# Torsion: From Geometry to Shaft Design > Torsion, built from the twist itself. We hang a torque on a round shaft and watch what happens to the material: cross sections stay plane and circular, radii stay straight, and a line ruled along the surface tilts through a shear angle. That one picture gives shear strain proportional to radius with no material property involved. An elastic law turns it into a linear shear stress, and integrating the moment of that stress over the cross section produces both the torque and the polar second moment of area as the integral it actually is, rather than as a symbol handed over in advance. The lecture closes on a design question: a solid bar and a tube of identical weight, side by side, and why every drive shaft you will ever meet is hollow. - Canonical watch page: [Torsion: From Geometry to Shaft Design](https://academa.ai/lectures/torsion-shafts) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Engineering - Published: 2026-08-28T19:38:08.598Z - Updated: 2026-08-28T19:38:08.598Z - Duration: PT777S (12 minutes 57 seconds) - Chapters: 4 - Views: 3 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TYG4BTT9C22N6DCGBTAD6/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TYG4BTT9C22N6DCGBTAD6/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TYG4BTT9C22N6DCGBTAD6/0/dark/poster.jpg) ## Description How twisting a round shaft gives shear strain linear in radius, the torque integral, the polar moment of area, and why drive shafts are tubes. ## Chapters - [00:00–02:40.996 · Twisting the Shaft](https://academa.ai/lectures/torsion-shafts?t=0) - [02:40.996–05:43.074 · Shear Strain from Geometry](https://academa.ai/lectures/torsion-shafts?t=160.99616666666665) - [05:43.074–09:31.854 · Stress, Torque and the Polar Moment](https://academa.ai/lectures/torsion-shafts?t=343.07395833333334) - [09:31.854–12:57 · Solid Shaft or Tube](https://academa.ai/lectures/torsion-shafts?t=571.8537916666667) ## Transcript ### [00:00 · Twisting the Shaft](https://academa.ai/lectures/torsion-shafts?t=0) Axial loading gave you a very comfortable answer. Pull a bar along its own axis and the stress is load over area, the same at every point of the section, and the strain is the same everywhere too. Torsion is the next load case, and almost none of that comfort survives. Here is the question. A round shaft carries a torque T. How is that torque shared out across the cross section, and how much can the shaft carry before the material gives way? We will answer it the way it should be answered, by looking at what the twist does to the material and letting the algebra follow. So here is the shaft. A solid circular bar, with its axis running along z, clamped at the left end and free at the right. Notice that the section is a circle. That matters more than you might think: a square shaft under torque warps out of plane, and everything we are about to do would fail for it. Now hang a torque T on the free end, and rule one straight line along the surface, running from a point A at the fixed end to a point B at the far end. Apply the torque, and the shaft twists. The far end rotates through an angle we will call phi. Watch the cross sections while it turns. Each one rotates in its own plane, and because it is a circle it lands straight back onto itself. Nothing warps out of plane, nothing changes diameter, and no section slides along the axis. Plane sections stay plane. Now paint four radii onto the end face. They swing round together, all through the same angle, and every one of them is still perfectly straight. The section turns like a rigid disc on a hub. The line along the surface is the one thing that does not survive. Here is where it started, in gray, and here is where it has gone. B has been carried round the rim while A stayed put, so the line has tilted over, and that tilt is a shear angle. Notice that the line has barely changed length. There is no stretching here, only shearing. So, three facts, and every one of them came out of the picture rather than out of a material property. Sections stay plane and circular. Each section rotates rigidly, so radii stay straight. And a line along the surface tilts through an angle, which we will call gamma. That last one is the one we can use, because gamma is a strain. Put a number on it, and the whole of torsion falls out. ### [02:40.996 · Shear Strain from Geometry](https://academa.ai/lectures/torsion-shafts?t=160.99616666666665) Now put a number on that tilt. Two pictures. On the left we are looking straight down the axis at the far face of the shaft. On the right I have taken the cylindrical surface at some radius rho, slit it along a line and unrolled it flat, which turns the whole question into plane trigonometry. Start on the left. The whole section has rotated through the angle phi. A material point P sitting at radius rho has been carried round to P prime, and the distance it has travelled is the arc rho phi. Radius times angle, with the angle measured in radians. And there is the key fact already. The distance a point moves is proportional to how far out it sits. Bring the point in toward the axis and it travels less. Push it out to the surface and it travels the most. Settle it back somewhere in between, and remember that a point sitting exactly on the axis does not move at all. Now the right hand picture. The line A B was ruled straight along the shaft, parallel to the axis, and its length is L. A sits at the fixed end and does not move. B sits on the far face, and B has just been carried sideways by exactly that same distance, rho phi. So the line has swung round to A B prime, and the angle at A between where it was and where it is now is the shear strain, gamma. Its tangent is that sideways movement divided by the length: rho phi over L. Torsional strains are small, well under a degree in any shaft you would actually use, so the tangent is the angle to any accuracy that matters. Shear strain is rho phi over L. Now look at what is fixed in that expression. Phi belongs to the whole shaft. So does L. Both of them are the same number for every single point in this cross section. The only thing that varies from point to point is rho. So shear strain is proportional to radius, and nothing but geometry went into that. Watch it happen. At the axis, where rho is nearly zero, the strain almost vanishes. Halfway out it is half of its largest value. And at the outer surface, at radius c, it reaches its maximum, c phi over L. Divide the two and the shaft's own dimensions cancel out. Gamma is rho over c times gamma max. A straight line, zero at the centre and largest at the surface. That is the strain distribution, and no material property has appeared yet. One last way to picture it. Think of the shaft as a nest of thin tubes, one inside the next. Each tube shears a little relative to its neighbour, and the further out you go the more of that sliding has piled up. The skin does the most work. The axis does none. ### [05:43.074 · Stress, Torque and the Polar Moment](https://academa.ai/lectures/torsion-shafts?t=343.07395833333334) Strain is geometry. Stress needs the material, and for that we need one constitutive law. Below the elastic limit, shear stress and shear strain are proportional. Tau equals G gamma, where G is the shear modulus, the shear counterpart of Young's modulus. Substitute the strain we just derived. Tau is G rho phi over L. G is a material constant, phi and L belong to the whole shaft, so once again the only quantity that varies across the section is rho. Shear stress grows linearly with radius. So here is the stress at one point, a distance rho out from the axis. It acts tangentially, at right angles to the radius, and its size follows that straight line rule. Move the point outward and the arrow grows in exact proportion. Take it right out to the outer surface, at radius c, and the stress reaches its largest value there. Every point on that rim sits at the same radius, so the entire outer skin is at the maximum together. And here is the whole distribution across a radius. Nothing at the axis, rising straight out to tau max at the surface. Written down, that is tau equals rho over c times tau max. Now collect what all of that stress adds up to. Take a small patch of area, d A, sitting at radius rho. The force on that patch is stress times area, tau d A. That force acts tangentially, at a distance rho from the axis, so its moment about the axis is rho times tau d A. That is the torque this one little patch carries. The total torque is the sum over every patch in the section, which is the integral of rho tau d A taken over the area. Now substitute the stress distribution. Tau max over c is the same number everywhere in this section, so it comes straight outside the integral, and what is left behind is the integral of rho squared d A. Stop and look at that integral, because it is the whole point. There is no load in it and no material property in it. It contains nothing but the shape of the cross section, and where that shape puts its area. That integral is the polar second moment of area, J. So work it out for a solid circle. The natural element is a thin ring, because every point on a ring sits at the same radius rho. Give it a thickness d rho, and its area is its circumference times that thickness, two pi rho d rho. Substitute that in and the double integral collapses to an ordinary one in a single variable, running from zero at the axis out to c at the surface. Sweep the ring outward and you are sweeping through exactly that. Integrate rho cubed, and J comes out as pi c to the fourth over two. Notice the fourth power. Doubling the radius multiplies J by sixteen. Put J back into the torque relation and rearrange. The shear stress at radius rho is T rho over J. That is the torsion formula, and it is the exact counterpart of stress equals load over area. And phi came along for the ride. Thread the same substitution back through the strain we derived, and the angle of twist is T L over G J. Strength and stiffness both hang on the same integral. So the whole of torsion sits in J, and J is a statement about where the material is. Which raises exactly one design question. Where should you put it? ### [09:31.854 · Solid Shaft or Tube](https://academa.ai/lectures/torsion-shafts?t=571.8537916666667) So where should the material be? Here is the cross section of a solid steel shaft, thirty millimetres in radius, carrying a torque about its own axis. Go back to what J was actually measuring, the integral of rho squared d A. Every scrap of area is weighted by the square of its distance from the axis. So it is worth asking what the middle of this bar is earning. Shade in everything inside half the radius, everything within fifteen millimetres of the axis. By area, that core is a quarter of the whole bar. A quarter of the weight you carry around, and a quarter of the steel you paid for. But J for that core alone is pi times fifteen to the fourth over two, and fifteen to the fourth is one sixteenth of thirty to the fourth. So a quarter of the material is contributing about six percent of the torsional stiffness. The middle of a shaft is very close to dead weight. So take it out, and put it where the rho squared can get at it. Here is a tube, outer radius fifty millimetres, inner radius forty. Its area is pi times fifty squared minus forty squared, which is pi times nine hundred, and that is exactly the area of the solid bar. Same steel per metre, same weight. Now put numbers against them. Both sections have an area of about two thousand eight hundred and thirty square millimetres, so a metre of each weighs the same. J for the solid bar is pi over two times thirty to the fourth, about one point two seven million millimetres to the fourth. For the tube it is pi over two times fifty to the fourth minus forty to the fourth, and that comes to five point eight million. Four and a half times as much, for the same weight of steel. That is stiffness. Strength is a slightly different sum, because the torsion formula puts the peak stress right at the outer surface, where rho equals c. The torque you can carry at a given allowable stress is tau times J over c, and the tube has the bigger c. Work it out. Forty two thousand cubic millimetres for the solid bar, one hundred and sixteen thousand for the tube. Two point seven times the torque, at exactly the same weight. That is why a drive shaft is a tube. There are limits, of course. Make the wall too thin and the tube buckles, or dents, long before the material anywhere near yields, and a hollow shaft costs more to make and more to join. But the trend is not subtle, and it is why almost every drive shaft, every bicycle frame and every aircraft control tube you will ever meet is hollow. So, the whole argument in four lines. Geometry alone gave us shear strain proportional to radius. An elastic material turned that into shear stress proportional to radius. Integrating the moment of that stress over the area gave the torque, and dragged the polar second moment out into the open as an integral. And because that integral weights area by rho squared, metal near the axis earns almost nothing. And all of it came out of one picture. A line ruled along the surface of a bar, and what happened to that line when you twisted it. ## About Academa, Inc. Academa makes technical knowledge easier to understand through visual lectures and lets learners request new lecture videos on the topics they need. ## Complete audiovisual record Immutable source: [semantic.json](https://academa.ai/media/l/01M14TYG4BTT9C22N6DCGBTAD6/0/semantic.json) Record version: 1. Render attempt: 0. ### How to read this timeline Each scene owns its object identifiers. A beat's board is the complete board when listed, empty when marked empty, and unchanged from the nearest earlier listed board in the same scene when marked unchanged. Action times are absolute positions in the published video. ### Scene 1: [Twisting the Shaft](https://academa.ai/lectures/torsion-shafts?t=0) Span: 00:00–02:40.996 (0s–160.99616666666665s). #### Objects - card: a Title that says "Mechanics of Materials — Torsion: From Geometry to Shaft Design" - facts: a Block \[text\] that says "Each cross section stays plane and stays circular. Each section rotates rigidly, so radii stay straight. A line along the surface tilts through the shear angle $gamma$." - frame: an Axes3D (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), z\_range=(0.0, 3.6)) - generator: a Line \[red\] drawn in frame (start=(1.0, 0.0, 0.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)) - head\_kinematics: a Heading that says "What the Twist Does" - mid\_section: a Circle \[green\] drawn in frame (center=(0.0, 0.0, 1.5), normal\_vector=(0.0, 0.0, 1.0)) - pt\_a: a Point \[text\] labelled "A" drawn in frame (location=(1.0, 0.0, 0.0)) - pt\_b: a Point \[text\] labelled "B" drawn in frame (location=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)) - quarter\_section: a Circle \[green\] drawn in frame (center=(0.0, 0.0, 0.75), normal\_vector=(0.0, 0.0, 1.0)) - question: a Panel that says "A round shaft carries a torque $T$. How is that torque shared out across the cross section, and how much can the shaft carry?" - reference: a Line \[gray\] drawn in frame (start=(1.0, 0.0, 0.0), end=(1.0, 0.0, 3.0), dashed=True) - shaft: a Cylinder \[blue\] drawn in frame (end=(0.0, 0.0, 3.0), opacity=0.16) - spoke\_east: a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)) - spoke\_north: a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 1.5707963267948966))), (1.0 \* sin((twist +…) - spoke\_south: a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 4.71238898038469))), (1.0 \* sin((twist + 4…) - spoke\_west: a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 3.141592653589793))), (1.0 \* sin((twist + …) - torque\_mark: a CurvedArrow \[magenta\] labelled "T" drawn in frame (start=(1.15, -0.5, 3.45), end=(-0.5, 1.15, 3.45)) - twist: a VariableNumber (format\_spec='.2f') #### Beats ##### [00:00](https://academa.ai/lectures/torsion-shafts?t=0) Narration: Axial loading gave you a very comfortable answer. Pull a bar along its own axis and the stress is load over area, the same at every point of the section, and the strain is the same everywhere too. Torsion is the next load case, and almost none of that comfort survives. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/torsion-shafts?t=0): card is shown on the screen, written out. - [00:1.5](https://academa.ai/lectures/torsion-shafts?t=1.5): card: enter:write-left-to-right. - [00:17.31](https://academa.ai/lectures/torsion-shafts?t=17.31): card is hidden from the screen — left the board. ##### [00:18.51](https://academa.ai/lectures/torsion-shafts?t=18.509999999999998) Narration: Here is the question. A round shaft carries a torque T. How is that torque shared out across the cross section, and how much can the shaft carry before the material gives way? We will answer it the way it should be answered, by looking at what the twist does to the material and letting the algebra follow. Board: Unchanged from the preceding beat in this scene. Actions: - [00:18.51](https://academa.ai/lectures/torsion-shafts?t=18.509999999999998): question is shown on the screen, written out. - [00:36.819](https://academa.ai/lectures/torsion-shafts?t=36.819): question moves to a new place on the board. ##### [00:38.019](https://academa.ai/lectures/torsion-shafts?t=38.019) Narration: So here is the shaft. A solid circular bar, with its axis running along z, clamped at the left end and free at the right. Notice that the section is a circle. That matters more than you might think: a square shaft under torque warps out of plane, and everything we are about to do would fail for it. Board: question — a Panel that says "A round shaft carries a torque $T$. How is that torque shared out across the cross section, and how much can the shaft carry?" Actions: - [00:38.019](https://academa.ai/lectures/torsion-shafts?t=38.019): frame is shown on the screen, written out. - [00:41.769](https://academa.ai/lectures/torsion-shafts?t=41.769): shaft is shown on the screen, written out. - [00:42.884](https://academa.ai/lectures/torsion-shafts?t=42.884): frame turns in its own slot. ##### [00:59.273](https://academa.ai/lectures/torsion-shafts?t=59.2735) Narration: Now hang a torque T on the free end, and rule one straight line along the surface, running from a point A at the fixed end to a point B at the far end. Apply the torque, and the shaft twists. The far end rotates through an angle we will call phi. Board: question — a Panel that says "A round shaft carries a torque $T$. How is that torque shared out across the cross section, and how much can the shaft carry?"; frame — an Axes3D (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), z\_range=(0.0, 3.6)); shaft — a Cylinder \[blue\] drawn in frame (end=(0.0, 0.0, 3.0), opacity=0.16) Actions: - [01:0.318](https://academa.ai/lectures/torsion-shafts?t=60.318): torque\_mark is shown on the screen, written out. - [01:2.315](https://academa.ai/lectures/torsion-shafts?t=62.315): reference is shown on the screen, written out. - [01:2.315](https://academa.ai/lectures/torsion-shafts?t=62.315): generator is shown on the screen, written out. - [01:6.135](https://academa.ai/lectures/torsion-shafts?t=66.13499999999999): pt\_a is shown on the screen, written out. - [01:6.435](https://academa.ai/lectures/torsion-shafts?t=66.43499999999999): pt\_b is shown on the screen, written out. - [01:11.638](https://academa.ai/lectures/torsion-shafts?t=71.638): generator is redrawn as the numbers it depends on change. - [01:11.638](https://academa.ai/lectures/torsion-shafts?t=71.638): pt\_b is redrawn as the numbers it depends on change. - [01:11.638](https://academa.ai/lectures/torsion-shafts?t=71.638): spoke\_east is redrawn as the numbers it depends on change. - [01:11.638](https://academa.ai/lectures/torsion-shafts?t=71.638): spoke\_north is redrawn as the numbers it depends on change. - [01:11.638](https://academa.ai/lectures/torsion-shafts?t=71.638): spoke\_west is redrawn as the numbers it depends on change. - [01:11.638](https://academa.ai/lectures/torsion-shafts?t=71.638): spoke\_south is redrawn as the numbers it depends on change. - [01:11.638](https://academa.ai/lectures/torsion-shafts?t=71.638): twist ticks to 0.3. ##### [01:17.346](https://academa.ai/lectures/torsion-shafts?t=77.346) Narration: Watch the cross sections while it turns. Each one rotates in its own plane, and because it is a circle it lands straight back onto itself. Nothing warps out of plane, nothing changes diameter, and no section slides along the axis. Plane sections stay plane. Board: question — a Panel that says "A round shaft carries a torque $T$. How is that torque shared out across the cross section, and how much can the shaft carry?"; frame — an Axes3D (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), z\_range=(0.0, 3.6)); shaft — a Cylinder \[blue\] drawn in frame (end=(0.0, 0.0, 3.0), opacity=0.16); torque\_mark — a CurvedArrow \[magenta\] labelled "T" drawn in frame (start=(1.15, -0.5, 3.45), end=(-0.5, 1.15, 3.45)); reference — a Line \[gray\] drawn in frame (start=(1.0, 0.0, 0.0), end=(1.0, 0.0, 3.0), dashed=True); generator — a Line \[red\] drawn in frame (start=(1.0, 0.0, 0.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.0, 0.0, 0.0)); pt\_b — a Point \[text\] labelled "B" drawn in frame (location=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)) Actions: - [01:18.148](https://academa.ai/lectures/torsion-shafts?t=78.148): quarter\_section is shown on the screen, written out. - [01:18.398](https://academa.ai/lectures/torsion-shafts?t=78.398): mid\_section is shown on the screen, written out. - [01:21.085](https://academa.ai/lectures/torsion-shafts?t=81.08500000000001): generator is redrawn as the numbers it depends on change. - [01:21.085](https://academa.ai/lectures/torsion-shafts?t=81.08500000000001): pt\_b is redrawn as the numbers it depends on change. - [01:21.085](https://academa.ai/lectures/torsion-shafts?t=81.08500000000001): spoke\_east is redrawn as the numbers it depends on change. - [01:21.085](https://academa.ai/lectures/torsion-shafts?t=81.08500000000001): spoke\_north is redrawn as the numbers it depends on change. - [01:21.085](https://academa.ai/lectures/torsion-shafts?t=81.08500000000001): spoke\_west is redrawn as the numbers it depends on change. - [01:21.085](https://academa.ai/lectures/torsion-shafts?t=81.08500000000001): spoke\_south is redrawn as the numbers it depends on change. - [01:21.085](https://academa.ai/lectures/torsion-shafts?t=81.08500000000001): twist ticks to 0.5. ##### [01:36.117](https://academa.ai/lectures/torsion-shafts?t=96.1165) Narration: Now paint four radii onto the end face. They swing round together, all through the same angle, and every one of them is still perfectly straight. The section turns like a rigid disc on a hub. Board: question — a Panel that says "A round shaft carries a torque $T$. How is that torque shared out across the cross section, and how much can the shaft carry?"; frame — an Axes3D (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), z\_range=(0.0, 3.6)); shaft — a Cylinder \[blue\] drawn in frame (end=(0.0, 0.0, 3.0), opacity=0.16); torque\_mark — a CurvedArrow \[magenta\] labelled "T" drawn in frame (start=(1.15, -0.5, 3.45), end=(-0.5, 1.15, 3.45)); reference — a Line \[gray\] drawn in frame (start=(1.0, 0.0, 0.0), end=(1.0, 0.0, 3.0), dashed=True); generator — a Line \[red\] drawn in frame (start=(1.0, 0.0, 0.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.0, 0.0, 0.0)); pt\_b — a Point \[text\] labelled "B" drawn in frame (location=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); quarter\_section — a Circle \[green\] drawn in frame (center=(0.0, 0.0, 0.75), normal\_vector=(0.0, 0.0, 1.0)); mid\_section — a Circle \[green\] drawn in frame (center=(0.0, 0.0, 1.5), normal\_vector=(0.0, 0.0, 1.0)) Actions: - [01:36.987](https://academa.ai/lectures/torsion-shafts?t=96.987): spoke\_east is shown on the screen, written out. - [01:37.107](https://academa.ai/lectures/torsion-shafts?t=97.107): spoke\_north is shown on the screen, written out. - [01:37.227](https://academa.ai/lectures/torsion-shafts?t=97.22699999999999): spoke\_west is shown on the screen, written out. - [01:37.347](https://academa.ai/lectures/torsion-shafts?t=97.347): spoke\_south is shown on the screen, written out. - [01:40.273](https://academa.ai/lectures/torsion-shafts?t=100.273): generator is redrawn as the numbers it depends on change. - [01:40.273](https://academa.ai/lectures/torsion-shafts?t=100.273): pt\_b is redrawn as the numbers it depends on change. - [01:40.273](https://academa.ai/lectures/torsion-shafts?t=100.273): spoke\_east is redrawn as the numbers it depends on change. - [01:40.273](https://academa.ai/lectures/torsion-shafts?t=100.273): spoke\_north is redrawn as the numbers it depends on change. - [01:40.273](https://academa.ai/lectures/torsion-shafts?t=100.273): spoke\_west is redrawn as the numbers it depends on change. - [01:40.273](https://academa.ai/lectures/torsion-shafts?t=100.273): spoke\_south is redrawn as the numbers it depends on change. - [01:40.273](https://academa.ai/lectures/torsion-shafts?t=100.273): twist ticks to 0.65. ##### [01:49.906](https://academa.ai/lectures/torsion-shafts?t=109.9055) Narration: The line along the surface is the one thing that does not survive. Here is where it started, in gray, and here is where it has gone. B has been carried round the rim while A stayed put, so the line has tilted over, and that tilt is a shear angle. Notice that the line has barely changed length. There is no stretching here, only shearing. Board: question — a Panel that says "A round shaft carries a torque $T$. How is that torque shared out across the cross section, and how much can the shaft carry?"; frame — an Axes3D (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), z\_range=(0.0, 3.6)); shaft — a Cylinder \[blue\] drawn in frame (end=(0.0, 0.0, 3.0), opacity=0.16); torque\_mark — a CurvedArrow \[magenta\] labelled "T" drawn in frame (start=(1.15, -0.5, 3.45), end=(-0.5, 1.15, 3.45)); reference — a Line \[gray\] drawn in frame (start=(1.0, 0.0, 0.0), end=(1.0, 0.0, 3.0), dashed=True); generator — a Line \[red\] drawn in frame (start=(1.0, 0.0, 0.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.0, 0.0, 0.0)); pt\_b — a Point \[text\] labelled "B" drawn in frame (location=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); quarter\_section — a Circle \[green\] drawn in frame (center=(0.0, 0.0, 0.75), normal\_vector=(0.0, 0.0, 1.0)); mid\_section — a Circle \[green\] drawn in frame (center=(0.0, 0.0, 1.5), normal\_vector=(0.0, 0.0, 1.0)); spoke\_east — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); spoke\_north — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 1.5707963267948966))), (1.0 \* sin((twist +…); spoke\_west — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 3.141592653589793))), (1.0 \* sin((twist + …); spoke\_south — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 4.71238898038469))), (1.0 \* sin((twist + 4…) Actions: - [01:56.001](https://academa.ai/lectures/torsion-shafts?t=116.00099999999999): reference is indicated — a transient flash. - [01:57.823](https://academa.ai/lectures/torsion-shafts?t=117.823): generator is indicated — a transient flash. - [01:59.739](https://academa.ai/lectures/torsion-shafts?t=119.73899999999999): generator is redrawn as the numbers it depends on change. - [01:59.739](https://academa.ai/lectures/torsion-shafts?t=119.73899999999999): pt\_b is redrawn as the numbers it depends on change. - [01:59.739](https://academa.ai/lectures/torsion-shafts?t=119.73899999999999): spoke\_east is redrawn as the numbers it depends on change. - [01:59.739](https://academa.ai/lectures/torsion-shafts?t=119.73899999999999): spoke\_north is redrawn as the numbers it depends on change. - [01:59.739](https://academa.ai/lectures/torsion-shafts?t=119.73899999999999): spoke\_west is redrawn as the numbers it depends on change. - [01:59.739](https://academa.ai/lectures/torsion-shafts?t=119.73899999999999): spoke\_south is redrawn as the numbers it depends on change. - [01:59.739](https://academa.ai/lectures/torsion-shafts?t=119.73899999999999): twist ticks to 0.8. - [02:12.069](https://academa.ai/lectures/torsion-shafts?t=132.06900000000002): frame moves to a new place on the board. - [02:12.069](https://academa.ai/lectures/torsion-shafts?t=132.06900000000002): question is hidden from the screen — left the board. ##### [02:12.669](https://academa.ai/lectures/torsion-shafts?t=132.669) Narration: So, three facts, and every one of them came out of the picture rather than out of a material property. Sections stay plane and circular. Each section rotates rigidly, so radii stay straight. And a line along the surface tilts through an angle, which we will call gamma. Board: frame — an Axes3D (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), z\_range=(0.0, 3.6)); shaft — a Cylinder \[blue\] drawn in frame (end=(0.0, 0.0, 3.0), opacity=0.16); torque\_mark — a CurvedArrow \[magenta\] labelled "T" drawn in frame (start=(1.15, -0.5, 3.45), end=(-0.5, 1.15, 3.45)); reference — a Line \[gray\] drawn in frame (start=(1.0, 0.0, 0.0), end=(1.0, 0.0, 3.0), dashed=True); generator — a Line \[red\] drawn in frame (start=(1.0, 0.0, 0.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.0, 0.0, 0.0)); pt\_b — a Point \[text\] labelled "B" drawn in frame (location=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); quarter\_section — a Circle \[green\] drawn in frame (center=(0.0, 0.0, 0.75), normal\_vector=(0.0, 0.0, 1.0)); mid\_section — a Circle \[green\] drawn in frame (center=(0.0, 0.0, 1.5), normal\_vector=(0.0, 0.0, 1.0)); spoke\_east — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); spoke\_north — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 1.5707963267948966))), (1.0 \* sin((twist +…); spoke\_west — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 3.141592653589793))), (1.0 \* sin((twist + …); spoke\_south — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 4.71238898038469))), (1.0 \* sin((twist + 4…) Actions: - [02:12.669](https://academa.ai/lectures/torsion-shafts?t=132.669): head\_kinematics is shown on the screen, written out. - [02:13.737](https://academa.ai/lectures/torsion-shafts?t=133.737): facts is shown on the screen, written out. - [02:20.703](https://academa.ai/lectures/torsion-shafts?t=140.703): facts (the "stays plane" part) is emphasized. - [02:24.581](https://academa.ai/lectures/torsion-shafts?t=144.581): facts (the "radii stay straight" part) is emphasized. - [02:24.581](https://academa.ai/lectures/torsion-shafts?t=144.581): facts (the "stays plane" part) is no longer emphasized. - [02:27.565](https://academa.ai/lectures/torsion-shafts?t=147.565): facts (the "radii stay straight" part) is no longer emphasized. - [02:27.565](https://academa.ai/lectures/torsion-shafts?t=147.565): facts (the "tilts through" part) is emphasized. ##### [02:31.81](https://academa.ai/lectures/torsion-shafts?t=151.81) Narration: That last one is the one we can use, because gamma is a strain. Put a number on it, and the whole of torsion falls out. Board: frame — an Axes3D (x\_range=(-1.4, 1.4), y\_range=(-1.4, 1.4), z\_range=(0.0, 3.6)); shaft — a Cylinder \[blue\] drawn in frame (end=(0.0, 0.0, 3.0), opacity=0.16); torque\_mark — a CurvedArrow \[magenta\] labelled "T" drawn in frame (start=(1.15, -0.5, 3.45), end=(-0.5, 1.15, 3.45)); reference — a Line \[gray\] drawn in frame (start=(1.0, 0.0, 0.0), end=(1.0, 0.0, 3.0), dashed=True); generator — a Line \[red\] drawn in frame (start=(1.0, 0.0, 0.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.0, 0.0, 0.0)); pt\_b — a Point \[text\] labelled "B" drawn in frame (location=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); quarter\_section — a Circle \[green\] drawn in frame (center=(0.0, 0.0, 0.75), normal\_vector=(0.0, 0.0, 1.0)); mid\_section — a Circle \[green\] drawn in frame (center=(0.0, 0.0, 1.5), normal\_vector=(0.0, 0.0, 1.0)); spoke\_east — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 0.0))), (1.0 \* sin((twist + 0.0))), 3.0)); spoke\_north — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 1.5707963267948966))), (1.0 \* sin((twist +…); spoke\_west — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 3.141592653589793))), (1.0 \* sin((twist + …); spoke\_south — a Line \[yellow\] drawn in frame (start=(0.0, 0.0, 3.0), end=((1.0 \* cos((twist + 4.71238898038469))), (1.0 \* sin((twist + 4…); facts — a Block \[text\] that says "Each cross section stays plane and stays circular. Each section rotates rigidly, so radii stay straight. A line along the surface tilts through the shear angle $gamma$."; head\_kinematics — a Heading that says "What the Twist Does" Actions: - [02:31.81](https://academa.ai/lectures/torsion-shafts?t=151.81): facts (the "tilts through" part) is no longer emphasized. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): facts is hidden from the screen — left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): frame is hidden from the screen — left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): shaft is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): torque\_mark is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): reference is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): generator is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): pt\_a is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): pt\_b is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): quarter\_section is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): mid\_section is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): spoke\_east is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): spoke\_north is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): spoke\_west is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): spoke\_south is hidden from the screen — frame left the board. - [02:39.954](https://academa.ai/lectures/torsion-shafts?t=159.9545): head\_kinematics is hidden from the screen — left the board. ### Scene 2: [Shear Strain from Geometry](https://academa.ai/lectures/torsion-shafts?t=160.99616666666665) Span: 02:40.996–05:43.074 (160.99616666666665s–343.07395833333334s). #### Objects - arm\_after: a Line \[red\] drawn in end\_view (end=((rho \* 0.9004471023526769), (rho \* 0.43496553411123023))) - arm\_before: a Line \[gray\] drawn in end\_view - axis\_line: a Line \[gray\] drawn in unrolled (end=(3.0, 0.0), dashed=True) - end\_view: a Figure (x\_range=(-1.3, 1.3), y\_range=(-1.3, 1.3), aspect=(1.0, 1.0)) - head\_strain: a Heading that says "Strain Grows With Radius" - head\_two: a Heading that says "Two Views of the Same Twist" - hub: a Point \[text\] labelled "O" drawn in end\_view - label\_left: a Tex \[text\] that says "Looking down the axis" (underline=True) - label\_right: a Tex \[text\] that says "The surface, unrolled flat" (underline=True) - length\_label: a Math \[text\] that says "$L$" drawn in unrolled - offset: a Line \[green\] labelled "rho phi" drawn in unrolled (start=(3.0, 0.0), end=(3.0, (rho \* 0.45))) - p\_after: a Point \[red\] labelled "P'" drawn in end\_view (location=((rho \* 0.9004471023526769), (rho \* 0.43496553411123023))) - p\_before: a Point \[text\] labelled "P" drawn in end\_view (location=(\, 0.0)) - point: a Point \[yellow\] drawn in end\_view - pt\_a2: a Point \[text\] labelled "A" drawn in unrolled - pt\_b2: a Point \[text\] labelled "B" drawn in unrolled (location=(3.0, 0.0)) - pt\_bp: a Point \[red\] labelled "B'" drawn in unrolled (location=(3.0, (rho \* 0.45))) - rho: a VariableNumber (initial\_value=0.6, format\_spec='.2f') - rim: a Circle \[blue\] drawn in end\_view - ring: a Circle \[gray\] drawn in end\_view - shear\_mark: an Angle \[yellow\] labelled "gamma" drawn in unrolled (sides=((3.0, 0.0), (3.0, (rho \* 0.45))), radius=0.5) - tilted: a Line \[red\] drawn in unrolled (end=(3.0, (rho \* 0.45))) - travel: a Line \[green\] labelled "rho phi" drawn in end\_view (start=(\, 0.0), end=((rho \* 0.9004471023526769), (rho \* 0.43496553411123023))) - turn\_mark: an Angle \[yellow\] labelled "phi" drawn in end\_view (sides=((\, 0.0), ((rho \* 0.9004471023526769)…, radius=0.22) - unrolled: a Figure (x\_range=(-0.4, 3.6), y\_range=(-0.6, 1.2), aspect=(4.0, 1.8)) - work: a Derivation \[text\] that says "$tan gamma &= frac(rho phi, L) \\ gamma &= frac(rho phi, L) \\ gamma\_upright("max") &= frac(c phi, L) \\ gamma &= frac(rho, c) gamma\_upright("max")$" #### Beats ##### [02:40.996](https://academa.ai/lectures/torsion-shafts?t=160.99616666666665) Narration: Now put a number on that tilt. Two pictures. On the left we are looking straight down the axis at the far face of the shaft. On the right I have taken the cylindrical surface at some radius rho, slit it along a line and unrolled it flat, which turns the whole question into plane trigonometry. Board: Empty. Actions: - [02:40.996](https://academa.ai/lectures/torsion-shafts?t=160.99616666666665): head\_two is shown on the screen, written out. - [02:45.118](https://academa.ai/lectures/torsion-shafts?t=165.11816666666664): end\_view is shown on the screen, written out. - [02:45.118](https://academa.ai/lectures/torsion-shafts?t=165.11816666666664): rim is shown on the screen, written out. - [02:45.318](https://academa.ai/lectures/torsion-shafts?t=165.31816666666666): hub is shown on the screen, written out. - [02:45.718](https://academa.ai/lectures/torsion-shafts?t=165.71816666666666): ring is shown on the screen, written out. - [02:46.318](https://academa.ai/lectures/torsion-shafts?t=166.31816666666666): arm\_before is shown on the screen, written out. - [02:47.118](https://academa.ai/lectures/torsion-shafts?t=167.11816666666664): p\_before is shown on the screen, written out. - [02:47.118](https://academa.ai/lectures/torsion-shafts?t=167.11816666666664): label\_left is shown on the screen, written out. - [02:49.762](https://academa.ai/lectures/torsion-shafts?t=169.76216666666664): unrolled is shown on the screen, written out. - [02:49.762](https://academa.ai/lectures/torsion-shafts?t=169.76216666666664): axis\_line is shown on the screen, written out. - [02:49.962](https://academa.ai/lectures/torsion-shafts?t=169.96216666666666): pt\_a2 is shown on the screen, written out. - [02:50.362](https://academa.ai/lectures/torsion-shafts?t=170.36216666666667): pt\_b2 is shown on the screen, written out. - [02:50.362](https://academa.ai/lectures/torsion-shafts?t=170.36216666666667): label\_right is shown on the screen, written out. ##### [03:0.974](https://academa.ai/lectures/torsion-shafts?t=180.97366666666665) Narration: Start on the left. The whole section has rotated through the angle phi. A material point P sitting at radius rho has been carried round to P prime, and the distance it has travelled is the arc rho phi. Radius times angle, with the angle measured in radians. Board: label\_left — a Tex \[text\] that says "Looking down the axis" (underline=True); end\_view — a Figure (x\_range=(-1.3, 1.3), y\_range=(-1.3, 1.3), aspect=(1.0, 1.0)); label\_right — a Tex \[text\] that says "The surface, unrolled flat" (underline=True); unrolled — a Figure (x\_range=(-0.4, 3.6), y\_range=(-0.6, 1.2), aspect=(4.0, 1.8)); head\_two — a Heading that says "Two Views of the Same Twist"; rim — a Circle \[blue\] drawn in end\_view; hub — a Point \[text\] labelled "O" drawn in end\_view; ring — a Circle \[gray\] drawn in end\_view; arm\_before — a Line \[gray\] drawn in end\_view; p\_before — a Point \[text\] labelled "P" drawn in end\_view (location=(\, 0.0)); axis\_line — a Line \[gray\] drawn in unrolled (end=(3.0, 0.0), dashed=True); pt\_a2 — a Point \[text\] labelled "A" drawn in unrolled; pt\_b2 — a Point \[text\] labelled "B" drawn in unrolled (location=(3.0, 0.0)) Actions: - [03:3.69](https://academa.ai/lectures/torsion-shafts?t=183.69016666666664): arm\_after is shown on the screen, written out. - [03:3.99](https://academa.ai/lectures/torsion-shafts?t=183.99016666666665): p\_after is shown on the screen, written out. - [03:4.886](https://academa.ai/lectures/torsion-shafts?t=184.88616666666667): turn\_mark is shown on the screen, written out. - [03:13.513](https://academa.ai/lectures/torsion-shafts?t=193.51316666666665): travel is shown on the screen, written out. ##### [03:19.895](https://academa.ai/lectures/torsion-shafts?t=199.89466666666667) Narration: And there is the key fact already. The distance a point moves is proportional to how far out it sits. Bring the point in toward the axis and it travels less. Push it out to the surface and it travels the most. Settle it back somewhere in between, and remember that a point sitting exactly on the axis does not move at all. Board: label\_left — a Tex \[text\] that says "Looking down the axis" (underline=True); end\_view — a Figure (x\_range=(-1.3, 1.3), y\_range=(-1.3, 1.3), aspect=(1.0, 1.0)); label\_right — a Tex \[text\] that says "The surface, unrolled flat" (underline=True); unrolled — a Figure (x\_range=(-0.4, 3.6), y\_range=(-0.6, 1.2), aspect=(4.0, 1.8)); head\_two — a Heading that says "Two Views of the Same Twist"; rim — a Circle \[blue\] drawn in end\_view; hub — a Point \[text\] labelled "O" drawn in end\_view; ring — a Circle \[gray\] drawn in end\_view; arm\_before — a Line \[gray\] drawn in end\_view; p\_before — a Point \[text\] labelled "P" drawn in end\_view (location=(\, 0.0)); axis\_line — a Line \[gray\] drawn in unrolled (end=(3.0, 0.0), dashed=True); pt\_a2 — a Point \[text\] labelled "A" drawn in unrolled; pt\_b2 — a Point \[text\] labelled "B" drawn in unrolled (location=(3.0, 0.0)); arm\_after — a Line \[red\] drawn in end\_view (end=((rho \* 0.9004471023526769), (rho \* 0.43496553411123023))); p\_after — a Point \[red\] labelled "P'" drawn in end\_view (location=((rho \* 0.9004471023526769), (rho \* 0.43496553411123023))); turn\_mark — an Angle \[yellow\] labelled "phi" drawn in end\_view (sides=((\, 0.0), ((rho \* 0.9004471023526769)…, radius=0.22); travel — a Line \[green\] labelled "rho phi" drawn in end\_view (start=(\, 0.0), end=((rho \* 0.9004471023526769), (rho \* 0.43496553411123023))) Actions: - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): ring is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): arm\_before is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): p\_before is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): arm\_after is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): p\_after is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): turn\_mark is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): travel is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): pt\_bp is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): offset is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): tilted is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): shear\_mark is redrawn as the numbers it depends on change. - [03:27.209](https://academa.ai/lectures/torsion-shafts?t=207.20916666666665): rho ticks to 0.22. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): ring is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): arm\_before is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): p\_before is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): arm\_after is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): p\_after is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): turn\_mark is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): travel is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): pt\_bp is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): offset is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): tilted is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): shear\_mark is redrawn as the numbers it depends on change. - [03:31.4](https://academa.ai/lectures/torsion-shafts?t=211.40016666666665): rho ticks to 1.0. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): ring is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): arm\_before is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): p\_before is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): arm\_after is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): p\_after is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): turn\_mark is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): travel is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): pt\_bp is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): offset is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): tilted is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): shear\_mark is redrawn as the numbers it depends on change. - [03:34.651](https://academa.ai/lectures/torsion-shafts?t=214.65116666666665): rho ticks to 0.6. - [03:38.215](https://academa.ai/lectures/torsion-shafts?t=218.21516666666665): point is shown on the screen, grown. - [03:40.215](https://academa.ai/lectures/torsion-shafts?t=220.21516666666665): point is hidden from the screen. ##### [03:41.892](https://academa.ai/lectures/torsion-shafts?t=221.89216666666667) Narration: Now the right hand picture. The line A B was ruled straight along the shaft, parallel to the axis, and its length is L. A sits at the fixed end and does not move. B sits on the far face, and B has just been carried sideways by exactly that same distance, rho phi. Board: Unchanged from the preceding beat in this scene. Actions: - [03:48.905](https://academa.ai/lectures/torsion-shafts?t=228.90516666666667): length\_label is shown on the screen, written out. - [03:55.964](https://academa.ai/lectures/torsion-shafts?t=235.96416666666664): pt\_bp is shown on the screen, written out. - [03:56.264](https://academa.ai/lectures/torsion-shafts?t=236.26416666666665): offset is shown on the screen, written out. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): unrolled moves to a new place on the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): end\_view is hidden from the screen — left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): rim is hidden from the screen — end\_view left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): hub is hidden from the screen — end\_view left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): ring is hidden from the screen — end\_view left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): arm\_before is hidden from the screen — end\_view left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): p\_before is hidden from the screen — end\_view left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): arm\_after is hidden from the screen — end\_view left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): p\_after is hidden from the screen — end\_view left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): turn\_mark is hidden from the screen — end\_view left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): travel is hidden from the screen — end\_view left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): head\_two is hidden from the screen — left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): label\_left is hidden from the screen — left the board. - [03:59.969](https://academa.ai/lectures/torsion-shafts?t=239.96866666666665): label\_right is hidden from the screen — left the board. ##### [04:0.569](https://academa.ai/lectures/torsion-shafts?t=240.56866666666667) Narration: So the line has swung round to A B prime, and the angle at A between where it was and where it is now is the shear strain, gamma. Its tangent is that sideways movement divided by the length: rho phi over L. Board: unrolled — a Figure (x\_range=(-0.4, 3.6), y\_range=(-0.6, 1.2), aspect=(4.0, 1.8)); axis\_line — a Line \[gray\] drawn in unrolled (end=(3.0, 0.0), dashed=True); pt\_a2 — a Point \[text\] labelled "A" drawn in unrolled; pt\_b2 — a Point \[text\] labelled "B" drawn in unrolled (location=(3.0, 0.0)); length\_label — a Math \[text\] that says "$L$" drawn in unrolled; pt\_bp — a Point \[red\] labelled "B'" drawn in unrolled (location=(3.0, (rho \* 0.45))); offset — a Line \[green\] labelled "rho phi" drawn in unrolled (start=(3.0, 0.0), end=(3.0, (rho \* 0.45))) Actions: - [04:0.569](https://academa.ai/lectures/torsion-shafts?t=240.56866666666667): head\_strain is shown on the screen, written out. - [04:1.73](https://academa.ai/lectures/torsion-shafts?t=241.73016666666666): tilted is shown on the screen, written out. - [04:6.501](https://academa.ai/lectures/torsion-shafts?t=246.50116666666665): shear\_mark is shown on the screen, written out. - [04:9.601](https://academa.ai/lectures/torsion-shafts?t=249.60116666666664): work is shown on the screen, written out. ##### [04:15.484](https://academa.ai/lectures/torsion-shafts?t=255.48366666666666) Narration: Torsional strains are small, well under a degree in any shaft you would actually use, so the tangent is the angle to any accuracy that matters. Shear strain is rho phi over L. Board: unrolled — a Figure (x\_range=(-0.4, 3.6), y\_range=(-0.6, 1.2), aspect=(4.0, 1.8)); axis\_line — a Line \[gray\] drawn in unrolled (end=(3.0, 0.0), dashed=True); pt\_a2 — a Point \[text\] labelled "A" drawn in unrolled; pt\_b2 — a Point \[text\] labelled "B" drawn in unrolled (location=(3.0, 0.0)); length\_label — a Math \[text\] that says "$L$" drawn in unrolled; pt\_bp — a Point \[red\] labelled "B'" drawn in unrolled (location=(3.0, (rho \* 0.45))); offset — a Line \[green\] labelled "rho phi" drawn in unrolled (start=(3.0, 0.0), end=(3.0, (rho \* 0.45))); head\_strain — a Heading that says "Strain Grows With Radius"; tilted — a Line \[red\] drawn in unrolled (end=(3.0, (rho \* 0.45))); shear\_mark — an Angle \[yellow\] labelled "gamma" drawn in unrolled (sides=((3.0, 0.0), (3.0, (rho \* 0.45))), radius=0.5) Actions: - [04:25.224](https://academa.ai/lectures/torsion-shafts?t=265.22416666666663): work is shown on the screen, written out. ##### [04:28.878](https://academa.ai/lectures/torsion-shafts?t=268.87766666666664) Narration: Now look at what is fixed in that expression. Phi belongs to the whole shaft. So does L. Both of them are the same number for every single point in this cross section. The only thing that varies from point to point is rho. Board: Unchanged from the preceding beat in this scene. Actions: - [04:32.175](https://academa.ai/lectures/torsion-shafts?t=272.17516666666666): work (the "phi" part) is emphasized. - [04:35.693](https://academa.ai/lectures/torsion-shafts?t=275.6931666666667): work (the "L" part) is emphasized. - [04:35.693](https://academa.ai/lectures/torsion-shafts?t=275.6931666666667): work (the "phi" part) is no longer emphasized. - [04:43.646](https://academa.ai/lectures/torsion-shafts?t=283.64616666666666): work (the "L" part) is no longer emphasized. - [04:43.646](https://academa.ai/lectures/torsion-shafts?t=283.64616666666666): work (the "rho" part) is emphasized. ##### [04:45.128](https://academa.ai/lectures/torsion-shafts?t=285.1281666666667) Narration: So shear strain is proportional to radius, and nothing but geometry went into that. Watch it happen. At the axis, where rho is nearly zero, the strain almost vanishes. Halfway out it is half of its largest value. And at the outer surface, at radius c, it reaches its maximum, c phi over L. Board: Unchanged from the preceding beat in this scene. Actions: - [04:46.684](https://academa.ai/lectures/torsion-shafts?t=286.68416666666667): work (the "rho" part) is no longer emphasized. - [04:53.197](https://academa.ai/lectures/torsion-shafts?t=293.1971666666667): pt\_bp is redrawn as the numbers it depends on change. - [04:53.197](https://academa.ai/lectures/torsion-shafts?t=293.1971666666667): offset is redrawn as the numbers it depends on change. - [04:53.197](https://academa.ai/lectures/torsion-shafts?t=293.1971666666667): tilted is redrawn as the numbers it depends on change. - [04:53.197](https://academa.ai/lectures/torsion-shafts?t=293.1971666666667): shear\_mark is redrawn as the numbers it depends on change. - [04:53.197](https://academa.ai/lectures/torsion-shafts?t=293.1971666666667): rho ticks to 0.1. - [04:58.05](https://academa.ai/lectures/torsion-shafts?t=298.05016666666666): pt\_bp is redrawn as the numbers it depends on change. - [04:58.05](https://academa.ai/lectures/torsion-shafts?t=298.05016666666666): offset is redrawn as the numbers it depends on change. - [04:58.05](https://academa.ai/lectures/torsion-shafts?t=298.05016666666666): tilted is redrawn as the numbers it depends on change. - [04:58.05](https://academa.ai/lectures/torsion-shafts?t=298.05016666666666): shear\_mark is redrawn as the numbers it depends on change. - [04:58.05](https://academa.ai/lectures/torsion-shafts?t=298.05016666666666): rho ticks to 0.5. - [05:1.811](https://academa.ai/lectures/torsion-shafts?t=301.8111666666667): pt\_bp is redrawn as the numbers it depends on change. - [05:1.811](https://academa.ai/lectures/torsion-shafts?t=301.8111666666667): offset is redrawn as the numbers it depends on change. - [05:1.811](https://academa.ai/lectures/torsion-shafts?t=301.8111666666667): tilted is redrawn as the numbers it depends on change. - [05:1.811](https://academa.ai/lectures/torsion-shafts?t=301.8111666666667): shear\_mark is redrawn as the numbers it depends on change. - [05:1.811](https://academa.ai/lectures/torsion-shafts?t=301.8111666666667): rho ticks to 1.0. - [05:3.773](https://academa.ai/lectures/torsion-shafts?t=303.77316666666667): work is shown on the screen, written out. ##### [05:7.444](https://academa.ai/lectures/torsion-shafts?t=307.4436666666667) Narration: Divide the two and the shaft's own dimensions cancel out. Gamma is rho over c times gamma max. A straight line, zero at the centre and largest at the surface. That is the strain distribution, and no material property has appeared yet. Board: Unchanged from the preceding beat in this scene. Actions: - [05:7.635](https://academa.ai/lectures/torsion-shafts?t=307.6351666666667): work is shown on the screen, written out. - [05:20.488](https://academa.ai/lectures/torsion-shafts?t=320.48816666666664): A box is drawn around work. ##### [05:24.571](https://academa.ai/lectures/torsion-shafts?t=324.5706666666667) Narration: One last way to picture it. Think of the shaft as a nest of thin tubes, one inside the next. Each tube shears a little relative to its neighbour, and the further out you go the more of that sliding has piled up. The skin does the most work. The axis does none. Board: Unchanged from the preceding beat in this scene. Actions: - [05:28.274](https://academa.ai/lectures/torsion-shafts?t=328.2741666666667): pt\_bp is redrawn as the numbers it depends on change. - [05:28.274](https://academa.ai/lectures/torsion-shafts?t=328.2741666666667): offset is redrawn as the numbers it depends on change. - [05:28.274](https://academa.ai/lectures/torsion-shafts?t=328.2741666666667): tilted is redrawn as the numbers it depends on change. - [05:28.274](https://academa.ai/lectures/torsion-shafts?t=328.2741666666667): shear\_mark is redrawn as the numbers it depends on change. - [05:28.274](https://academa.ai/lectures/torsion-shafts?t=328.2741666666667): rho ticks to 0.25. - [05:34.59](https://academa.ai/lectures/torsion-shafts?t=334.59016666666673): pt\_bp is redrawn as the numbers it depends on change. - [05:34.59](https://academa.ai/lectures/torsion-shafts?t=334.59016666666673): offset is redrawn as the numbers it depends on change. - [05:34.59](https://academa.ai/lectures/torsion-shafts?t=334.59016666666673): tilted is redrawn as the numbers it depends on change. - [05:34.59](https://academa.ai/lectures/torsion-shafts?t=334.59016666666673): shear\_mark is redrawn as the numbers it depends on change. - [05:34.59](https://academa.ai/lectures/torsion-shafts?t=334.59016666666673): rho ticks to 1.0. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): head\_strain is hidden from the screen — left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): unrolled is hidden from the screen — left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): axis\_line is hidden from the screen — unrolled left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): pt\_a2 is hidden from the screen — unrolled left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): pt\_b2 is hidden from the screen — unrolled left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): length\_label is hidden from the screen — unrolled left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): pt\_bp is hidden from the screen — unrolled left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): offset is hidden from the screen — unrolled left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): tilted is hidden from the screen — unrolled left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): shear\_mark is hidden from the screen — unrolled left the board. - [05:42.032](https://academa.ai/lectures/torsion-shafts?t=342.03229166666665): work is hidden from the screen — left the board. ### Scene 3: [Stress, Torque and the Polar Moment](https://academa.ai/lectures/torsion-shafts?t=343.07395833333334) Span: 05:43.074–09:31.854 (343.07395833333334s–571.8537916666667s). #### Objects - area\_label: a Math \[yellow\] that says "$dif A = 2 pi rho thin dif rho$" drawn in section - head\_j: a Heading that says "What That Integral Is" - head\_stress: a Heading that says "From Strain to Stress" - head\_torque: a Heading that says "Adding Up the Torque" - hub: a Point \[text\] labelled "O" drawn in section - j\_work: a Derivation \[text\] that says "$J &= integral\_A rho^2 thin dif A \\ &= integral\_0^c rho^2 thin 2 pi rho thin dif rho \\ &= 2 pi frac(c^4, 4) = frac(pi c^4, 2)$" - probe: a VariableNumber (initial\_value=0.25, format\_spec='.2f') - probe\_arrow: a Vector \[red\] labelled "tau" drawn in section (start=(\, 0.0), end=(\, (0.65 \* probe))) - probe\_dot: a Point \[yellow\] drawn in section (location=(\, 0.0)) - radius\_line: a Line \[gray\] labelled "c" drawn in section (dashed=True) - rim: a Circle \[blue\] drawn in section - ring\_inner: a Circle \[yellow\] drawn in section (radius=\) - ring\_outer: a Circle \[yellow\] drawn in section (radius=(ring\_radius + 0.08)) - ring\_radius: a VariableNumber (initial\_value=0.3, format\_spec='.2f') - section: a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)) - stress\_law: a Math \[text\] that says "$tau = frac(rho, c) tau\_upright("max")$" - stress\_work: a Derivation \[text\] that says "$tau &= G gamma \\ tau &= G frac(rho phi, L)$" - tau\_1: a Vector \[red\] drawn in section (start=(0.25, 0.0), end=(0.25, 0.1625)) - tau\_2: a Vector \[red\] drawn in section (start=(0.5, 0.0), end=(0.5, 0.325)) - tau\_3: a Vector \[red\] drawn in section (start=(0.75, 0.0), end=(0.75, 0.4875)) - tau\_4: a Vector \[red\] labelled "tau\_upright("max")" drawn in section (start=(1.0, 0.0), end=(1.0, 0.65)) - torque\_law: a Math \[text\] that says "$T = frac(tau\_upright("max"), c) integral\_A rho^2 thin dif A$" - torque\_work: a Derivation \[text\] that says "$dif F &= tau thin dif A \\ dif T &= rho thin tau thin dif A \\ T &= integral\_A rho thin tau thin dif A$" - torsion\_formula: a Math \[text\] that says "$tau = frac(T rho, J)$" - twist\_formula: a Math \[text\] that says "$phi = frac(T L, G J)$" #### Beats ##### [05:43.074](https://academa.ai/lectures/torsion-shafts?t=343.07395833333334) Narration: Strain is geometry. Stress needs the material, and for that we need one constitutive law. Below the elastic limit, shear stress and shear strain are proportional. Tau equals G gamma, where G is the shear modulus, the shear counterpart of Young's modulus. Board: Empty. Actions: - [05:43.074](https://academa.ai/lectures/torsion-shafts?t=343.07395833333334): head\_stress is shown on the screen, written out. - [05:43.074](https://academa.ai/lectures/torsion-shafts?t=343.07395833333334): section is shown on the screen, written out. - [05:45.617](https://academa.ai/lectures/torsion-shafts?t=345.61695833333334): rim is shown on the screen, written out. - [05:45.617](https://academa.ai/lectures/torsion-shafts?t=345.61695833333334): hub is shown on the screen, written out. - [05:45.817](https://academa.ai/lectures/torsion-shafts?t=345.81695833333333): radius\_line is shown on the screen, written out. - [05:54.231](https://academa.ai/lectures/torsion-shafts?t=354.2309583333333): section moves to a new place on the board. - [05:54.231](https://academa.ai/lectures/torsion-shafts?t=354.2309583333333): stress\_work is shown on the screen, written out. ##### [06:1.077](https://academa.ai/lectures/torsion-shafts?t=361.0774583333333) Narration: Substitute the strain we just derived. Tau is G rho phi over L. G is a material constant, phi and L belong to the whole shaft, so once again the only quantity that varies across the section is rho. Shear stress grows linearly with radius. Board: section — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); head\_stress — a Heading that says "From Strain to Stress"; rim — a Circle \[blue\] drawn in section; hub — a Point \[text\] labelled "O" drawn in section; radius\_line — a Line \[gray\] labelled "c" drawn in section (dashed=True) Actions: - [06:1.426](https://academa.ai/lectures/torsion-shafts?t=361.4259583333333): stress\_work is shown on the screen, written out. - [06:15.985](https://academa.ai/lectures/torsion-shafts?t=375.98495833333334): stress\_work (the "rho" part) is emphasized. - [06:19.479](https://academa.ai/lectures/torsion-shafts?t=379.4789583333333): stress\_work (the "rho" part) is no longer emphasized. ##### [06:21.066](https://academa.ai/lectures/torsion-shafts?t=381.06595833333336) Narration: So here is the stress at one point, a distance rho out from the axis. It acts tangentially, at right angles to the radius, and its size follows that straight line rule. Move the point outward and the arrow grows in exact proportion. Board: Unchanged from the preceding beat in this scene. Actions: - [06:22.842](https://academa.ai/lectures/torsion-shafts?t=382.8419583333333): probe\_dot is shown on the screen, written out. - [06:23.042](https://academa.ai/lectures/torsion-shafts?t=383.04195833333335): probe\_arrow is shown on the screen, written out. - [06:33.733](https://academa.ai/lectures/torsion-shafts?t=393.73295833333333): probe\_dot is redrawn as the numbers it depends on change. - [06:33.733](https://academa.ai/lectures/torsion-shafts?t=393.73295833333333): probe\_arrow is redrawn as the numbers it depends on change. - [06:33.733](https://academa.ai/lectures/torsion-shafts?t=393.73295833333333): probe ticks to 0.55. ##### [06:37.27](https://academa.ai/lectures/torsion-shafts?t=397.26995833333336) Narration: Take it right out to the outer surface, at radius c, and the stress reaches its largest value there. Every point on that rim sits at the same radius, so the entire outer skin is at the maximum together. Board: section — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); head\_stress — a Heading that says "From Strain to Stress"; rim — a Circle \[blue\] drawn in section; hub — a Point \[text\] labelled "O" drawn in section; radius\_line — a Line \[gray\] labelled "c" drawn in section (dashed=True); probe\_dot — a Point \[yellow\] drawn in section (location=(\, 0.0)); probe\_arrow — a Vector \[red\] labelled "tau" drawn in section (start=(\, 0.0), end=(\, (0.65 \* probe))) Actions: - [06:38.036](https://academa.ai/lectures/torsion-shafts?t=398.0359583333333): probe\_dot is redrawn as the numbers it depends on change. - [06:38.036](https://academa.ai/lectures/torsion-shafts?t=398.0359583333333): probe\_arrow is redrawn as the numbers it depends on change. - [06:38.036](https://academa.ai/lectures/torsion-shafts?t=398.0359583333333): probe ticks to 1.0. ##### [06:50.095](https://academa.ai/lectures/torsion-shafts?t=410.09545833333334) Narration: And here is the whole distribution across a radius. Nothing at the axis, rising straight out to tau max at the surface. Written down, that is tau equals rho over c times tau max. Board: Unchanged from the preceding beat in this scene. Actions: - [06:50.095](https://academa.ai/lectures/torsion-shafts?t=410.09545833333334): probe\_dot is hidden from the screen. - [06:50.095](https://academa.ai/lectures/torsion-shafts?t=410.09545833333334): probe\_arrow is hidden from the screen. - [06:51.419](https://academa.ai/lectures/torsion-shafts?t=411.41895833333336): tau\_1 is shown on the screen, written out. - [06:51.569](https://academa.ai/lectures/torsion-shafts?t=411.56895833333334): tau\_2 is shown on the screen, written out. - [06:51.719](https://academa.ai/lectures/torsion-shafts?t=411.7189583333334): tau\_3 is shown on the screen, written out. - [06:51.869](https://academa.ai/lectures/torsion-shafts?t=411.86895833333335): tau\_4 is shown on the screen, written out. - [06:58.536](https://academa.ai/lectures/torsion-shafts?t=418.5359583333333): stress\_law is shown on the screen, written out. - [07:3.551](https://academa.ai/lectures/torsion-shafts?t=423.55145833333336): section moves to a new place on the board. - [07:3.551](https://academa.ai/lectures/torsion-shafts?t=423.55145833333336): stress\_law moves to a new place on the board. - [07:3.551](https://academa.ai/lectures/torsion-shafts?t=423.55145833333336): head\_stress is hidden from the screen — left the board. - [07:3.551](https://academa.ai/lectures/torsion-shafts?t=423.55145833333336): stress\_work is hidden from the screen — left the board. ##### [07:4.151](https://academa.ai/lectures/torsion-shafts?t=424.1514583333333) Narration: Now collect what all of that stress adds up to. Take a small patch of area, d A, sitting at radius rho. The force on that patch is stress times area, tau d A. Board: stress\_law — a Math \[text\] that says "$tau = frac(rho, c) tau\_upright("max")$"; section — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); rim — a Circle \[blue\] drawn in section; hub — a Point \[text\] labelled "O" drawn in section; radius\_line — a Line \[gray\] labelled "c" drawn in section (dashed=True); tau\_1 — a Vector \[red\] drawn in section (start=(0.25, 0.0), end=(0.25, 0.1625)); tau\_2 — a Vector \[red\] drawn in section (start=(0.5, 0.0), end=(0.5, 0.325)); tau\_3 — a Vector \[red\] drawn in section (start=(0.75, 0.0), end=(0.75, 0.4875)); tau\_4 — a Vector \[red\] labelled "tau\_upright("max")" drawn in section (start=(1.0, 0.0), end=(1.0, 0.65)) Actions: - [07:4.151](https://academa.ai/lectures/torsion-shafts?t=424.1514583333333): head\_torque is shown on the screen, written out. - [07:13.393](https://academa.ai/lectures/torsion-shafts?t=433.39295833333335): torque\_work is shown on the screen, written out. ##### [07:17.812](https://academa.ai/lectures/torsion-shafts?t=437.81195833333334) Narration: That force acts tangentially, at a distance rho from the axis, so its moment about the axis is rho times tau d A. That is the torque this one little patch carries. Board: stress\_law — a Math \[text\] that says "$tau = frac(rho, c) tau\_upright("max")$"; section — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); rim — a Circle \[blue\] drawn in section; hub — a Point \[text\] labelled "O" drawn in section; radius\_line — a Line \[gray\] labelled "c" drawn in section (dashed=True); tau\_1 — a Vector \[red\] drawn in section (start=(0.25, 0.0), end=(0.25, 0.1625)); tau\_2 — a Vector \[red\] drawn in section (start=(0.5, 0.0), end=(0.5, 0.325)); tau\_3 — a Vector \[red\] drawn in section (start=(0.75, 0.0), end=(0.75, 0.4875)); tau\_4 — a Vector \[red\] labelled "tau\_upright("max")" drawn in section (start=(1.0, 0.0), end=(1.0, 0.65)); head\_torque — a Heading that says "Adding Up the Torque" Actions: - [07:23.083](https://academa.ai/lectures/torsion-shafts?t=443.08295833333335): torque\_work is shown on the screen, written out. ##### [07:30.881](https://academa.ai/lectures/torsion-shafts?t=450.88095833333335) Narration: The total torque is the sum over every patch in the section, which is the integral of rho tau d A taken over the area. Board: Unchanged from the preceding beat in this scene. Actions: - [07:34.921](https://academa.ai/lectures/torsion-shafts?t=454.9209583333334): torque\_work is shown on the screen, written out. ##### [07:39.19](https://academa.ai/lectures/torsion-shafts?t=459.1899583333333) Narration: Now substitute the stress distribution. Tau max over c is the same number everywhere in this section, so it comes straight outside the integral, and what is left behind is the integral of rho squared d A. Board: Unchanged from the preceding beat in this scene. Actions: - [07:39.887](https://academa.ai/lectures/torsion-shafts?t=459.8869583333333): torque\_law is shown on the screen, written out. - [07:51.601](https://academa.ai/lectures/torsion-shafts?t=471.6009583333333): torque\_law (the "integral\_A rho^2 thin dif A" part) is emphasized. - [07:52.924](https://academa.ai/lectures/torsion-shafts?t=472.9239583333333): torque\_law moves to a new place on the board. - [07:52.924](https://academa.ai/lectures/torsion-shafts?t=472.9239583333333): head\_torque is hidden from the screen — left the board. - [07:52.924](https://academa.ai/lectures/torsion-shafts?t=472.9239583333333): stress\_law is hidden from the screen — left the board. - [07:52.924](https://academa.ai/lectures/torsion-shafts?t=472.9239583333333): torque\_work is hidden from the screen — left the board. ##### [07:53.524](https://academa.ai/lectures/torsion-shafts?t=473.5239583333333) Narration: Stop and look at that integral, because it is the whole point. There is no load in it and no material property in it. It contains nothing but the shape of the cross section, and where that shape puts its area. That integral is the polar second moment of area, J. Board: section — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); rim — a Circle \[blue\] drawn in section; hub — a Point \[text\] labelled "O" drawn in section; radius\_line — a Line \[gray\] labelled "c" drawn in section (dashed=True); tau\_1 — a Vector \[red\] drawn in section (start=(0.25, 0.0), end=(0.25, 0.1625)); tau\_2 — a Vector \[red\] drawn in section (start=(0.5, 0.0), end=(0.5, 0.325)); tau\_3 — a Vector \[red\] drawn in section (start=(0.75, 0.0), end=(0.75, 0.4875)); tau\_4 — a Vector \[red\] labelled "tau\_upright("max")" drawn in section (start=(1.0, 0.0), end=(1.0, 0.65)); torque\_law — a Math \[text\] that says "$T = frac(tau\_upright("max"), c) integral\_A rho^2 thin dif A$" Actions: - [07:53.524](https://academa.ai/lectures/torsion-shafts?t=473.5239583333333): head\_j is shown on the screen, written out. - [07:53.524](https://academa.ai/lectures/torsion-shafts?t=473.5239583333333): tau\_1 is hidden from the screen. - [07:53.524](https://academa.ai/lectures/torsion-shafts?t=473.5239583333333): tau\_2 is hidden from the screen. - [07:53.524](https://academa.ai/lectures/torsion-shafts?t=473.5239583333333): tau\_3 is hidden from the screen. - [07:53.524](https://academa.ai/lectures/torsion-shafts?t=473.5239583333333): tau\_4 is hidden from the screen. - [08:8.176](https://academa.ai/lectures/torsion-shafts?t=488.1759583333333): j\_work is shown on the screen, written out. - [08:8.176](https://academa.ai/lectures/torsion-shafts?t=488.1759583333333): torque\_law (the "integral\_A rho^2 thin dif A" part) is no longer emphasized. ##### [08:11.359](https://academa.ai/lectures/torsion-shafts?t=491.35945833333335) Narration: So work it out for a solid circle. The natural element is a thin ring, because every point on a ring sits at the same radius rho. Give it a thickness d rho, and its area is its circumference times that thickness, two pi rho d rho. Board: section — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); rim — a Circle \[blue\] drawn in section; hub — a Point \[text\] labelled "O" drawn in section; radius\_line — a Line \[gray\] labelled "c" drawn in section (dashed=True); torque\_law — a Math \[text\] that says "$T = frac(tau\_upright("max"), c) integral\_A rho^2 thin dif A$"; head\_j — a Heading that says "What That Integral Is" Actions: - [08:15.742](https://academa.ai/lectures/torsion-shafts?t=495.7419583333333): ring\_inner is shown on the screen, written out. - [08:15.892](https://academa.ai/lectures/torsion-shafts?t=495.8919583333333): ring\_outer is shown on the screen, written out. - [08:23.3](https://academa.ai/lectures/torsion-shafts?t=503.2999583333333): area\_label is shown on the screen, written out. ##### [08:27.534](https://academa.ai/lectures/torsion-shafts?t=507.5339583333333) Narration: Substitute that in and the double integral collapses to an ordinary one in a single variable, running from zero at the axis out to c at the surface. Sweep the ring outward and you are sweeping through exactly that. Board: section — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); rim — a Circle \[blue\] drawn in section; hub — a Point \[text\] labelled "O" drawn in section; radius\_line — a Line \[gray\] labelled "c" drawn in section (dashed=True); torque\_law — a Math \[text\] that says "$T = frac(tau\_upright("max"), c) integral\_A rho^2 thin dif A$"; head\_j — a Heading that says "What That Integral Is"; ring\_inner — a Circle \[yellow\] drawn in section (radius=\); ring\_outer — a Circle \[yellow\] drawn in section (radius=(ring\_radius + 0.08)); area\_label — a Math \[yellow\] that says "$dif A = 2 pi rho thin dif rho$" drawn in section Actions: - [08:27.882](https://academa.ai/lectures/torsion-shafts?t=507.8819583333333): j\_work is shown on the screen, written out. - [08:37.67](https://academa.ai/lectures/torsion-shafts?t=517.6699583333333): ring\_inner is redrawn as the numbers it depends on change. - [08:37.67](https://academa.ai/lectures/torsion-shafts?t=517.6699583333333): ring\_outer is redrawn as the numbers it depends on change. - [08:37.67](https://academa.ai/lectures/torsion-shafts?t=517.6699583333333): ring\_radius ticks to 0.9. ##### [08:42.275](https://academa.ai/lectures/torsion-shafts?t=522.2749583333333) Narration: Integrate rho cubed, and J comes out as pi c to the fourth over two. Notice the fourth power. Doubling the radius multiplies J by sixteen. Board: Unchanged from the preceding beat in this scene. Actions: - [08:42.623](https://academa.ai/lectures/torsion-shafts?t=522.6229583333334): j\_work is shown on the screen, written out. - [08:46.164](https://academa.ai/lectures/torsion-shafts?t=526.1639583333333): j\_work (the "c^4" part) is emphasized. - [08:51.749](https://academa.ai/lectures/torsion-shafts?t=531.7489583333333): j\_work (the "c^4" part) is no longer emphasized. ##### [08:53.301](https://academa.ai/lectures/torsion-shafts?t=533.3009583333333) Narration: Put J back into the torque relation and rearrange. The shear stress at radius rho is T rho over J. That is the torsion formula, and it is the exact counterpart of stress equals load over area. Board: Unchanged from the preceding beat in this scene. Actions: - [08:55.855](https://academa.ai/lectures/torsion-shafts?t=535.8549583333333): torsion\_formula is shown on the screen, written out. - [09:4.202](https://academa.ai/lectures/torsion-shafts?t=544.2019583333333): A box is drawn around torsion\_formula. ##### [09:7.705](https://academa.ai/lectures/torsion-shafts?t=547.7049583333333) Narration: And phi came along for the ride. Thread the same substitution back through the strain we derived, and the angle of twist is T L over G J. Strength and stiffness both hang on the same integral. Board: section — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); rim — a Circle \[blue\] drawn in section; hub — a Point \[text\] labelled "O" drawn in section; radius\_line — a Line \[gray\] labelled "c" drawn in section (dashed=True); torque\_law — a Math \[text\] that says "$T = frac(tau\_upright("max"), c) integral\_A rho^2 thin dif A$"; torsion\_formula — a Math \[text\] that says "$tau = frac(T rho, J)$"; head\_j — a Heading that says "What That Integral Is"; ring\_inner — a Circle \[yellow\] drawn in section (radius=\); ring\_outer — a Circle \[yellow\] drawn in section (radius=(ring\_radius + 0.08)); area\_label — a Math \[yellow\] that says "$dif A = 2 pi rho thin dif rho$" drawn in section Actions: - [09:13.986](https://academa.ai/lectures/torsion-shafts?t=553.9859583333333): twist\_formula is shown on the screen, written out. ##### [09:20.576](https://academa.ai/lectures/torsion-shafts?t=560.5759583333333) Narration: So the whole of torsion sits in J, and J is a statement about where the material is. Which raises exactly one design question. Where should you put it? Board: section — a Figure (x\_range=(-1.35, 1.35), y\_range=(-1.35, 1.35), aspect=(1.0, 1.0)); rim — a Circle \[blue\] drawn in section; hub — a Point \[text\] labelled "O" drawn in section; radius\_line — a Line \[gray\] labelled "c" drawn in section (dashed=True); torque\_law — a Math \[text\] that says "$T = frac(tau\_upright("max"), c) integral\_A rho^2 thin dif A$"; torsion\_formula — a Math \[text\] that says "$tau = frac(T rho, J)$"; twist\_formula — a Math \[text\] that says "$phi = frac(T L, G J)$"; head\_j — a Heading that says "What That Integral Is"; ring\_inner — a Circle \[yellow\] drawn in section (radius=\); ring\_outer — a Circle \[yellow\] drawn in section (radius=(ring\_radius + 0.08)); area\_label — a Math \[yellow\] that says "$dif A = 2 pi rho thin dif rho$" drawn in section Actions: - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): head\_j is hidden from the screen — left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): j\_work is hidden from the screen — left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): section is hidden from the screen — left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): rim is hidden from the screen — section left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): hub is hidden from the screen — section left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): radius\_line is hidden from the screen — section left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): ring\_inner is hidden from the screen — section left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): ring\_outer is hidden from the screen — section left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): area\_label is hidden from the screen — section left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): torque\_law is hidden from the screen — left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): torsion\_formula is hidden from the screen — left the board. - [09:30.812](https://academa.ai/lectures/torsion-shafts?t=570.8121249999999): twist\_formula is hidden from the screen — left the board. ### Scene 4: [Solid Shaft or Tube](https://academa.ai/lectures/torsion-shafts?t=571.8537916666667) Span: 09:31.854–12:56.604 (571.8537916666667s–776.6042291666666s). #### Objects - head\_numbers: a Heading that says "Same Weight, Very Different Shaft" - head\_recap: a Heading that says "The Argument in Four Lines" - head\_where: a Heading that says "Where Should the Metal Be?" - hollow\_bore: a Circle \[blue\] drawn in hollow\_view (radius=40.0) - hollow\_fill: a Region \[blue\] drawn in hollow\_view (predicates=(\.inside at 0x2a800ec5bba0\>,), x\_range=(-50.000475963963275, 50.000475963963275), y\_range=(-50.000475963963275, 50.000475963963275)) - hollow\_inner\_radius: a Line \[gray\] labelled "c\_i = 40 thin upright("mm")" drawn in hollow\_view (end=(0.0, 40.0), dashed=True) - hollow\_outer\_radius: a Line \[gray\] labelled "c\_o = 50 thin upright("mm")" drawn in hollow\_view (end=(50.0, 0.0), dashed=True) - hollow\_rim: a Circle \[blue\] drawn in hollow\_view (radius=50.0) - hollow\_view: a Figure (x\_range=(-58.0, 58.0), y\_range=(-58.0, 58.0), aspect=(1.0, 1.0)) - label\_hollow: a Tex \[text\] that says "Tube, same weight" (underline=True) - label\_solid: a Tex \[text\] that says "Solid bar" (underline=True) - numbers: a Table \[text\] that says "Solid bar Tube Radius $c = 30$ $c\_o = 50$, $c\_i = 40$ Area, $upright("mm")^2$ 2830 2830 $J$, $upright("mm")^4$ $1.27 times 10^6$ $5.80 times 10^6$ $frac(J, c)$, $upright("mm")^3$ $4.24 times 10^4$ $1.16 times 10^5$" (rows=(('', 'Solid bar', 'Tube'), ('Radius', '$c = 30$', '$c\_o = 50$,…, header=True) - point: a Point \[yellow\] drawn in solid\_view - solid\_core: a Region \[red\] drawn in solid\_view (predicates=(\.inside at 0x2a800ca8e840\>,), x\_range=(-15.000142789188983, 15.000142789188983), y\_range=(-15.000142789188983, 15.000142789188983)) - solid\_fill: a Region \[blue\] drawn in solid\_view (predicates=(\.inside at 0x2a800ca8fc40\>,), x\_range=(-30.000285578377966, 30.000285578377966), y\_range=(-30.000285578377966, 30.000285578377966)) - solid\_radius: a Line \[gray\] labelled "c = 30 thin upright("mm")" drawn in solid\_view (end=(30.0, 0.0), dashed=True) - solid\_rim: a Circle \[blue\] drawn in solid\_view (radius=30.0) - solid\_view: a Figure (x\_range=(-58.0, 58.0), y\_range=(-58.0, 58.0), aspect=(1.0, 1.0)) - takeaways: a Block \[text\] that says "Geometry alone gives $gamma = frac(rho phi, L)$, linear in $rho$. An elastic material gives $tau = G gamma$, linear in $rho$ too. Integrating $rho tau thin dif A$ gives $T$, and $J = integral\_A rho^2 thin dif A$. Weighting by $rho^2$ means…" - verdict: a Panel that says "Same steel per metre: the tube is 4.6 times as stiff in twist, and carries 2.7 times the torque at the same allowable stress." #### Beats ##### [09:31.854](https://academa.ai/lectures/torsion-shafts?t=571.8537916666667) Narration: So where should the material be? Here is the cross section of a solid steel shaft, thirty millimetres in radius, carrying a torque about its own axis. Board: Empty. Actions: - [09:31.854](https://academa.ai/lectures/torsion-shafts?t=571.8537916666667): head\_where is shown on the screen, written out. - [09:31.854](https://academa.ai/lectures/torsion-shafts?t=571.8537916666667): solid\_view is shown on the screen, written out. - [09:34.896](https://academa.ai/lectures/torsion-shafts?t=574.8957916666667): solid\_fill is shown on the screen, written out. - [09:34.896](https://academa.ai/lectures/torsion-shafts?t=574.8957916666667): solid\_rim is shown on the screen, written out. - [09:37.241](https://academa.ai/lectures/torsion-shafts?t=577.2407916666666): solid\_radius is shown on the screen, written out. ##### [09:42.148](https://academa.ai/lectures/torsion-shafts?t=582.1482916666666) Narration: Go back to what J was actually measuring, the integral of rho squared d A. Every scrap of area is weighted by the square of its distance from the axis. So it is worth asking what the middle of this bar is earning. Board: solid\_view — a Figure (x\_range=(-58.0, 58.0), y\_range=(-58.0, 58.0), aspect=(1.0, 1.0)); head\_where — a Heading that says "Where Should the Metal Be?"; solid\_fill — a Region \[blue\] drawn in solid\_view (predicates=(\.inside at 0x2a800ca8fc40\>,), x\_range=(-30.000285578377966, 30.000285578377966), y\_range=(-30.000285578377966, 30.000285578377966)); solid\_rim — a Circle \[blue\] drawn in solid\_view (radius=30.0); solid\_radius — a Line \[gray\] labelled "c = 30 thin upright("mm")" drawn in solid\_view (end=(30.0, 0.0), dashed=True) Actions: - [09:54.374](https://academa.ai/lectures/torsion-shafts?t=594.3737916666666): point is shown on the screen, grown. - [09:56.374](https://academa.ai/lectures/torsion-shafts?t=596.3737916666666): point is hidden from the screen. ##### [09:56.808](https://academa.ai/lectures/torsion-shafts?t=596.8077916666666) Narration: Shade in everything inside half the radius, everything within fifteen millimetres of the axis. By area, that core is a quarter of the whole bar. A quarter of the weight you carry around, and a quarter of the steel you paid for. Board: Unchanged from the preceding beat in this scene. Actions: - [09:57.156](https://academa.ai/lectures/torsion-shafts?t=597.1557916666667): solid\_core is shown on the screen, written out. ##### [10:11.653](https://academa.ai/lectures/torsion-shafts?t=611.6532916666666) Narration: But J for that core alone is pi times fifteen to the fourth over two, and fifteen to the fourth is one sixteenth of thirty to the fourth. So a quarter of the material is contributing about six percent of the torsional stiffness. The middle of a shaft is very close to dead weight. Board: solid\_view — a Figure (x\_range=(-58.0, 58.0), y\_range=(-58.0, 58.0), aspect=(1.0, 1.0)); head\_where — a Heading that says "Where Should the Metal Be?"; solid\_fill — a Region \[blue\] drawn in solid\_view (predicates=(\.inside at 0x2a800ca8fc40\>,), x\_range=(-30.000285578377966, 30.000285578377966), y\_range=(-30.000285578377966, 30.000285578377966)); solid\_rim — a Circle \[blue\] drawn in solid\_view (radius=30.0); solid\_radius — a Line \[gray\] labelled "c = 30 thin upright("mm")" drawn in solid\_view (end=(30.0, 0.0), dashed=True); solid\_core — a Region \[red\] drawn in solid\_view (predicates=(\.inside at 0x2a800ca8e840\>,), x\_range=(-15.000142789188983, 15.000142789188983), y\_range=(-15.000142789188983, 15.000142789188983)) Actions: - [10:23.24](https://academa.ai/lectures/torsion-shafts?t=623.2397916666666): solid\_core is indicated — a transient flash. ##### [10:29.29](https://academa.ai/lectures/torsion-shafts?t=629.2902916666667) Narration: So take it out, and put it where the rho squared can get at it. Here is a tube, outer radius fifty millimetres, inner radius forty. Its area is pi times fifty squared minus forty squared, which is pi times nine hundred, and that is exactly the area of the solid bar. Same steel per metre, same weight. Board: Unchanged from the preceding beat in this scene. Actions: - [10:33.998](https://academa.ai/lectures/torsion-shafts?t=633.9977916666667): solid\_view moves to a new place on the board. - [10:33.998](https://academa.ai/lectures/torsion-shafts?t=633.9977916666667): hollow\_view is shown on the screen, written out. - [10:33.998](https://academa.ai/lectures/torsion-shafts?t=633.9977916666667): hollow\_fill is shown on the screen, written out. - [10:33.998](https://academa.ai/lectures/torsion-shafts?t=633.9977916666667): label\_solid is shown on the screen, written out. - [10:33.998](https://academa.ai/lectures/torsion-shafts?t=633.9977916666667): label\_hollow is shown on the screen, written out. - [10:34.198](https://academa.ai/lectures/torsion-shafts?t=634.1977916666667): hollow\_rim is shown on the screen, written out. - [10:34.598](https://academa.ai/lectures/torsion-shafts?t=634.5977916666667): hollow\_bore is shown on the screen, written out. - [10:35.763](https://academa.ai/lectures/torsion-shafts?t=635.7627916666667): hollow\_outer\_radius is shown on the screen, written out. - [10:37.76](https://academa.ai/lectures/torsion-shafts?t=637.7597916666666): hollow\_inner\_radius is shown on the screen, written out. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): head\_where is hidden from the screen — left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): hollow\_view is hidden from the screen — left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): hollow\_fill is hidden from the screen — hollow\_view left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): hollow\_rim is hidden from the screen — hollow\_view left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): hollow\_bore is hidden from the screen — hollow\_view left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): hollow\_outer\_radius is hidden from the screen — hollow\_view left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): hollow\_inner\_radius is hidden from the screen — hollow\_view left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): label\_hollow is hidden from the screen — left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): label\_solid is hidden from the screen — left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): solid\_view is hidden from the screen — left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): solid\_fill is hidden from the screen — solid\_view left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): solid\_rim is hidden from the screen — solid\_view left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): solid\_radius is hidden from the screen — solid\_view left the board. - [10:51.599](https://academa.ai/lectures/torsion-shafts?t=651.5987916666667): solid\_core is hidden from the screen — solid\_view left the board. ##### [10:52.199](https://academa.ai/lectures/torsion-shafts?t=652.1987916666667) Narration: Now put numbers against them. Both sections have an area of about two thousand eight hundred and thirty square millimetres, so a metre of each weighs the same. Board: Empty. Actions: - [10:52.199](https://academa.ai/lectures/torsion-shafts?t=652.1987916666667): head\_numbers is shown on the screen, written out. - [10:53.175](https://academa.ai/lectures/torsion-shafts?t=653.1747916666667): numbers is shown on the screen, written out. - [10:53.575](https://academa.ai/lectures/torsion-shafts?t=653.5747916666667): numbers is shown on the screen, written out. - [10:55.891](https://academa.ai/lectures/torsion-shafts?t=655.8907916666667): numbers is shown on the screen, written out. ##### [11:2.401](https://academa.ai/lectures/torsion-shafts?t=662.4007916666667) Narration: J for the solid bar is pi over two times thirty to the fourth, about one point two seven million millimetres to the fourth. For the tube it is pi over two times fifty to the fourth minus forty to the fourth, and that comes to five point eight million. Four and a half times as much, for the same weight of steel. Board: head\_numbers — a Heading that says "Same Weight, Very Different Shaft" Actions: - [11:4.189](https://academa.ai/lectures/torsion-shafts?t=664.1887916666667): numbers is shown on the screen, written out. - [11:18.933](https://academa.ai/lectures/torsion-shafts?t=678.9327916666666): numbers (the "row=4" part) is emphasized. - [11:22.462](https://academa.ai/lectures/torsion-shafts?t=682.4622916666667): numbers (the "row=4" part) is no longer emphasized. ##### [11:23.062](https://academa.ai/lectures/torsion-shafts?t=683.0622916666666) Narration: That is stiffness. Strength is a slightly different sum, because the torsion formula puts the peak stress right at the outer surface, where rho equals c. The torque you can carry at a given allowable stress is tau times J over c, and the tube has the bigger c. Board: Unchanged from the preceding beat in this scene. Actions: - [11:35.102](https://academa.ai/lectures/torsion-shafts?t=695.1017916666667): numbers is shown on the screen, written out. ##### [11:41.577](https://academa.ai/lectures/torsion-shafts?t=701.5767916666666) Narration: Work it out. Forty two thousand cubic millimetres for the solid bar, one hundred and sixteen thousand for the tube. Two point seven times the torque, at exactly the same weight. That is why a drive shaft is a tube. Board: Unchanged from the preceding beat in this scene. Actions: - [11:43.388](https://academa.ai/lectures/torsion-shafts?t=703.3877916666667): numbers (the "row=5" part) is emphasized. - [11:52.653](https://academa.ai/lectures/torsion-shafts?t=712.6527916666666): verdict is shown on the screen, written out. - [11:54.324](https://academa.ai/lectures/torsion-shafts?t=714.3237916666667): numbers (the "row=5" part) is no longer emphasized. ##### [11:56.416](https://academa.ai/lectures/torsion-shafts?t=716.4162916666667) Narration: There are limits, of course. Make the wall too thin and the tube buckles, or dents, long before the material anywhere near yields, and a hollow shaft costs more to make and more to join. But the trend is not subtle, and it is why almost every drive shaft, every bicycle frame and every aircraft control tube you will ever meet is hollow. Board: verdict — a Panel that says "Same steel per metre: the tube is 4.6 times as stiff in twist, and carries 2.7 times the torque at the same allowable stress."; head\_numbers — a Heading that says "Same Weight, Very Different Shaft" Actions: - [12:9.426](https://academa.ai/lectures/torsion-shafts?t=729.4257916666667): verdict (the "4.6 times as stiff" part) is emphasized. - [12:14.197](https://academa.ai/lectures/torsion-shafts?t=734.1967916666666): verdict (the "4.6 times as stiff" part) is no longer emphasized. - [12:19.305](https://academa.ai/lectures/torsion-shafts?t=739.3052916666667): head\_numbers is hidden from the screen — left the board. - [12:19.305](https://academa.ai/lectures/torsion-shafts?t=739.3052916666667): numbers is hidden from the screen — left the board. - [12:19.305](https://academa.ai/lectures/torsion-shafts?t=739.3052916666667): verdict is hidden from the screen — left the board. ##### [12:19.905](https://academa.ai/lectures/torsion-shafts?t=739.9052916666667) Narration: So, the whole argument in four lines. Geometry alone gave us shear strain proportional to radius. An elastic material turned that into shear stress proportional to radius. Integrating the moment of that stress over the area gave the torque, and dragged the polar second moment out into the open as an integral. And because that integral weights area by rho squared, metal near the axis earns almost nothing. Board: Empty. Actions: - [12:19.905](https://academa.ai/lectures/torsion-shafts?t=739.9052916666667): head\_recap is shown on the screen, written out. - [12:21.717](https://academa.ai/lectures/torsion-shafts?t=741.7167916666667): takeaways is shown on the screen, written out. - [12:23.075](https://academa.ai/lectures/torsion-shafts?t=743.0747916666667): takeaways (the "Geometry alone" part) is emphasized. - [12:27.313](https://academa.ai/lectures/torsion-shafts?t=747.3127916666666): takeaways (the "Geometry alone" part) is no longer emphasized. - [12:27.313](https://academa.ai/lectures/torsion-shafts?t=747.3127916666666): takeaways (the "elastic material" part) is emphasized. - [12:31.806](https://academa.ai/lectures/torsion-shafts?t=751.8057916666667): takeaways (the "Integrating" part) is emphasized. - [12:31.806](https://academa.ai/lectures/torsion-shafts?t=751.8057916666667): takeaways (the "elastic material" part) is no longer emphasized. - [12:44.066](https://academa.ai/lectures/torsion-shafts?t=764.0657916666667): takeaways (the "Integrating" part) is no longer emphasized. - [12:44.066](https://academa.ai/lectures/torsion-shafts?t=764.0657916666667): takeaways (the "near the axis" part) is emphasized. ##### [12:47.208](https://academa.ai/lectures/torsion-shafts?t=767.2082916666667) Narration: And all of it came out of one picture. A line ruled along the surface of a bar, and what happened to that line when you twisted it. Board: takeaways — a Block \[text\] that says "Geometry alone gives $gamma = frac(rho phi, L)$, linear in $rho$. An elastic material gives $tau = G gamma$, linear in $rho$ too. Integrating $rho tau thin dif A$ gives $T$, and $J = integral\_A rho^2 thin dif A$. Weighting by $rho^2$ means…"; head\_recap — a Heading that says "The Argument in Four Lines" Actions: - [12:47.208](https://academa.ai/lectures/torsion-shafts?t=767.2082916666667): takeaways (the "near the axis" part) is no longer emphasized. - [12:55.563](https://academa.ai/lectures/torsion-shafts?t=775.5625625): head\_recap is hidden from the screen — left the board. - [12:55.563](https://academa.ai/lectures/torsion-shafts?t=775.5625625): takeaways is hidden from the screen — left the board.