{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Pressure fields\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/pressure-fields). 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": [ "Pressure describes normal force per area in a fluid. It is a scalar field: one pressure\n", "value belongs to each position and time. The force it produces on a surface also depends\n", "on the surface area and normal direction, so pressure itself is not a vector force.\n", "\n", "## Pressure and its units\n", "\n", "For a small flat surface on which normal force is approximately uniform,\n", "\n", "\n", "$$\n", "p=\\frac{F_\\perp}{A}\n", "$$\n", "\n", "\n", "The SI unit is the pascal:\n", "\n", "\n", "$$\n", "1\\ \\mathrm{Pa}=1\\ \\mathrm{N/m^2}\n", "$$\n", "\n", "\n", "Pressure acts normally on any imagined surface in a fluid at rest. At one point in an\n", "ordinary static fluid, it has the same value for every surface orientation; otherwise an\n", "infinitesimal fluid element would experience an unbalanced torque or shear.\n", "\n", "For a finite curved or nonuniformly loaded surface, force is an integral:\n", "\n", "\n", "$$\n", "\\mathbf F=-\\int_A p\\,\\hat{\\mathbf n}\\,dA\n", "$$\n", "\n", "\n", "Here $\\hat{\\mathbf n}$ points\n", "out of the chosen fluid control volume, so surrounding pressure acts inward. Uniform\n", "pressure over a flat surface reduces to magnitude\n", "$pA$.\n", "\n", "## Hydrostatic balance\n", "\n", "Consider a fluid at rest with vertical coordinate\n", "$z$ positive upward. Force\n", "balance on a thin fluid element gives\n", "\n", "\n", "$$\n", "\\frac{dp}{dz}=-\\rho g\n", "$$\n", "\n", "\n", "Pressure decreases upward because lower fluid must support the weight of fluid above it.\n", "In vector form for constant downward gravitational acceleration,\n", "\n", "\n", "$$\n", "\\boldsymbol\\nabla p=\\rho\\mathbf g\n", "$$\n", "\n", "\n", "This is a local equation. It remains useful when density or gravity changes with\n", "position, provided the fluid is in mechanical equilibrium." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "density = 1000.0\n", "gravity = np.array([0.0, 0.0, -9.81])\n", "pressure_gradient = density * gravity\n", "assert np.allclose(-pressure_gradient + density * gravity, 0)\n", "print(\"Pressure gradient (Pa/m):\", pressure_gradient)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "## Constant-density liquid\n", "\n", "Let depth $h$ be measured downward from a\n", "surface where pressure is\n", "$p_0$. For constant density and\n", "gravity, integration gives\n", "\n", "\n", "$$\n", "p(h)=p_0+\\rho gh\n", "$$\n", "\n", "\n", "The pressure difference between two depths is\n", "\n", "\n", "$$\n", "\\Delta p=\\rho g\\Delta h\n", "$$\n", "\n", "\n", "Only vertical separation matters, not container shape or horizontal distance. Points at\n", "the same height in the same connected, uniform, resting fluid have equal pressure. This\n", "is sometimes called the hydrostatic paradox: differently shaped containers can exert\n", "the same bottom pressure at equal depth even though they contain different fluid masses.\n", "Forces from sloping walls complete the overall force balance." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "depth, surface_pressure = 2.0, 101325.0\n", "pressure_increase = density * 9.81 * depth\n", "absolute_pressure = surface_pressure + pressure_increase\n", "print(f\"Absolute pressure: {absolute_pressure:.1f} Pa\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "The formula uses depth downward. If using height upward, the sign changes through the\n", "hydrostatic gradient equation.\n", "\n", "## Absolute and gauge pressure\n", "\n", "Absolute pressure is measured relative to vacuum and cannot be negative for an ordinary\n", "equilibrium fluid. Gauge pressure is measured relative to local atmospheric pressure:\n", "\n", "\n", "$$\n", "p_{\\mathrm{gauge}}=p_{\\mathrm{absolute}}-p_{\\mathrm{atmosphere}}\n", "$$\n", "\n", "\n", "A tire-pressure reading is usually gauge pressure. A pressure below atmospheric has\n", "negative gauge pressure but still positive absolute pressure. Equations involving only\n", "pressure differences can use either convention consistently. Thermodynamic equations\n", "and gas laws generally require absolute pressure.\n", "\n", "Mixing an absolute pressure at one location with a gauge pressure at another introduces\n", "an error of approximately one atmospheric pressure. Store the reference explicitly.\n", "\n", "## Pascal's principle and hydraulics\n", "\n", "An externally applied pressure change to a confined static fluid is transmitted\n", "throughout the fluid, neglecting hydrostatic differences when points are at equal height.\n", "In a hydraulic device with piston areas\n", "$A_1$ and\n", "$A_2$,\n", "\n", "\n", "$$\n", "\\frac{F_1}{A_1}=\\frac{F_2}{A_2}\n", "$$\n", "\n", "\n", "so a larger output area produces a larger output force. Energy is not multiplied. Volume\n", "conservation for an incompressible fluid gives\n", "\n", "\n", "$$\n", "A_1\\Delta x_1=A_2\\Delta x_2\n", "$$\n", "\n", "\n", "The larger-force piston moves a proportionally smaller distance, so ideal input and\n", "output work agree.\n", "\n", "## Manometers and pressure differences\n", "\n", "A manometer turns pressure difference into a fluid-height difference. For a simple\n", "U-tube containing one liquid of density\n", "$\\rho$ with one side exposed\n", "to atmosphere,\n", "\n", "\n", "$$\n", "p_{\\mathrm{gauge}}=\\rho g\\Delta h\n", "$$\n", "\n", "\n", "For multiple fluids, move through each vertical column and add pressure when moving\n", "downward, subtract when moving upward. Points at the same elevation in the same connected\n", "static liquid have equal pressure." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "pressure = 101325.0\n", "# Traverse down 0.2 m through water, then up 0.1 m through oil.\n", "for density, height_change in [(1000.0, -0.2), (800.0, 0.1)]:\n", " pressure -= density * 9.81 * height_change\n", "print(f\"Final pressure: {pressure:.1f} Pa\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "This path method prevents memorized sign mistakes and handles several immiscible liquids.\n", "\n", "## Variable density and atmosphere\n", "\n", "The constant-density formula is excellent for modest depths in liquids. A gas is more\n", "compressible, so density often changes substantially with height. The correct relation\n", "remains\n", "$dp/dz=-\\rho g$,\n", "but it must be combined with an equation of state and temperature model.\n", "\n", "For an ideal gas at constant temperature,\n", "$p=\\rho R_{\\mathrm s}T$.\n", "Integration gives an exponential barometric profile:\n", "\n", "\n", "$$\n", "p(z)=p(z_0)\\exp\\left[-\\frac{g(z-z_0)}{R_{\\mathrm s}T}\\right]\n", "$$\n", "\n", "\n", "Real atmospheric temperature changes with height, so this is another approximation.\n", "\n", "## Force on submerged surfaces\n", "\n", "Pressure increases with depth, so force on a vertical wall is not uniform. For a plane\n", "surface in a constant-density liquid, integrate\n", "$p_0+\\rho gh$ over the\n", "area. The resultant acts deeper than the geometric centroid because deeper portions carry\n", "greater pressure. This point is called the center of pressure.\n", "\n", "Uniform atmospheric pressure acting on both sides of a surface cancels in the net force.\n", "Gauge pressure is therefore convenient for many open-tank force calculations. If one\n", "side is vacuum or a different gas pressure, it must remain in the balance.\n", "\n", "## Accelerating fluids and other limits\n", "\n", "The familiar depth law assumes rest in an inertial frame. In a container accelerating\n", "uniformly, the free surface becomes perpendicular to the effective gravity vector. In\n", "rotating flow, pressure can vary radially. These cases still follow momentum balance but\n", "not the simple vertical formula without modification.\n", "\n", "Surface tension produces pressure jumps across curved interfaces, especially at small\n", "scales. Flowing viscous fluids need momentum equations and can lose mechanical energy.\n", "Sensors also have finite size, response time, calibration, and elevation, all of which\n", "matter in precise comparisons.\n", "\n", "## Verification checks\n", "\n", "At zero depth, recover the boundary pressure. Doubling constant-density depth should\n", "double only the pressure increase, not necessarily the absolute pressure. Check units:\n", "$\\rho gh$ reduces to\n", "force per area. Compare equal-height points in a connected resting liquid.\n", "\n", "For a discretized pressure field, calculate the numerical gradient and verify that it\n", "balances density times gravity. Refine vertical spacing and compare pressure at the same\n", "physical heights. When converting pressure to force, verify surface normals and integrate\n", "rather than multiplying one arbitrary sample by total area.\n", "\n", "Pressure fields are simple only after the coordinate, reference pressure, density model,\n", "and mechanical state are declared. Those details make the difference between a reusable\n", "physical law and an unexplained number.\n", "\n", "## A worked Python notebook\n", "\n", "Calculate hydrostatic pressure in water under a surface pressure of 101325 pascals. The question for this calculation is precise: how does **absolute pressure**\n", "change when we change **depth below surface**, 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", "Density and gravity are held constant and the fluid is at rest. 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 10 m. 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 depth below surface, expressed in m. The output is\n", "absolute pressure, expressed in Pa. 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", "$$\n", "f(q) = 9810.0 q + 101325\n", "$$\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 m, the expression gives 120945 Pa. 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-009", "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(\"101325+1000*9.81*q\", locals={\"q\": q})\n", "reference_input = 2.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted absolute pressure:\", reference_output, \"Pa\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "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", "101325 Pa. The midpoint, at 5 m, gives\n", "150375 Pa. The upper endpoint gives 199425 Pa.\n", "Read these together with the assumptions: they describe different choices of depth below surface\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-011", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.0, 5.0, 10.0])\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} m -> {output_value:.6g} Pa\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "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 depth below surface; the vertical axis shows absolute pressure. 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-013", "metadata": {}, "source": [ "input_grid = np.linspace(0.0, 10.0, 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(\"Depth below surface (m)\")\n", "ax.set_ylabel(\"Absolute pressure (Pa)\")\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-014", "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", "absolute pressure change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 2 m is\n", "9810 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", "$$\n", "f'(q) = 9810.0\n", "$$\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-015", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 2.05\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, \"Pa\")\n", "print(\"Full expression:\", full_prediction, \"Pa\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"Pa\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 2.05 m. The tangent estimate is\n", "121436 Pa, while the full expression gives\n", "121436 Pa. 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-017", "metadata": {}, "source": [ "input_span = 10.0\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-018", "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", "depth below surface inside the stated interval, predict whether absolute pressure 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: Density and gravity are held constant and the fluid is at rest. 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 Pa:\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-019", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(10.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 10.0, 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-020", "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", "\n", "- [OpenStax, *University Physics Volume 1*, Section 14.1, \"Fluids, Density, and Pressure\"](https://openstax.org/books/university-physics-volume-1/pages/14-1-fluids-density-and-pressure).\n", "- [OpenStax, *University Physics Volume 1*, Section 14.2, \"Measuring Pressure\"](https://openstax.org/books/university-physics-volume-1/pages/14-2-measuring-pressure).\n", "- [OpenStax, *University Physics Volume 1*, Section 14.3, \"Pascal's Principle and Hydraulics\"](https://openstax.org/books/university-physics-volume-1/pages/14-3-pascals-principle-and-hydraulics).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 28, \"Fluid Dynamics\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter28/).\n", "- [Benjamin Crowell, *Light and Matter*, pressure and fluid statics discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Pressure\"](https://physics.info/pressure/).\n", "- [Physics LibreTexts, \"Pressure at a Depth in a Fluid\"](https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/14%3A_Fluid_Mechanics/14.03%3A_Pressure_at_a_Depth_in_a_Fluid)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "pressure-fields.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/pressure-fields", "source_sha256": "7804884e7862a737b7a1d7fb46169d48f2c43c770d4cb30f7b415e7ecbd30b94" } }, "nbformat": 4, "nbformat_minor": 5 }