A Byte of PhysicsLogo

Dimensional analysis with code

Dimensional analysis checks the structure of a physical relationship before numerical values are substituted. If an equation adds a length to a time, or claims that an acceleration is a velocity, no choice of numbers can repair it. A program can propagate dimensions through an expression and report the first incompatible operation.

Units and dimensions answer different questions

A unit chooses a measurement scale; a dimension identifies the kind of physical quantity. Metres and kilometres are different units of the same length dimension. Metres per second and kilometres per hour are different units of the same velocity dimension.

For mechanics, three base dimensions are sufficient for many examples:

Velocity, acceleration, force, and energy then reduce to

\[[v]=LT^{-1},\quad[a]=LT^{-2},\quad[F]=MLT^{-2},\quad[E]=ML^2T^{-2}\]

Square brackets here mean "dimensions of," not array indexing. Numerical conversion factors are a separate concern. A dimension checker can confirm that feet and metres are both lengths, but it still needs the correct scale factor to convert them.

The algebra of dimension vectors

Represent a mechanical dimension with one exponent per base dimension.

from dataclasses import dataclass


@dataclass(frozen=True)
class Dimension:
    mass: int
    length: int
    time: int


distance = Dimension(mass=0, length=1, time=0)
duration = Dimension(mass=0, length=0, time=1)

The product of quantities adds their exponent vectors. Dividing distance by duration therefore subtracts the duration exponents:

\[L/T=LT^{-1}\] 
def divide_dimensions(
    numerator: Dimension,
    denominator: Dimension,
) -> Dimension:
    return Dimension(
        mass=numerator.mass - denominator.mass,
        length=numerator.length - denominator.length,
        time=numerator.time - denominator.time,
    )

Addition and subtraction do not combine exponents. They require both operands to have the same dimension vector, because only like physical quantities can be added.

Catch a wrong kinematics equation

For constant acceleration, the displacement relation contains a term \(at^2\). Its dimensions are

\[[at^2]=(LT^{-2})T^2=L\]

so it can be added to another displacement. If an implementation accidentally uses \(at^3\), then

\[[at^3]=(LT^{-2})T^3=LT\]

which is not a length. This detects the wrong exponent without evaluating a trajectory. The Physics Hypertextbook includes exactly this kind of consistency exercise, and both OpenStax and MIT use dimensional checks as an early test of equations.

length = Dimension(mass=0, length=1, time=0)
time = Dimension(mass=0, length=0, time=1)
velocity = Dimension(mass=0, length=1, time=-1)

velocity_times_time = Dimension(
    mass=velocity.mass + time.mass,
    length=velocity.length + time.length,
    time=velocity.time + time.time,
)

assert velocity_times_time == length

velocity_times_time_cubed = Dimension(
    mass=velocity.mass + 3 * time.mass,
    length=velocity.length + 3 * time.length,
    time=velocity.time + 3 * time.time,
)

assert velocity_times_time_cubed != length

Build an expression tree, not a string trick

A symbolic checker should parse an expression into operations. Each node returns a dimension or a precise error. A string replacement system is likely to confuse variable names, powers, grouping, and functions.

from dataclasses import dataclass
from typing import Literal


@dataclass(frozen=True)
class Expression:
    kind: Literal["symbol", "multiply", "divide", "add", "power"]
    name: str | None = None
    left: "Expression | None" = None
    right: "Expression | None" = None
    base: "Expression | None" = None
    exponent: int | None = None

Evaluation walks the tree from the leaves upward. A symbol looks up its declared dimension. Multiplication and division combine exponents. Addition compares the two child results and should report the location and both dimensions when they differ.

def add_dimensions(left, right):
    return Dimension(
        left.mass + right.mass,
        left.length + right.length,
        left.time + right.time,
    )


def evaluate_dimension(node, symbols):
    if node.kind == "symbol":
        return symbols[node.name]
    if node.kind == "power":
        base = evaluate_dimension(node.base, symbols)
        return Dimension(
            base.mass * node.exponent,
            base.length * node.exponent,
            base.time * node.exponent,
        )
    left = evaluate_dimension(node.left, symbols)
    right = evaluate_dimension(node.right, symbols)
    if node.kind == "multiply":
        return add_dimensions(left, right)
    if node.kind == "divide":
        return divide_dimensions(left, right)
    if node.kind == "add":
        if left != right:
            raise ValueError(f"Cannot add {left} and {right}.")
        return left
    raise ValueError(f"Unknown operation: {node.kind}")


product = Expression(
    kind="multiply",
    left=Expression(kind="symbol", name="v"),
    right=Expression(kind="symbol", name="t"),
)
assert evaluate_dimension(product, {"v": velocity, "t": time}) == length

The useful diagnostic is not merely "false." A reader needs to know which addition, function, or equality first became inconsistent.

Functions impose dimensional rules

The argument of an exponential, logarithm, sine, or cosine must be dimensionless. The power series defining these functions adds terms of different powers; those terms could not be added if the argument retained a physical dimension.

For simple harmonic motion,

\[x(t)=A\cos(\omega t+\phi)\]

the angle \(\omega t+\phi\) must be dimensionless. Therefore angular frequency has dimension \(T^{-1}\) and phase is dimensionless. The amplitude must have the same dimension as position.

A square-root node divides every dimension exponent by two. A general power requires care: fractional powers can produce fractional exponents, and a quantity raised to a variable dimensional exponent is ordinarily invalid.

Derive the form of a pendulum period

Dimensional analysis can do more than reject equations. Suppose the period \(P\) of a small-angle simple pendulum depends on length \(\ell\) and gravitational acceleration \(g\). Assume a product of powers:

\[P=C\ell^a g^b\]

The constant \(C\) is assumed dimensionless. Matching dimensions gives

\[T=L^a(LT^{-2})^b=L^{a+b}T^{-2b}\]

Equating exponents requires \(a+b=0\) and \(-2b=1\). Thus \(b=-1/2\) and \(a=1/2\), so

\[P=C\sqrt{\frac{\ell}{g}}\]

Dimensional analysis cannot determine \(C=2\pi\) and cannot establish the small-angle assumption. It narrows the possible form, while dynamics supplies the dimensionless constant and physical conditions.

Consistent is not necessarily correct

Many wrong equations pass a dimensional check. Both \(E=mv^2\) and

\(E=\tfrac12 mv^2\)

have energy dimensions, but only the second is the nonrelativistic kinetic-energy formula. Dimensional analysis cannot determine dimensionless numerical factors.

It also cannot distinguish quantities that share dimensions. Work and torque both have force-times-length dimensions. Frequency and angular frequency both have inverse-time dimensions but differ by a factor of \(2\pi\) under common definitions. Semantic quantity types and explicit conventions must supplement the dimension vector.

Use dimensional analysis as an early compiler-like gate. Then continue with algebra, limiting cases, independent calculations, and comparison with observation. Passing one gate should make an equation eligible for further review, not automatically correct.

Dimensionless groups support comparison

Combining variables into a dimensionless group can reveal when apparently different systems behave in the same way. For a simple example, distance travelled at constant speed can be compared with a characteristic length by forming

\[\Pi=\frac{vt}{L_0}\]

The numerator and denominator are both lengths, so \(\Pi\) has no remaining dimension. A value below one means the travelled distance is shorter than the selected reference length; a value above one means it is longer. The conclusion no longer depends on whether the underlying distances were expressed in metres or feet, provided the conversion was correct.

More advanced fluid and heat problems use dimensionless groups to organize competing effects and scale laboratory results. A program should store the definition and conventions for each group rather than relying only on its conventional name. Two sources can use slightly different characteristic lengths and therefore compute different numerical values for a group with the same label.

Dimensionless analysis also helps choose test fixtures. Changing every length in a geometrically similar problem by the same scale may preserve some dimensionless ratios while changing dimensional outputs. If a program claims similarity, tests can compare those ratios instead of demanding identical raw values.

Preserve rational exponents when possible

A symbolic checker should avoid introducing floating-point noise into dimension exponents. The square root of an area has length dimension exactly; storing the exponent as the decimal approximation 0.5 works in this case, but repeated operations can make an exponent that should be zero become a tiny nonzero value. Rational exponent pairs or normalized fractions keep identities exact.

If floating-point exponents are unavoidable, equality needs a carefully chosen tolerance and normalization rule. That tolerance applies to the internal symbolic representation, not to measured physical uncertainty. Mixing those two ideas would let a software workaround masquerade as scientific error analysis.

Finally, the checker should preserve unknown dimensions as errors or symbolic unknowns. Treating an undeclared symbol as dimensionless can cause a malformed equation to pass. A strict failure with a source location is more useful than a confident result built from missing declarations.

A worked Python notebook

Combine an initial speed of 2 metres per second with an acceleration of 1 metre per second squared. The question for this calculation is precise: how does distance change when we change duration, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Each term must have length dimensions before the terms can be added. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 4 s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for duration, expressed in s. The output is distance, expressed in m. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = \frac{q^{2}}{2} + 2 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 2 s, the expression gives 6 m. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("2*q+q**2/2", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted distance:", reference_output, "m")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0 m. The midpoint, at 2 s, gives 6 m. The upper endpoint gives 16 m. Read these together with the assumptions: they describe different choices of duration within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 2.0, 4.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} s -> {output_value:.6g} m")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows duration; the vertical axis shows distance. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 4.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Duration (s)")
ax.set_ylabel("Distance (m)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does distance change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 s is 4 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = q + 2\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 2.02
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 2.02 s. The tangent estimate is 6.08 m, while the full expression gives 6.0802 m. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 4.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of duration inside the stated interval, predict whether distance will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Each term must have length dimensions before the terms can be added. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of m: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(4.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 4.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

Share to: