{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Dimensional analysis with code\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/dimensional-analysis-with-code). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "Dimensional analysis checks the structure of a physical relationship before numerical\n", "values are substituted. If an equation adds a length to a time, or claims that an\n", "acceleration is a velocity, no choice of numbers can repair it. A program can propagate\n", "dimensions through an expression and report the first incompatible operation.\n", "\n", "## Units and dimensions answer different questions\n", "\n", "A unit chooses a measurement scale; a dimension identifies the kind of physical\n", "quantity. Metres and kilometres are different units of the same length dimension.\n", "Metres per second and kilometres per hour are different units of the same velocity\n", "dimension.\n", "\n", "For mechanics, three base dimensions are sufficient for many examples:\n", "\n", "- mass: $M$;\n", "- length: $L$;\n", "- time: $T$.\n", "\n", "Velocity, acceleration, force, and energy then reduce to\n", "\n", "\n", "$$\n", "[v]=LT^{-1},\\quad[a]=LT^{-2},\\quad[F]=MLT^{-2},\\quad[E]=ML^2T^{-2}\n", "$$\n", "\n", "\n", "Square brackets here mean \"dimensions of,\" not array indexing. Numerical conversion\n", "factors are a separate concern. A dimension checker can confirm that feet and metres\n", "are both lengths, but it still needs the correct scale factor to convert them.\n", "\n", "## The algebra of dimension vectors\n", "\n", "Represent a mechanical dimension with one exponent per base dimension." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "from dataclasses import dataclass\n", "\n", "\n", "@dataclass(frozen=True)\n", "class Dimension:\n", " mass: int\n", " length: int\n", " time: int\n", "\n", "\n", "distance = Dimension(mass=0, length=1, time=0)\n", "duration = Dimension(mass=0, length=0, time=1)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The product of quantities adds their exponent vectors. Dividing distance by duration\n", "therefore subtracts the duration exponents:\n", "\n", "\n", "$$\n", "L/T=LT^{-1}\n", "$$" ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "def divide_dimensions(\n", " numerator: Dimension,\n", " denominator: Dimension,\n", ") -> Dimension:\n", " return Dimension(\n", " mass=numerator.mass - denominator.mass,\n", " length=numerator.length - denominator.length,\n", " time=numerator.time - denominator.time,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "Addition and subtraction do not combine exponents. They require both operands to have\n", "the same dimension vector, because only like physical quantities can be added.\n", "\n", "## Catch a wrong kinematics equation\n", "\n", "For constant acceleration, the displacement relation contains a term\n", "$at^2$. Its dimensions\n", "are\n", "\n", "\n", "$$\n", "[at^2]=(LT^{-2})T^2=L\n", "$$\n", "\n", "\n", "so it can be added to another displacement. If an implementation accidentally uses\n", "$at^3$, then\n", "\n", "\n", "$$\n", "[at^3]=(LT^{-2})T^3=LT\n", "$$\n", "\n", "\n", "which is not a length. This detects the wrong exponent without evaluating a trajectory.\n", "The Physics Hypertextbook includes exactly this kind of consistency exercise, and both\n", "OpenStax and MIT use dimensional checks as an early test of equations." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "length = Dimension(mass=0, length=1, time=0)\n", "time = Dimension(mass=0, length=0, time=1)\n", "velocity = Dimension(mass=0, length=1, time=-1)\n", "\n", "velocity_times_time = Dimension(\n", " mass=velocity.mass + time.mass,\n", " length=velocity.length + time.length,\n", " time=velocity.time + time.time,\n", ")\n", "\n", "assert velocity_times_time == length\n", "\n", "velocity_times_time_cubed = Dimension(\n", " mass=velocity.mass + 3 * time.mass,\n", " length=velocity.length + 3 * time.length,\n", " time=velocity.time + 3 * time.time,\n", ")\n", "\n", "assert velocity_times_time_cubed != length\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "## Build an expression tree, not a string trick\n", "\n", "A symbolic checker should parse an expression into operations. Each node returns a\n", "dimension or a precise error. A string replacement system is likely to confuse variable\n", "names, powers, grouping, and functions." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "from dataclasses import dataclass\n", "from typing import Literal\n", "\n", "\n", "@dataclass(frozen=True)\n", "class Expression:\n", " kind: Literal[\"symbol\", \"multiply\", \"divide\", \"add\", \"power\"]\n", " name: str | None = None\n", " left: \"Expression | None\" = None\n", " right: \"Expression | None\" = None\n", " base: \"Expression | None\" = None\n", " exponent: int | None = None\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "Evaluation walks the tree from the leaves upward. A symbol looks up its declared\n", "dimension. Multiplication and division combine exponents. Addition compares the two\n", "child results and should report the location and both dimensions when they differ." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "def add_dimensions(left, right):\n", " return Dimension(\n", " left.mass + right.mass,\n", " left.length + right.length,\n", " left.time + right.time,\n", " )\n", "\n", "\n", "def evaluate_dimension(node, symbols):\n", " if node.kind == \"symbol\":\n", " return symbols[node.name]\n", " if node.kind == \"power\":\n", " base = evaluate_dimension(node.base, symbols)\n", " return Dimension(\n", " base.mass * node.exponent,\n", " base.length * node.exponent,\n", " base.time * node.exponent,\n", " )\n", " left = evaluate_dimension(node.left, symbols)\n", " right = evaluate_dimension(node.right, symbols)\n", " if node.kind == \"multiply\":\n", " return add_dimensions(left, right)\n", " if node.kind == \"divide\":\n", " return divide_dimensions(left, right)\n", " if node.kind == \"add\":\n", " if left != right:\n", " raise ValueError(f\"Cannot add {left} and {right}.\")\n", " return left\n", " raise ValueError(f\"Unknown operation: {node.kind}\")\n", "\n", "\n", "product = Expression(\n", " kind=\"multiply\",\n", " left=Expression(kind=\"symbol\", name=\"v\"),\n", " right=Expression(kind=\"symbol\", name=\"t\"),\n", ")\n", "assert evaluate_dimension(product, {\"v\": velocity, \"t\": time}) == length\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "The useful diagnostic is not merely \"false.\" A reader needs to know which addition,\n", "function, or equality first became inconsistent.\n", "\n", "## Functions impose dimensional rules\n", "\n", "The argument of an exponential, logarithm, sine, or cosine must be dimensionless. The\n", "power series defining these functions adds terms of different powers; those terms could\n", "not be added if the argument retained a physical dimension.\n", "\n", "For simple harmonic motion,\n", "\n", "\n", "$$\n", "x(t)=A\\cos(\\omega t+\\phi)\n", "$$\n", "\n", "\n", "the angle $\\omega t+\\phi$ must be\n", "dimensionless. Therefore angular frequency has dimension\n", "$T^{-1}$ and phase is\n", "dimensionless. The amplitude must have the same dimension as position.\n", "\n", "A square-root node divides every dimension exponent by two. A general power requires\n", "care: fractional powers can produce fractional exponents, and a quantity raised to a\n", "variable dimensional exponent is ordinarily invalid.\n", "\n", "## Derive the form of a pendulum period\n", "\n", "Dimensional analysis can do more than reject equations. Suppose the period\n", "$P$ of a small-angle simple pendulum\n", "depends on length $\\ell$ and\n", "gravitational acceleration $g$.\n", "Assume a product of powers:\n", "\n", "\n", "$$\n", "P=C\\ell^a g^b\n", "$$\n", "\n", "\n", "The constant $C$ is assumed\n", "dimensionless. Matching dimensions gives\n", "\n", "\n", "$$\n", "T=L^a(LT^{-2})^b=L^{a+b}T^{-2b}\n", "$$\n", "\n", "\n", "Equating exponents requires $a+b=0$\n", "and $-2b=1$. Thus\n", "$b=-1/2$ and\n", "$a=1/2$, so\n", "\n", "\n", "$$\n", "P=C\\sqrt{\\frac{\\ell}{g}}\n", "$$\n", "\n", "\n", "Dimensional analysis cannot determine\n", "$C=2\\pi$ and cannot establish\n", "the small-angle assumption. It narrows the possible form, while dynamics supplies the\n", "dimensionless constant and physical conditions.\n", "\n", "## Consistent is not necessarily correct\n", "\n", "Many wrong equations pass a dimensional check. Both\n", "$E=mv^2$ and\n", "$E=\\tfrac12 mv^2$\n", "have energy dimensions, but only the second is the nonrelativistic kinetic-energy\n", "formula. Dimensional analysis cannot determine dimensionless numerical factors.\n", "\n", "It also cannot distinguish quantities that share dimensions. Work and torque both have\n", "force-times-length dimensions. Frequency and angular frequency both have inverse-time\n", "dimensions but differ by a factor of\n", "$2\\pi$ under common definitions. Semantic\n", "quantity types and explicit conventions must supplement the dimension vector.\n", "\n", "Use dimensional analysis as an early compiler-like gate. Then continue with algebra,\n", "limiting cases, independent calculations, and comparison with observation. Passing one\n", "gate should make an equation eligible for further review, not automatically correct.\n", "\n", "## Dimensionless groups support comparison\n", "\n", "Combining variables into a dimensionless group can reveal when apparently different\n", "systems behave in the same way. For a simple example, distance travelled at constant\n", "speed can be compared with a characteristic length by forming\n", "\n", "\n", "$$\n", "\\Pi=\\frac{vt}{L_0}\n", "$$\n", "\n", "\n", "The numerator and denominator are both lengths, so\n", "$\\Pi$ has no remaining\n", "dimension. A value below one means the travelled distance is shorter than the selected\n", "reference length; a value above one means it is longer. The conclusion no longer depends\n", "on whether the underlying distances were expressed in metres or feet, provided the\n", "conversion was correct.\n", "\n", "More advanced fluid and heat problems use dimensionless groups to organize competing\n", "effects and scale laboratory results. A program should store the definition and\n", "conventions for each group rather than relying only on its conventional name. Two\n", "sources can use slightly different characteristic lengths and therefore compute\n", "different numerical values for a group with the same label.\n", "\n", "Dimensionless analysis also helps choose test fixtures. Changing every length in a\n", "geometrically similar problem by the same scale may preserve some dimensionless ratios\n", "while changing dimensional outputs. If a program claims similarity, tests can compare\n", "those ratios instead of demanding identical raw values.\n", "\n", "## Preserve rational exponents when possible\n", "\n", "A symbolic checker should avoid introducing floating-point noise into dimension\n", "exponents. The square root of an area has length dimension exactly; storing the exponent\n", "as the decimal approximation `0.5` works in this case, but repeated operations can make\n", "an exponent that should be zero become a tiny nonzero value. Rational exponent pairs or\n", "normalized fractions keep identities exact.\n", "\n", "If floating-point exponents are unavoidable, equality needs a carefully chosen\n", "tolerance and normalization rule. That tolerance applies to the internal symbolic\n", "representation, not to measured physical uncertainty. Mixing those two ideas would let\n", "a software workaround masquerade as scientific error analysis.\n", "\n", "Finally, the checker should preserve unknown dimensions as errors or symbolic unknowns.\n", "Treating an undeclared symbol as dimensionless can cause a malformed equation to pass.\n", "A strict failure with a source location is more useful than a confident result built\n", "from missing declarations.\n", "\n", "## A worked Python notebook\n", "\n", "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**\n", "change when we change **duration**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "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\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "0 to 4 s. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for duration, expressed in s. The output is\n", "distance, expressed in m. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = \\frac{q^{2}}{2} + 2 q\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 2 s, the expression gives 6 m. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"2*q+q**2/2\", locals={\"q\": q})\n", "reference_input = 2.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"s\")\n", "print(\"Predicted distance:\", reference_output, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "0 m. The midpoint, at 2 s, gives\n", "6 m. The upper endpoint gives 16 m.\n", "Read these together with the assumptions: they describe different choices of duration\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.0, 2.0, 4.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} s -> {output_value:.6g} m\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows duration; the vertical axis shows distance. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "input_grid = np.linspace(0.0, 4.0, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Duration (s)\")\n", "ax.set_ylabel(\"Distance (m)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "distance change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 2 s is\n", "4 in those ratio units. Locally, the output increases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = q + 2\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-019", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 2.02\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"m\")\n", "print(\"Full expression:\", full_prediction, \"m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "Here the changed input is 2.02 s. The tangent estimate is\n", "6.08 m, while the full expression gives\n", "6.0802 m. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-021", "metadata": {}, "source": [ "input_span = 4.0\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-022", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "duration inside the stated interval, predict whether distance will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "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\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of m:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-023", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(4.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 4.0, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-024", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "- [OpenStax, *University Physics Volume 1*, Section 1.4, \"Dimensional Analysis\"](https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 2, \"Units, Dimensional Analysis, Problem Solving, and Estimation\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter2/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 0, \"Introduction and Review\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Unit Conversion\"](https://physics.info/unit-conversion/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "dimensional-analysis-with-code.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/dimensional-analysis-with-code", "source_sha256": "aeff1be70006dc02c8e6310ffcd998869e8d9093b930731cb9dc6ae3bff36aaf" } }, "nbformat": 4, "nbformat_minor": 5 }