and B measure the same type of quantity (same dimensions). Absolute error for sum/difference: ( Z = A + B ) Z = A + B Note how dimensional thinking aligns with these data rules: the arithmetic mean has the same dimension as A; adding absolute errors is valid only when A and B share the same dimensions; and relative errors are dimensionless ratios. This harmony is why dimensional analysis sits naturally with error propagation in measurement science. It is okay to take logarithm of a quantity with units as long as you use SI. Logarithm, exponential, and trigonometric functions require dimensionless arguments. Always non-dimensionalize first (e.g., divide by a reference value) before applying log or exp. Angles measured in radians are dimensionless (ratio of arc length to radius). That is why sin(θ) and exp(iθ) are valid only when θ is in radians. tip Limitations of dimensional analysis: (1) It cannot predict pure numbers like 2, π, or √2. (2) It fails when multiple independent dimensionless groups exist (you may get underdetermined exponents). (3) It cannot distinguish between additive combinations that share the same dimensions (e.g., a + b t and a e kt both can yield the same dimensions for certain outputs). (4) It assumes a power-law form; if the true law involves sums of different power-laws or non-algebraic functions, dimensions alone will not fix it. Check if an equation is dimensionally homogeneous Find numerical constants (2, π, 1/2, 6π, etc.) Convert between units and estimate magnitudes Resolve multiple dimensionless groups without extra info Predict exponents in power-law relations Validate non-algebraic forms (e.g., exponential vs sinusoidal) Reveal dimensions of physical constants Prove causation or detailed dynamics What dimensional analysis can do What it cannot do Memory aid for the seven SI base dimension symbols. Make Long Trips In The Night Joyfully → M, L, T, I, Θ, N, J Fractional and negative dimensions commonly appear. For example, frequency f has [ T -1 ]; surface mass density σ (mass per area) has [M L -2 ]; thermal conductivity k has [M L T -3 Θ -1]. Resistivity ρ e has [M L 3 T -3 I -2 ], and permittivity ε 0 has [ M -1 L -3 T 4 I 2 ]. Such exponents carry real physical meaning: they show how a measurement scales when you stretch length or speed up time. Radians are dimensionless; degrees are an alternative unit for the same dimensionless quantity. Angle as a dimensionless ratio This ratio determines the angle in radians, which is a key dimensionless quantity used to ensure geometric similarity when scaling models. Model testing and similarity: when building small-scale models (like a car in a wind tunnel), keep the key dimensionless numbers the same as in real life to ensure similarity. Reynolds number Re = ρ v L / η is dimensionless; by matching Re between model and full-scale, you can predict drag behavior without full-size tests. Dimensional analysis leads you to such groups and clarifies which combinations matter. Trap: Mixing up g (9.8 m s -2 ) and G (6.67× 10 -11 N m 2 kg -2). Their symbols are similar but their dimensions are very different. Always check [g] = L T -2 versus [G] = M -1 L 3 T -2 . neet-alert Dimensional consistency also guides algebraic rearrangements. When you isolate a variable, verify that its dimensions match what you expect. For example, from v 2 = u 2 + 2 a s, solving for s yields s = ( v 2 − u 2 )/(2 a). Check: [s] = ( L 2 T -2 )/(L T -2 ) = L. If a derived expression yields a different dimension (say, L T or L 0 ), you have made an algebraic or conceptual error. Another common check is for logarithms in data fitting. If an experimental law suggests y = A x n, taking logs gives ln y = ln A + n ln x. This step is valid only if y and x have been scaled to dimensionless forms or if you interpret ln y as ln(y/y0) with a reference y0 of the same dimension. In practical curve fitting, the intercept ln A involves your choice of units; be consistent. log y Log–log plotting converts a power law to a straight line; the slope reveals the exponent. logy Power n (slope) slope log A (intercept) intercept logy = slope logx + intercept logx 2D PLOT Log–log plot of a power law y = A·xⁿ log x A straight line with slope equal to the power n in y = A x n; the intercept equals log A in the chosen base. custom control dependent slope (n) derived Log–log method to estimate exponents Assume a power-law y = A x n based on dimensional reasoning. Take logs: log y = log A + n log x. Plot log y against log x; fit a straight line. Read the slope n and intercept log A. Compare with dimensional prediction and refine the model if needed. Worked conversion strategy: when faced with a weird unit, break it into base units first. Replace gram by 10 -3 kg, centimeter by 10 -2 m, dyne by 10 -5 N, poise by 0.1 Pa s, and bar by 10 5 Pa. Then collect powers and regroup to the SI derived unit you want. This is faster and safer than memorizing dozens of direct conversion factors. Strain = ΔL/L Refractive index n = c/v Coefficient of friction μ = F friction / N Reynolds number Re = ρ v L / η Mach number Ma = v / c s Common dimensionless combinations Dimensional analysis can quickly check the plausibility of answers. If a calculation for time gives units like m s -1 , you know an algebraic slip occurred. In NEET-style problems, adopt a habit: before plugging numbers, do a dimension check; after finishing, verify the unit of the result. This double-check costs seconds and saves marks. Angles and solid angles are special. Plane angle θ is dimensionless (radian), and solid angle Ω is also dimensionless (steradian). However, you still quote them with units to clarify interpretation (rad, sr). Keep this in mind when differentiating or integrating with respect to angle; the derivative d/dθ does not introduce any new dimension by itself. Dimensions of key electromagnetic constants: Permittivity ε 0 has [ M -1 L -3 T 4 I 2 ], permeability μ 0 has [M L T -2 I -2 ]. The speed of light c emerges from c = 1/√(μ 0 ε 0), which is dimensionally consistent: multiply their dimensions to get [ L -2 T 2 ], invert and take square root to obtain [L T -1 ] for speed. Planck constant has the dimensions of action (energy × time). Planck constant h Dimensional prediction for spring–mass oscillation: if T depends only on mass m and spring constant k, assume T = k 1 m a k b. With [T] = T, [m] = M, [k] = M T -2 . Matching exponents gives a = 1/2 and b =$-1/2, so T ∝ √(m/k). Again, the dimensionless factor (2π) needs full dynamics to determine. Assume T = C m a k b [T] = T 1 , [m] = M 1 , [k] = M 1 T -2 medium Use dimensional analysis to find the form of the period T for a mass m attached to a spring with force constant k (ignore damping and amplitude). Exponents a and b Use this quick checklist: same kind? (dimensions match), same scale? (units consistent), same sense? (vector/scalar nature preserved). If any answer is “no,” pause and fix before proceeding. remember Worked example of sanity check: Suppose kinetic energy is written as E = (1/3) m v. Dimensions give [E] = M L T -1 , which is not M L 2 T -2 . So the formula must be wrong. The correct dependence needs v 2 and a dimensionless factor (1/2): E = (1/2) m v 2 has [M L 2 T -2 ]. Dimensional reasoning with data: When experiments show that a period increases with string length, dimensions suggest a square-root trend, helping you linearize data (plot T versus √L) for efficient analysis. This is not a proof but a guide that speeds up both thinking and computation. Before calculation: verify both sides of an equation have identical dimensions. While converting: reduce to base units, then regroup. After calculation: confirm the final unit matches the target quantity. For functional forms: ensure arguments of sin, cos, exp, log are dimensionless. For means/errors: check that operations obey additivity rules for dimensions. Exam quick checks using dimensions Dimensional decomposition practice: Pressure p = F/A has [M L -1 T -2 ]. Surface tension S = force per unit length has [M T -2 ]. Viscosity η (dynamic) has [M L -1 T -1 ], while kinematic viscosity ν = η/ρ has [ L 2 T -1 ]. Energy density u = E/V has [M L -1 T -2 ]. Learning these common patterns helps you recognize errors at sight. In problems that mix many variables, count how many independent dimensionless groups can be formed (Buckingham π-theorem idea). If the number of variables minus the number of base dimensions is greater than one, you cannot determine a unique power-law without extra physics or data. This explains why drag in fluids can depend on Reynolds number as well as surface roughness. neet-alert Do not average quantities with different dimensions or units. Averaging 2 m and 200 cm is valid only after converting to the same unit. Averaging 2 m and 3 s is meaningless—dimensions must match. Dimensional cleaning in multi-step derivations: Keep a side-column where you track the dimension of each new intermediate variable. If any mismatch appears, stop and re-derive that step. This habit often catches sign errors, missed factors of length or time, and confusion between mass and weight. Order-of-magnitude checks: If you estimate a person’s walking speed as 3 m s -1 , then kinetic energy of a 60 kg person is roughly (1/2)×60× 3 2 ≈ 270 J. The steps use consistent dimensions and give a plausible scale. On NEET, a rough magnitude within a factor of 2–3 is often enough to eliminate wrong options before precise calculation. Dimensional relations in thermodynamics: The gas constant R has [M L 2 T -2 N -1 Θ -1]. Specific heat c carries [ L 2 T -2 Θ -1]. Boltzmann constant k B = R/ N A shares [M L 2 T -2 Θ -1] per particle. These patterns ensure that pV and nRT both evaluate to energy dimensions, preserving homogeneity in the ideal gas law. Electromagnetism cross-check: In the energy density of an electric field, u = (1/2) ε E 2 , verify [u] = [ε][E] 2. With [E] = M L T -3 I -1 and [ε] = M -1 L -3 T 4 I 2 , you recover [u] = M L -1 T -2 $, agreeing with energy per volume. Such checks quickly validate complex-looking formulas. Dimensional behavior under scaling: If you double all lengths in a mechanical system while keeping material properties fixed, quantities with L n scale by 2 n. For example, area-related forces (like pressure × area) scale differently from line-related forces (like surface tension × length). These insights tell you which effects dominate at small vs large scales. When an expression combines several terms, check each term separately. For example, y = A e -x/λ + B cos(ω t) is acceptable only if x/λ and ω t are dimensionless. Also, A and B must have the same dimensions as y, because they are coefficients of dimensionless functions. Dimensional analysis of wave speed: For a string under tension T (dimension of force) with mass per unit length μ (dimension M L -1 ), the speed v must satisfy [v] = [T/μ] 1/2 = [M L T -2 / (M L -1 )] 1/2 = [ L 2 T -2 ] 1/2 = L T -1 . The exact constant is 1, hence v =$√(T/μ). Dimensional analysis identifies the correct square-root dependence cleanly. Dimensional calibration of experimental slopes: If a graph of y versus x is a straight line with slope m, then [m] = [y]/[x]. For example, plotting displacement (m) vs time 2 ( s 2 ) under constant acceleration gives slope = (1/2) a, and [slope] = m s -2 $. This ensures that reading slopes returns a physically meaningful quantity. Recap: Key terms [Q] Dimensional formula Expression of a quantity in base dimensions as [M a L b T c I d Θ e N f J g]. Homogeneity All additive terms in a valid equation share identical dimensions. Quantity with overall dimensions [1]; necessary as arguments of sin, cos, exp, log. Dimensionless quantity pure number Unit conversion Rewriting a quantity in a different unit system by mapping base units and collecting powers. Ratio ΔA/A (dimensionless); adds in products and quotients. Relative error Absolute error Error with the same units as the quantity; adds in sums and differences. Mean value is the unbiased best estimate and preserves the dimension of the measured variable. In multiplicative relations, worst-case relative errors add. Raising to a power multiplies relative error by |n|. For sum/difference, absolute errors add directly.