{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Torque as cross product\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/torque-as-cross-product). 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": [ "Torque measures how a force changes angular momentum about a chosen point or axis. The\n", "same force can produce a large torque, a small torque, or zero torque depending on where\n", "and in which direction it acts.\n", "\n", "## Definition about a point\n", "\n", "If a force\n", "$\\mathbf F$ acts at a point whose\n", "position relative to origin\n", "$O$ is\n", "$\\mathbf r$, its torque about that\n", "origin is\n", "\n", "\n", "$$\n", "\\boldsymbol\\tau_O=\\mathbf r\\times\\mathbf F\n", "$$\n", "\n", "\n", "Its magnitude is\n", "\n", "\n", "$$\n", "\\tau=rF\\sin\\phi\n", "$$\n", "\n", "\n", "where $\\phi$ is the\n", "smaller angle between the vectors. The SI unit is newton metre. Although this unit has\n", "the same base dimensions as a joule, torque and energy are different quantities: torque\n", "is an axial vector associated with rotation, while energy is a scalar.\n", "\n", "The right-hand rule gives torque direction. Curl the fingers of the right hand from\n", "$\\mathbf r$ toward\n", "$\\mathbf F$ through the smaller angle;\n", "the thumb points along\n", "$\\boldsymbol\\tau$.\n", "\n", "## Lever arm and line of action\n", "\n", "The perpendicular lever arm\n", "$r_\\perp$ is the shortest\n", "distance from the origin to the force's line of action. Therefore,\n", "\n", "\n", "$$\n", "\\tau=F r_\\perp\n", "$$\n", "\n", "\n", "Equivalently, the component of force perpendicular to\n", "$\\mathbf r$ produces the torque:\n", "\n", "\n", "$$\n", "\\tau=rF_\\perp\n", "$$\n", "\n", "\n", "Use one of these interpretations, not both at once. Multiplying perpendicular force by\n", "perpendicular distance again would count the sine factor twice.\n", "\n", "A force whose line of action passes through the origin has zero lever arm and zero torque\n", "about that origin. The force can still accelerate the center of mass. Torque and net\n", "force answer different questions." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "origin_m = np.zeros(3)\n", "application_m = np.array([0.4, 0.0, 0.0])\n", "force_n = np.array([0.0, 20.0, 0.0])\n", "torque_nm = np.cross(application_m - origin_m, force_n)\n", "print(\"Torque (N m):\", torque_nm)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "## Component calculation\n", "\n", "For Cartesian components,\n", "\n", "\n", "$$\n", "\\mathbf r\\times\\mathbf F=\\begin{bmatrix}r_yF_z-r_zF_y\\\\r_zF_x-r_xF_z\\\\r_xF_y-r_yF_x\\end{bmatrix}\n", "$$\n", "\n", "\n", "The order matters:\n", "$\\mathbf F\\times\\mathbf r=-\\mathbf r\\times\\mathbf F$.\n", "Swapping operands produces the opposite torque.\n", "\n", "For motion in the\n", "$xy$, only the perpendicular component is\n", "needed:\n", "\n", "\n", "$$\n", "\\tau_z=r_xF_y-r_yF_x\n", "$$\n", "\n", "\n", "If positive\n", "$z$ points out of the page, positive\n", "$\\tau_z$ tends to produce\n", "counterclockwise angular acceleration under the usual convention.\n", "\n", "## Example: pushing a door\n", "\n", "Suppose a force of\n", "$20\\,\\mathrm N$ is applied\n", "perpendicularly at a point\n", "$0.80\\,\\mathrm m$ from a\n", "door hinge. The torque magnitude is\n", "\n", "\n", "$$\n", "\\tau=(0.80\\ \\mathrm m)(20\\ \\mathrm N)=16\\ \\mathrm{N\\,m}\n", "$$\n", "\n", "\n", "Figure 1(a) shows the geometry of this calculation. The upward force's line of action\n", "misses the hinge, and its perpendicular distance from the hinge equals the door's lever\n", "length. The right-hand rule gives torque out of the page, corresponding to the curved\n", "counterclockwise arrow. Applying the same force halfway from the hinge halves this\n", "distance and therefore halves the torque.\n", "\n", "Now follow the dashed line through the force in Figure 1(b): it passes through the\n", "hinge. The perpendicular lever arm is zero even though the handle is still far from\n", "the hinge. This is why pushing toward the hinge produces no torque about it.\n", "\n", "## Net torque from several forces\n", "\n", "Torques about the same origin add vectorially:\n", "\n", "\n", "$$\n", "\\boldsymbol\\tau_{\\mathrm{net},O}=\\sum_k(\\mathbf r_k\\times\\mathbf F_k)\n", "$$\n", "\n", "\n", "For the door in Figure 1, keep O at the hinge when adding further applied forces.\n", "Each force needs its own application point and line of action, just as the two panels\n", "use the same origin but obtain different torques from different force directions.\n", "\n", "Every lever vector must begin at the same origin. Mixing torques calculated about\n", "different points has no clear meaning. If a force is distributed over a body, replace\n", "the sum with an integral:\n", "\n", "\n", "$$\n", "\\boldsymbol\\tau_O=\\int\\mathbf r\\times d\\mathbf F\n", "$$\n", "\n", "\n", "For uniform gravity near Earth's surface, the net gravitational torque on a rigid body\n", "can be calculated as if its weight acted at the center of mass. In a nonuniform\n", "gravitational field, this shortcut is not generally exact because tidal variation across\n", "the body matters." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "forces = [\n", " (np.array([0.4, 0, 0]), np.array([0, 20, 0])),\n", " (np.array([-0.2, 0, 0]), np.array([0, -10, 0])),\n", "]\n", "net_torque = sum(\n", " (np.cross(point - origin_m, force) for point, force in forces), np.zeros(3)\n", ")\n", "print(\"Net torque (N m):\", net_torque)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "## A force couple\n", "\n", "Two equal and opposite forces separated by a distance have zero net force but can have\n", "nonzero net torque. This arrangement is a couple. For ideal equal-and-opposite forces,\n", "the couple torque is independent of the chosen origin because shifting the origin adds\n", "opposite terms that cancel.\n", "\n", "A couple can change rotation without accelerating the center of mass. Turning a steering\n", "wheel with opposite tangential forces from two hands approximates this situation.\n", "\n", "## Changing the origin\n", "\n", "Torque generally depends on origin. If origin is shifted by vector\n", "$\\mathbf a$, net torque changes as\n", "\n", "\n", "$$\n", "\\boldsymbol\\tau_{O'}=\\boldsymbol\\tau_O-\\mathbf a\\times\\mathbf F_{\\mathrm{net}}\n", "$$\n", "\n", "\n", "When net force is zero, net torque is independent of origin. Otherwise, always state the\n", "point about which torque is calculated. In statics, choosing a pivot strategically can\n", "make unknown reaction forces contribute zero torque, but it does not make those forces\n", "physically disappear.\n", "\n", "## Torque and angular momentum\n", "\n", "The general rotational balance law about a fixed point in an inertial frame is\n", "\n", "\n", "$$\n", "\\boldsymbol\\tau_{\\mathrm{net}}=\\frac{d\\mathbf L}{dt}\n", "$$\n", "\n", "\n", "This is more general than\n", "$\\tau=I\\alpha$.\n", "The scalar relation applies to rotation about a fixed principal axis with constant moment\n", "of inertia, or other circumstances where the required components reduce consistently.\n", "\n", "In unrestricted three-dimensional rigid-body motion, angular momentum and angular\n", "velocity are related through the inertia tensor:\n", "\n", "\n", "$$\n", "\\mathbf L=\\mathbf I\\boldsymbol\\omega\n", "$$\n", "\n", "\n", "Torque need not be parallel to angular acceleration, and angular momentum need not be\n", "parallel to angular velocity. Dividing a three-dimensional torque vector by one scalar\n", "inertia is generally incorrect.\n", "\n", "## Torque, work, and power\n", "\n", "For fixed-axis rotation, a torque component about the axis does differential work\n", "\n", "\n", "$$\n", "dW=\\tau\\,d\\theta\n", "$$\n", "\n", "\n", "and instantaneous power is\n", "\n", "\n", "$$\n", "P=\\tau\\omega\n", "$$\n", "\n", "\n", "In vector form, rotational power is\n", "$P=\\boldsymbol\\tau\\cdot\\boldsymbol\\omega$.\n", "A torque perpendicular to angular velocity changes its direction without instantaneous\n", "rotational power, a key idea in gyroscopic precession.\n", "\n", "## Verification strategy\n", "\n", "Test geometry first. Parallel and antiparallel force and lever vectors give zero torque.\n", "Perpendicular unit vectors produce unit magnitude with the right-hand-rule direction.\n", "Reversing force reverses torque. Translating origin and application point together leaves\n", "the lever vector unchanged.\n", "\n", "Then test dynamics separately. For a fixed positive inertia, doubling torque doubles\n", "angular acceleration. Zero net torque preserves angular momentum, not necessarily zero\n", "angular velocity. For numerical force data, compare the cross-product calculation with\n", "the perpendicular-distance calculation in nondegenerate cases.\n", "\n", "## Reliability checks\n", "\n", "- Name the origin or axis and use it for every torque contribution.\n", "- Express position and force in one coordinate frame.\n", "- Preserve cross-product order and right-hand-rule direction.\n", "- Distinguish lever-arm distance from position-vector magnitude.\n", "- Sum vector torques, not their magnitudes.\n", "- Use center-of-mass weight placement only for an effectively uniform gravity field.\n", "- Apply the scalar relation only for a justified fixed-axis model.\n", "- Use the inertia tensor for general three-dimensional rigid-body dynamics.\n", "- Keep torque and energy distinct despite their shared base units.\n", "- Test zero, sign reversal, perpendicular, and origin-shift cases.\n", "\n", "Torque is the geometry-aware form of rotational influence. The cross product preserves\n", "the origin, direction, and lever-arm information needed to connect applied forces with\n", "angular momentum and fixed-axis acceleration.\n", "\n", "## A worked Python notebook\n", "\n", "Apply a 20 newton force at a lever arm of 0.4 metres. The question for this calculation is precise: how does **torque about pivot**\n", "change when we change **force angle**, 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 angle is between the lever arm and force; the scalar sign refers to the chosen normal axis. 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 3.14159 rad. 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 force angle, expressed in rad. The output is\n", "torque about pivot, expressed in N m. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = 8 \\sin{\\left(q \\right)}\n", "$$\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 1.5708 rad, the expression gives 8 N m. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-007", "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(\"8*sin(q)\", locals={\"q\": q})\n", "reference_input = 1.5707963267948966\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"rad\")\n", "print(\"Predicted torque about pivot:\", reference_output, \"N m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "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 m. The midpoint, at 1.5708 rad, gives\n", "8 N m. The upper endpoint gives 9.79717e-16 N m.\n", "Read these together with the assumptions: they describe different choices of force angle\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-009", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.0, 1.5707963267948966, 3.141592653589793])\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} rad -> {output_value:.6g} N m\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "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 force angle; the vertical axis shows torque about pivot. 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-011", "metadata": {}, "source": [ "input_grid = np.linspace(0.0, 3.141592653589793, 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(\"Force angle (rad)\")\n", "ax.set_ylabel(\"Torque about pivot (N m)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "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", "torque about pivot change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 1.5708 rad is\n", "4.89859e-16 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", "$$\n", "f'(q) = 8 \\cos{\\left(q \\right)}\n", "$$\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-013", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 1.5865042900628454\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 m\")\n", "print(\"Full expression:\", full_prediction, \"N m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"N m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Here the changed input is 1.5865 rad. The tangent estimate is\n", "8 N m, while the full expression gives\n", "7.99901 N m. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "input_span = 3.141592653589793\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-016", "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", "force angle inside the stated interval, predict whether torque about pivot 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 angle is between the lever arm and force; the scalar sign refers to the chosen normal axis. 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 m:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(3.141592653589793) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 3.141592653589793, 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-018", "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 10.6, \"Torque\"](https://openstax.org/books/university-physics-volume-1/pages/10-6-torque).\n", "- [OpenStax, *University Physics Volume 1*, Section 10.7, \"Newton's Second Law for Rotation\"](https://openstax.org/books/university-physics-volume-1/pages/10-7-newtons-second-law-for-rotation).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 17, \"Two Dimensional Rotational Dynamics\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter17/).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 19, \"Angular Momentum\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter19/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 8, \"Rotation\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Torque\"](https://physics.info/torque/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "torque-as-cross-product.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/torque-as-cross-product", "source_sha256": "154dd2761a6b2d8671abf567742cc004737951b8a478c4e5daa937547575dc41" } }, "nbformat": 4, "nbformat_minor": 5 }