{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Solving coupled equations of motion\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/solving-coupled-equations-motion). 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": [ "Equations of motion are coupled when the acceleration of one coordinate or body depends\n", "on another coordinate or body. A spring between two carts, masses connected by a rope,\n", "and planets attracting one another are coupled systems. They must be solved as one\n", "state: advancing one body before evaluating the others mixes different physical times.\n", "\n", "## One state for the whole system\n", "\n", "For two particles in one dimension, a useful state is\n", "\n", "\n", "$$\n", "\\mathbf y=(x_1,v_1,x_2,v_2)^T\n", "$$\n", "\n", "\n", "Its derivative is\n", "\n", "\n", "$$\n", "\\dot{\\mathbf y}=(v_1,a_1,v_2,a_2)^T\n", "$$\n", "\n", "\n", "The accelerations may depend on both positions and both velocities. State order is a\n", "data contract: a solver that interprets the second component as another position rather\n", "than velocity will return numerically smooth nonsense. Named records can improve\n", "readability, but the mathematical coupling remains the same." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "masses = np.array([1.0, 2.0])\n", "\n", "\n", "def derivative(state):\n", " x1, x2, v1, v2 = state\n", " force = 10.0 * (x2 - x1 - 1.0)\n", " return np.array([v1, v2, force / masses[0], -force / masses[1]])\n", "\n", "\n", "state = np.array([0.0, 1.2, 0.0, 0.0])\n", "print(\"State derivative:\", derivative(state))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The function returns rates; it does not update state. Time stepping is a separate\n", "operation.\n", "\n", "## Example: two carts joined by a spring\n", "\n", "Let cart positions be\n", "$x_1$ and\n", "$x_2$. A spring of rest\n", "length $L_0$ has extension\n", "\n", "\n", "$$\n", "\\Delta L=x_2-x_1-L_0\n", "$$\n", "\n", "\n", "assuming cart two remains to the right. Hooke's law gives equal and opposite forces:\n", "\n", "\n", "$$\n", "m_1\\ddot x_1=k\\Delta L,\\qquad m_2\\ddot x_2=-k\\Delta L\n", "$$\n", "\n", "\n", "If the spring is stretched, it pulls the first cart right and the second cart left. The\n", "signs follow from geometry and Newton's third law; they should not be assigned by two\n", "unrelated functions.\n", "\n", "The center of mass\n", "\n", "\n", "$$\n", "X=\\frac{m_1x_1+m_2x_2}{m_1+m_2}\n", "$$\n", "\n", "\n", "moves at constant velocity when no external horizontal force acts. The relative\n", "coordinate\n", "$q=x_2-x_1-L_0$ obeys\n", "\n", "\n", "$$\n", "\\ddot q=-k\\left(\\frac{1}{m_1}+\\frac{1}{m_2}\\right)q\n", "$$\n", "\n", "\n", "This reduction separates uniform center-of-mass motion from internal oscillation. It\n", "also supplies two independent checks: center-of-mass momentum remains constant, and the\n", "relative motion has angular frequency\n", "\n", "\n", "$$\n", "\\omega=\\sqrt{k\\left(\\frac{1}{m_1}+\\frac{1}{m_2}\\right)}\n", "$$\n", "\n", "\n", "## Evaluate pair forces once\n", "\n", "A robust interaction loop evaluates each unordered pair once and applies opposite\n", "vectors to both bodies." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "pair_force = 10.0 * (state[1] - state[0] - 1.0)\n", "net_forces = np.array([pair_force, -pair_force])\n", "accelerations = net_forces / masses\n", "assert np.isclose(net_forces.sum(), 0.0)\n", "print(\"Accelerations (m/s^2):\", accelerations)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "If the second force is calculated independently, roundoff or inconsistent branches can\n", "violate the third-law pair. If body one is moved before the pair force on body two is\n", "calculated, the interaction uses two separations in one nominal instant.\n", "\n", "## Coupling through a rope constraint\n", "\n", "Consider two masses connected by an ideal massless inextensible rope over an ideal\n", "pulley. With each coordinate chosen positive in its direction of motion, fixed rope\n", "length implies equal acceleration magnitudes. For a simple Atwood machine with\n", "$m_2>m_1$,\n", "\n", "\n", "$$\n", "T-m_1g=m_1a,\\qquad m_2g-T=m_2a\n", "$$\n", "\n", "\n", "Adding eliminates tension:\n", "\n", "\n", "$$\n", "a=\\frac{(m_2-m_1)g}{m_1+m_2}\n", "$$\n", "\n", "\n", "and substitution gives\n", "\n", "\n", "$$\n", "T=\\frac{2m_1m_2}{m_1+m_2}g\n", "$$\n", "\n", "\n", "These results depend on the pulley and rope assumptions. Pulley rotational inertia makes\n", "the two tensions differ; rope stretch adds another dynamical degree of freedom. The\n", "kinematic rope constraint must be written alongside the force equations." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "first_mass, second_mass, gravity = 2.0, 3.0, 9.81\n", "acceleration = (second_mass - first_mass) * gravity / (first_mass + second_mass)\n", "tension = first_mass * (gravity + acceleration)\n", "print(f\"Acceleration: {acceleration:.3f} m/s^2; tension: {tension:.3f} N\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "The expression for tension uses the first mass equation and the chosen sign convention.\n", "Substituting it into the second equation is an independent algebra check.\n", "\n", "## Matrix form for linear coupled motion\n", "\n", "Many small-oscillation systems can be written\n", "\n", "\n", "$$\n", "\\mathbf M\\ddot{\\mathbf q}+\\mathbf C\\dot{\\mathbf q}+\\mathbf K\\mathbf q=\\mathbf f(t)\n", "$$\n", "\n", "\n", "The mass matrix\n", "$\\mathbf M$ couples accelerations,\n", "$\\mathbf C$ couples velocities, and\n", "$\\mathbf K$ couples displacements.\n", "Units differ between these matrices, so treating them as interchangeable arrays is\n", "incorrect.\n", "\n", "To calculate acceleration,\n", "\n", "\n", "$$\n", "\\ddot{\\mathbf q}=\\mathbf M^{-1}(\\mathbf f-\\mathbf C\\dot{\\mathbf q}-\\mathbf K\\mathbf q)\n", "$$\n", "\n", "\n", "but software should normally solve the linear system rather than form an explicit\n", "matrix inverse. A singular mass matrix can indicate redundant coordinates, algebraic\n", "constraints, or a malformed model.\n", "\n", "## Normal modes\n", "\n", "For an undamped unforced linear system, try\n", "$\\mathbf q=\\mathbf u e^{i\\omega t}$.\n", "This leads to\n", "\n", "\n", "$$\n", "(\\mathbf K-\\omega^2\\mathbf M)\\mathbf u=\\mathbf 0\n", "$$\n", "\n", "\n", "The eigenvectors give collective motion patterns, and the eigenvalues give squared\n", "angular frequencies when the physical matrices satisfy the required positivity\n", "conditions. Normal modes do not mean the original coordinates were uncoupled; they are\n", "a transformed basis that decouples this linear idealization.\n", "\n", "## Numerical integration of a coupled state\n", "\n", "Each stage of a Runge-Kutta method must evaluate the complete derivative from one staged\n", "whole-system state. It is wrong to complete all stages for one body while keeping the\n", "others at the old state." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "dt = 0.01\n", "for _ in range(100):\n", " first_slope = derivative(state)\n", " midpoint = state + 0.5 * dt * first_slope\n", " state = state + dt * derivative(midpoint)\n", "print(\"State after 1 s:\", state)\n", "assert np.isclose(np.dot(masses, state[2:]), 0.0, atol=1e-10)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "The same principle applies to implicit methods: solve the coupled unknown next state to\n", "the requested tolerance rather than performing one undocumented sequential pass.\n", "\n", "Coupling can create fast and slow modes. The step must resolve the fastest relevant\n", "mode even if the plotted coordinate mainly shows slow motion. A stable-looking coarse\n", "result may suppress or distort an unresolved fast oscillation.\n", "\n", "## Scaling and solver diagnostics\n", "\n", "Coupled states can contain quantities with very different numerical scales. A position\n", "measured in kilometres, a small angle in radians, and a velocity in metres per second\n", "should not be compared with one raw absolute tolerance. Choose characteristic scales\n", "for each kind of variable or use unit-aware absolute tolerances together with a relative\n", "tolerance. Rescaling changes numerical conditioning but must not change the physical\n", "solution after units are converted back.\n", "\n", "An implicit or algebraic solve should report more than a returned state. Record the\n", "residual of the equations, iteration count, and whether convergence was achieved. A\n", "solver that reaches its iteration limit has not proved the equations are satisfied. If\n", "the residual is a vector, scale its components before reducing them to one norm so that\n", "the largest numerical unit does not hide failure elsewhere.\n", "\n", "Parameter changes can also reveal structural mistakes. Make one mass extremely large\n", "and verify that it approaches a fixed support in the appropriate limit. Make the spring\n", "constant small and recover nearly independent motion. Exchange labels on two identical\n", "bodies and confirm that the solution exchanges in the same way. These tests examine the\n", "coupling itself, while time-step refinement examines the numerical approximation.\n", "\n", "For long calculations, store derived diagnostics separately from the evolving state.\n", "Rounding a displayed center-of-mass position and feeding it back into the next update\n", "silently changes the system. The integrator should retain full internal precision, while\n", "output routines decide how many digits are justified for reporting.\n", "\n", "## Initial conditions must satisfy constraints\n", "\n", "Positions and velocities should satisfy any rope, joint, or manifold constraint at the\n", "initial time. Starting with an inextensible rope at the wrong length or with incompatible\n", "endpoint velocities can create an artificial impulse or persistent solver failure.\n", "\n", "For differential-algebraic systems, not every arbitrary state vector is admissible.\n", "Initialization may require solving constraint equations for dependent coordinates,\n", "velocities, and multipliers.\n", "\n", "## Checks for coupled systems\n", "\n", "- With the coupling parameter set to zero, recover independent single-body equations.\n", "- Pair forces are equal, opposite, and based on one snapshot.\n", "- Symmetric masses and symmetric initial conditions preserve the expected symmetry.\n", "- A closed system preserves total momentum when only internal forces act.\n", "- Conservative spring systems preserve total energy in the continuous model.\n", "- The two-cart center of mass follows the external-force prediction.\n", "- The Atwood result becomes zero acceleration for equal masses.\n", "- Matrix dimensions, units, and symmetry match the physical model.\n", "- Initial states satisfy every geometric and velocity constraint.\n", "- Step refinement converges at shared physical times and resolves the fastest mode.\n", "\n", "Coupled motion is not a collection of unrelated one-body updates. It is one larger\n", "initial-value problem whose forces, constraints, and numerical stages must all refer to\n", "the same system state.\n", "\n", "## A worked Python notebook\n", "\n", "Keep one Atwood-machine mass at 2 kilograms and change the other mass. The question for this calculation is precise: how does **acceleration**\n", "change when we change **second hanging mass**, 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", "The rope and pulley are ideal, and positive acceleration follows the second mass downward. 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", "0.5 to 5 kg. 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 second hanging mass, expressed in kg. The output is\n", "acceleration, expressed in m/s^2. 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", "$$\n", "f(q) = \\frac{9.81 q - 19.62}{q + 2}\n", "$$\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 kg, the expression gives 1.962 m/s^2. 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-011", "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(\"9.81*(q-2)/(q+2)\", locals={\"q\": q})\n", "reference_input = 3.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"kg\")\n", "print(\"Predicted acceleration:\", reference_output, \"m/s^2\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "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", "-5.886 m/s^2. The midpoint, at 2.75 kg, gives\n", "1.54895 m/s^2. The upper endpoint gives 4.20429 m/s^2.\n", "Read these together with the assumptions: they describe different choices of second hanging mass\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-013", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.5, 2.75, 5.0])\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} kg -> {output_value:.6g} m/s^2\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "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 second hanging mass; the vertical axis shows acceleration. 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-015", "metadata": {}, "source": [ "input_grid = np.linspace(0.5, 5.0, 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(\"Second hanging mass (kg)\")\n", "ax.set_ylabel(\"Acceleration (m/s^2)\")\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-016", "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", "acceleration change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 3 kg is\n", "1.5696 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", "$$\n", "f'(q) = \\frac{9.81}{q + 2} - \\frac{9.81 q - 19.62}{\\left(q + 2\\right)^{2}}\n", "$$\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-017", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 3.0225\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/s^2\")\n", "print(\"Full expression:\", full_prediction, \"m/s^2\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m/s^2\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "Here the changed input is 3.0225 kg. The tangent estimate is\n", "1.99732 m/s^2, while the full expression gives\n", "1.99716 m/s^2. 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-019", "metadata": {}, "source": [ "input_span = 4.5\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-020", "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", "second hanging mass inside the stated interval, predict whether acceleration 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: The rope and pulley are ideal, and positive acceleration follows the second mass downward. 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/s^2:\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-021", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(5.0) - evaluate(0.5))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.5, 5.0, 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-022", "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 6.1, \"Solving Problems with Newton's Laws\"](https://openstax.org/books/university-physics-volume-1/pages/6-1-solving-problems-with-newtons-laws).\n", "- [OpenStax, *University Physics Volume 1*, Section 15.1, \"Simple Harmonic Motion\"](https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 8, worked pulley and cart examples](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)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "solving-coupled-equations-motion.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/solving-coupled-equations-motion", "source_sha256": "d2a59acbc9a5de8d2f8d1038237cd75d2372451cb159b857eb9a82c12bf60f0f" } }, "nbformat": 4, "nbformat_minor": 5 }