{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Force models as functions\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/force-models-as-functions). 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": [ "A force model maps a physical state and parameters to a vector force. Writing that map\n", "as a small function makes assumptions, units, reference frames, and singularities easier\n", "to inspect. It also keeps the interaction law separate from the numerical method that\n", "advances motion.\n", "\n", "## A force function contract\n", "\n", "For a body state\n", "$\\mathbf y=(\\mathbf r,\\mathbf v)$,\n", "a general force may be written\n", "\n", "\n", "$$\n", "\\mathbf F=\\mathbf F(\\mathbf r,\\mathbf v,t;\\boldsymbol\\alpha)\n", "$$\n", "\n", "\n", "The parameter set\n", "$\\boldsymbol\\alpha$\n", "can include masses, charges, spring constants, fluid properties, or field data. A useful\n", "software contract states:\n", "\n", "- target and source bodies or field;\n", "- required position, velocity, time, and frame;\n", "- parameter units and allowed ranges;\n", "- output basis and force units;\n", "- valid spatial and speed regime;\n", "- behavior at singularities and outside the domain." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "\n", "def spring_force(position, stiffness=10.0):\n", " position = np.asarray(position, dtype=float)\n", " if stiffness < 0 or not np.all(np.isfinite(position)):\n", " raise ValueError(\"Use nonnegative stiffness and finite positions.\")\n", " return -stiffness * position\n", "\n", "\n", "print(\"Spring force (N):\", spring_force([0.1, 0.0]))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "## Uniform gravitational force\n", "\n", "Near Earth's surface over a modest height range, gravitational field can be\n", "approximated as constant:\n", "\n", "\n", "$$\n", "\\mathbf F_g=m\\mathbf g\n", "$$\n", "\n", "\n", "If vertical is positive upward,\n", "$\\mathbf g=(0,-g)$.\n", "The model needs body mass and a field vector but not velocity. Its force function should\n", "return force, not acceleration; the dynamics layer divides the summed force by mass." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "mass_kg = 2.0\n", "gravitational_field = np.array([0.0, -9.81])\n", "weight_n = mass_kg * gravitational_field\n", "print(\"Weight (N):\", weight_n)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "This approximation fails over planetary-scale distance or where the field varies\n", "materially. The failure belongs in model metadata rather than being hidden inside a\n", "constant named `gravity` with unclear units.\n", "\n", "## Central inverse-square gravity\n", "\n", "For point masses\n", "$m_1$ and\n", "$m_2$ at distinct positions,\n", "\n", "\n", "$$\n", "\\mathbf F_{2\\to1}=Gm_1m_2\\frac{\\mathbf r_2-\\mathbf r_1}{|\\mathbf r_2-\\mathbf r_1|^3}\n", "$$\n", "\n", "\n", "The displacement points from target to source, so the positive coefficient produces\n", "attraction. The formula is undefined at zero separation. Returning zero there would\n", "replace a singular model with a false no-force claim." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "G = 6.67430e-11\n", "source_position = np.array([0.0, 0.0])\n", "target_position = np.array([6.371e6, 0.0])\n", "source_mass, target_mass = 5.972e24, 2.0\n", "displacement = source_position - target_position\n", "distance = np.linalg.norm(displacement)\n", "if distance == 0:\n", " raise ValueError(\"Point masses cannot overlap.\")\n", "force_n = G * source_mass * target_mass * displacement / distance**3\n", "print(\"Gravitational force (N):\", force_n)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "If finite-size bodies are intended, replace the point model with an appropriate interior\n", "or contact model. Adding a softening length changes the law and must be reported.\n", "\n", "## Ideal spring force\n", "\n", "For a one-dimensional ideal spring,\n", "\n", "\n", "$$\n", "F_s=-k(x-x_{\\mathrm{eq}})\n", "$$\n", "\n", "\n", "The spring constant\n", "$k>0$ has units of force per\n", "length. The minus sign makes the force restoring: positive displacement produces\n", "negative force and vice versa.\n", "\n", "For a spring between two points in multiple dimensions, let\n", "$\\mathbf d=\\mathbf r_B-\\mathbf r_A$,\n", "length $\\ell=|\\mathbf d|$, rest\n", "length $\\ell_0$, and\n", "direction $\\hat{\\mathbf d}=\\mathbf d/\\ell$.\n", "The force on A is\n", "\n", "\n", "$$\n", "\\mathbf F_{B\\to A}=k(\\ell-\\ell_0)\\hat{\\mathbf d}\n", "$$\n", "\n", "\n", "The direction is undefined when endpoints coincide. A zero rest length does not remove\n", "the need for an explicit policy at zero separation. Real springs also have elastic\n", "limits, mass, damping, and geometry absent from Hooke's linear model.\n", "\n", "## Drag depends on relative velocity\n", "\n", "Linear drag is often written\n", "\n", "\n", "$$\n", "\\mathbf F_D=-b(\\mathbf v-\\mathbf v_{\\mathrm{fluid}})\n", "$$\n", "\n", "\n", "Quadratic drag can be written\n", "\n", "\n", "$$\n", "\\mathbf F_D=-\\frac12\\rho C_DA|\\mathbf v_{\\mathrm{rel}}|\\mathbf v_{\\mathrm{rel}}\n", "$$\n", "\n", "\n", "where\n", "$\\mathbf v_{\\mathrm{rel}}=\\mathbf v-\\mathbf v_{\\mathrm{fluid}}$.\n", "The force opposes motion through the fluid, not necessarily motion relative to ground.\n", "The quadratic coefficient depends on density, drag coefficient, and reference area.\n", "The drag coefficient can itself vary with Reynolds number, orientation, and flow regime." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "body_velocity = np.array([3.0, 0.0])\n", "fluid_velocity = np.array([1.0, 0.0])\n", "density, drag_coefficient, area = 1.2, 0.5, 0.1\n", "relative_velocity = body_velocity - fluid_velocity\n", "drag_n = (\n", " -0.5\n", " * density\n", " * drag_coefficient\n", " * area\n", " * np.linalg.norm(relative_velocity)\n", " * relative_velocity\n", ")\n", "print(\"Drag force (N):\", drag_n)\n", "assert np.dot(drag_n, relative_velocity) <= 0\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "At zero relative speed this expression naturally returns zero without normalizing the\n", "zero vector.\n", "\n", "## Contact and friction are state-dependent modes\n", "\n", "Normal contact is not generally a force defined everywhere in space. It is active when\n", "surfaces touch and the unconstrained motion would violate nonpenetration. Its magnitude\n", "is solved from dynamics and the contact constraint.\n", "\n", "Static friction obeys an inequality\n", "\n", "\n", "$$\n", "|f_s|\\le\\mu_sN\n", "$$\n", "\n", "\n", "rather than a single magnitude formula. A friction function must know whether contact is\n", "sticking or sliding. It first computes the force required for sticking and compares it\n", "with the limit. If the limit is exceeded, the state changes to sliding and a kinetic law\n", "applies. This is a piecewise constraint problem, not a smooth function of velocity with\n", "a guessed sign at zero.\n", "\n", "## Superposition and force accumulation\n", "\n", "When a model supports superposition, net force is the vector sum\n", "\n", "\n", "$$\n", "\\mathbf F_{\\mathrm{net},i}=\\sum_j\\mathbf F_{j\\to i}+\\sum_k\\mathbf F_{k,\\mathrm{external}}\n", "$$\n", "\n", "\n", "Evaluate every term from the same snapshot. Pair interactions should return equal and\n", "opposite forces to their two targets. Summing magnitudes and assigning one direction is\n", "wrong except in a one-dimensional case where signs already encode direction." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "contributions = {\"weight\": weight_n, \"drag\": drag_n}\n", "net_force = sum(contributions.values(), start=np.zeros(2))\n", "for name, force in contributions.items():\n", " print(name, force, \"N\")\n", "print(\"Net force (N):\", net_force)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "Keeping the audit trail permits a free-body diagram and allows unexpectedly large or\n", "canceling forces to be inspected.\n", "\n", "## Conservative-force check\n", "\n", "When a force comes from a time-independent potential energy,\n", "\n", "\n", "$$\n", "\\mathbf F=-\\nabla U\n", "$$\n", "\n", "\n", "the force implementation can be compared with a symbolic or finite-difference gradient\n", "of the implemented potential. For one coordinate,\n", "\n", "\n", "$$\n", "F(x)\\approx-\\frac{U(x+h)-U(x-h)}{2h}\n", "$$\n", "\n", "\n", "Agreement over a useful range of step sizes checks sign and derivative consistency. It\n", "does not prove that the chosen potential models the real system.\n", "\n", "## Determinism, caching, and state\n", "\n", "A force function should not mutate its input state. Otherwise the result can depend on\n", "the order in which bodies or forces are evaluated. Caching is safe only when the cache\n", "key includes every relevant position, velocity, time, parameter, and environmental\n", "state. A field that changes with time makes a position-only cache stale.\n", "\n", "Random microscopic forcing requires an explicit stochastic model and controlled random\n", "stream. It should not appear as hidden nondeterminism in an otherwise ordinary force\n", "function.\n", "\n", "## Validation tests\n", "\n", "- Check direction using simple axis-aligned configurations.\n", "- Check units and parameter ranges for every model.\n", "- Confirm equal and opposite pair forces on distinct bodies.\n", "- Verify inverse-square magnitude changes by a factor of four when distance doubles.\n", "- Verify a spring force is zero at rest length and reverses across equilibrium.\n", "- Verify drag is zero at zero relative velocity and always removes kinetic energy in a\n", " stationary passive fluid model.\n", "- Compare conservative forces with the negative potential gradient.\n", "- Reject or explicitly treat singular and out-of-domain states.\n", "- Evaluate bodies in different orders and obtain the same forces from one snapshot.\n", "- Keep the force model fixed while refining the integrator step, so numerical and model\n", " errors are not confused.\n", "\n", "A force function is trustworthy when its physical source, target, domain, units, and\n", "assumptions are as explicit as its returned numbers. The dynamics layer can then combine\n", "forces and test alternative integrators without rewriting the physics.\n", "\n", "## A worked Python notebook\n", "\n", "Use a quadratic drag model along the positive direction of relative motion. The question for this calculation is precise: how does **drag force**\n", "change when we change **speed relative to air**, 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, area, and drag coefficient are fixed; the signed expression shown is for nonnegative relative speed. 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/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 speed relative to air, expressed in m/s. The output is\n", "drag force, expressed in N. 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", "$$\n", "f(q) = - 0.03 q^{2}\n", "$$\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 3 m/s, the expression gives -0.27 N. 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(\"-0.03*q**2\", locals={\"q\": q})\n", "reference_input = 3.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m/s\")\n", "print(\"Predicted drag force:\", reference_output, \"N\")\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 N. The midpoint, at 5 m/s, gives\n", "-0.75 N. The upper endpoint gives -3 N.\n", "Read these together with the assumptions: they describe different choices of speed relative to air\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, 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/s -> {output_value:.6g} N\")\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 speed relative to air; the vertical axis shows drag force. 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, 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(\"Speed relative to air (m/s)\")\n", "ax.set_ylabel(\"Drag force (N)\")\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", "drag force change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 3 m/s is\n", "-0.18 in those ratio units. Locally, the output decreases 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", "$$\n", "f'(q) = - 0.06 q\n", "$$\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 = 3.05\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, \"N\")\n", "print(\"Full expression:\", full_prediction, \"N\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"N\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "Here the changed input is 3.05 m/s. The tangent estimate is\n", "-0.279 N, while the full expression gives\n", "-0.279075 N. 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 = 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-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", "speed relative to air inside the stated interval, predict whether drag force 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, area, and drag coefficient are fixed; the signed expression shown is for nonnegative relative speed. 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 N:\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(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-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", "\n", "- [OpenStax, *University Physics Volume 1*, Section 6.4, \"Drag Force and Terminal Speed\"](https://openstax.org/books/university-physics-volume-1/pages/6-4-drag-force-and-terminal-speed).\n", "- [OpenStax, *University Physics Volume 1*, Section 13.1, \"Newton's Law of Universal Gravitation\"](https://openstax.org/books/university-physics-volume-1/pages/13-1-newtons-law-of-universal-gravitation).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 8, \"Applications of Newton's Second Law\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter8/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 5, \"Analysis of Forces\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Hooke's Law\"](https://physics.info/elasticity/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "force-models-as-functions.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/force-models-as-functions", "source_sha256": "e4bad79863d24ce76bd820ad755ed085a7bbdd02f238db8320336f371fb9de7f" } }, "nbformat": 4, "nbformat_minor": 5 }