# Kinematics of a Body: The Instantaneous Axis > A rigid body carries exactly one restriction: the distances between its points never change. We differentiate that restriction to obtain the velocity relation v\_B = v\_A + omega cross r, then watch the body move under its translation and rotation terms. A rolling wheel makes the instantaneous centre visible and shows why a different material point occupies it from one instant to the next. Finally, one continuous three-dimensional chapter turns the point at rest into a tilted instantaneous axis, sweeps the family of points along it, and follows a material point through Chasles' screw motion. Live limiting cases reduce the screw to pure rotation and pure translation. - Canonical watch page: [Kinematics of a Body: The Instantaneous Axis](https://academa.ai/lectures/instantaneous-axis-of-rotation) - Publisher: [Academa, Inc.](https://academa.ai) - Subject: Engineering - Published: 2026-08-28T19:20:12.654Z - Updated: 2026-08-28T19:20:12.654Z - Duration: PT481S (8 minutes 1 second) - Chapters: 3 - Views: 0 - Language: en-US - Access: Free - Video stream: [HLS content](https://academa.ai/media/l/01M14TYA5P7YWR2CMWJYKMN214/0/dark/master.m3u8) - Audiovisual record: [Semantic JSON](https://academa.ai/media/l/01M14TYA5P7YWR2CMWJYKMN214/0/semantic.json) - Thumbnail: [Image](https://academa.ai/media/l/01M14TYA5P7YWR2CMWJYKMN214/0/dark/poster.jpg) ## Description The rigid-body velocity relation, the instantaneous centre of a rolling wheel, and the screw axis that replaces it in three dimensions. ## Chapters - [00:00–01:52.46 · The Rigid Body Relation](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=0) - [01:52.46–03:46.226 · The Instantaneous Centre](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=112.45983333333334) - [03:46.226–08:01 · The Instantaneous Axis and Screw Motion](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=226.22579166666668) ## Transcript ### [00:00 · The Rigid Body Relation](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=0) Kinematics of a rigid body begins with one promise: distances inside the body never change. That promise will give us its complete velocity law. Here is the body itself, a collection of material points locked together. Pick any two of them and call them A and B. The vector r runs from A to B, and rigidity says its length never changes. Write that fixed length as r dotted with itself, equal to the constant L squared. Differentiate. The constant disappears, leaving r dotted with its own rate of change equal to zero. So that rate points square to r. Here is the whole family. Change the length and even reverse the arrow; it stays perpendicular. Every member can be written as omega crossed with r. For a rigid body, one angular velocity omega works for every pair of points at once. It belongs to the body, not to A or B. Now compose the positions. Start at the position of A, add r, and land at the position of B. Differentiate that addition. The first velocity is v A. The second term is the rate of r, which becomes omega cross r. Their sum is v B. A translates with v A while the body turns about A with angular velocity omega. B carries both effects, so its green arrow follows the sum we just built. Now let the body move through the translation and rotation we just composed. Translation plus rotation is the entire instantaneous freedom of a rigid body. Every construction that follows is this one relation read in a different way. ### [01:52.46 · The Instantaneous Centre](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=112.45983333333334) In the plane, the rigid-body relation has a consequence you can watch. Here is a wheel rolling along the ground without slipping. The material point touching the ground is not sliding. At this instant C has velocity zero, so it is momentarily at rest. That is the instantaneous centre. Now take A in the relation to be C. Its velocity term is zero. What remains says that every point P moves as omega crossed with the vector from C to P. The hub is one radius from C, so it moves at omega R. The top material point is two radii away, so its speed is twice the hub speed. At Q, the same rule gives a velocity perpendicular to C Q. Its length grows in direct proportion to Q's distance from C. At one instant, then, the whole velocity field looks exactly like a wheel pinned at C and rotating about it. But C is not one fixed material point. Watch C zero leave the ground as the wheel rolls. A moment later, a different material point is touching down, and that new point is the one at rest. The same idea gives a ruler construction. Here is a moving bar with the velocity direction already attached at A and B. Each point circles C, so draw a perpendicular to each velocity. The centre must lie on both lines. The two lines meet here, at C. From that one point, every speed is omega times the distance out to the material point. There is one limiting case. If the two velocities are equal and parallel, their perpendiculars are parallel too. They never meet, so C is at infinity and the body is purely translating. ### [03:46.226 · The Instantaneous Axis and Screw Motion](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=226.22579166666668) Lift the body into three dimensions. This solid extends through all three coordinate directions, with its material points locked into one shape. Which of those points, if any, are momentarily at rest? Ask the rigid-body relation. Choose A on the body, draw its velocity, and keep the body's one angular velocity omega in view. We are hunting for a point P whose velocity is zero. The cross-product term must cancel v A exactly. It can do that only with the part square to omega, because every omega cross r is perpendicular to omega. Here the magenta arrow is that exact cancellation. For the moment v A is entirely square to omega. Solving the cancellation gives this particular displacement r zero. But r zero is only one answer. Add any multiple lambda omega and the cross product does not change. Watch lambda sweep through its values: every point it reaches is another solution, so the solutions fill a line. That tilted line is the instantaneous axis of rotation. Every green location on it has zero velocity at this instant. A point off the axis swings around it in a circle lying square to omega. Its speed is omega times its perpendicular distance from the axis. The same construction at B uses the same omega. Turn the view and the geometry separates cleanly: the red axis is one tilted line in space, and both green velocities stand square to it. Omega belongs to the whole body. A and B do not get different angular velocities. What changes from point to point is r, and therefore the cross-product contribution. Now remove the assumption we just used. If v A is not square to omega, split it into a perpendicular piece and a parallel piece. Let those two pieces land. Complete their parallelogram, and its green diagonal is the original velocity v A. The cross product can cancel the perpendicular piece, exactly as before. It can never touch the parallel piece, whose projection formula is this. So the axis still exists, but its points now slide along omega instead of resting. This yellow arrow is the velocity shared by every point on it. Take a point away from the axis. Its velocity is the sum of a turn around the axis and that same slide along it. Follow one material point. The combined motion traces this yellow helix, winding around the tilted axis while advancing along it. This is Chasles' theorem. At every instant, rigid-body motion is a screw: a rotation about an axis together with a translation along that axis. Watch the numbered point perform both parts. The slide per unit turn is the pitch. First limiting case: let the slide shrink to nothing. The advancing helix closes into a circle, the axis velocity vanishes, and the motion becomes pure rotation. Second limiting case: restore the slide and let omega shrink to nothing. The winding opens into a straight path, the red omega arrow disappears, and equal green velocities show pure translation. Put turn and slide back together. The tilted axis, the helix and the travelling point return as the one picture that contains the general case. Three ideas carry the lecture. Keep the moving screw beside them while we read the list. First, any two points share one omega, and their velocities differ by omega cross the vector between them. Second, plane motion has an instantaneous centre, but the material point occupying it changes as the body moves. Third, spatial motion has an instantaneous axis. Rotation about it plus translation along it is the general screw motion, every instant. ## 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/01M14TYA5P7YWR2CMWJYKMN214/0/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: [The Rigid Body Relation](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=0) Span: 00:00–01:52.46 (0s–112.45983333333334s). #### Objects - arrow\_rate: a Vector \[magenta\] labelled "frac(dif arrow(r), dif t)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…, end=((((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1…) - arrow\_va: an Arrow \[green\] labelled "arrow(v)\_A" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=((((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0…) - arrow\_vb: an Arrow \[green\] labelled "arrow(v)\_B" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…, end=((((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1…) - body: a Polygon \[blue\] drawn in plane (vertices=((((2.1 + (0.55 \* motion)) + ((-1.2000000000000002 \* cos((0.35 …, fill\_opacity=0.12) - card: a Title that says "Engineering Dynamics — Kinematics of a Body: The Instantaneous Axis" - curl\_omega: a CurvedArrow \[red\] labelled "arrow(omega)" drawn in plane (start=((((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0…, end=((((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0…, bend=0.45) - family: a VariableNumber (initial\_value=0.48, format\_spec='.1f') - heading: a Heading that says "Two Points, One Angular Velocity" - heading\_sum: a Heading that says "Position Adds, So Velocity Adds" - motion: a VariableNumber (format\_spec='.1f') - plane: a Figure (x\_range=(0.5, 6.0), y\_range=(0.5, 4.8), aspect=(5.5, 4.3)) - point: a Point \[yellow\] drawn in plane (location=(2.1, 1.7)) - point\_2: a Point \[yellow\] drawn in plane (location=(4.0, 3.3)) - position\_sum: a Math \[text\] that says "$arrow(x)\_A + arrow(r) = arrow(x)\_B$" - pt\_a: a Point \[text\] labelled "A" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…) - pt\_b: a Point \[text\] labelled "B" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…) - rate\_law: a Math \[text\] that says "$frac(dif arrow(r), dif t)$" - relation: a Math \[text\] that says "$arrow(v)\_B = arrow(v)\_A + arrow(omega) times arrow(r)$" - right\_rate: an Angle \[yellow\] drawn in plane (vertex=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…, sides=((((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0…, right\_angle=True) - seg\_r: a Line \[yellow\] labelled "arrow(r)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…) - velocity\_sum: a Math \[text\] that says "$arrow(v)\_A + arrow(omega) times arrow(r) = arrow(v)\_B$" - work: a Derivation \[text\] that says "$arrow(r) dot arrow(r) &= L^2 \\ 2 thin arrow(r) dot frac(dif arrow(r), dif t) &= 0$" #### Beats ##### [00:00](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=0) Narration: Kinematics of a rigid body begins with one promise: distances inside the body never change. That promise will give us its complete velocity law. Board: Empty. Actions: - [00:00](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=0): card is shown on the screen, written out. - [00:1.5](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=1.5): card: enter:write-left-to-right. - [00:10.182](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=10.182): card is hidden from the screen — left the board. ##### [00:11.382](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=11.382) Narration: Here is the body itself, a collection of material points locked together. Pick any two of them and call them A and B. The vector r runs from A to B, and rigidity says its length never changes. Board: Unchanged from the preceding beat in this scene. Actions: - [00:11.382](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=11.382): heading is shown on the screen, written out. - [00:11.382](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=11.382): plane is shown on the screen, written out. - [00:11.382](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=11.382): body is shown on the screen, written out. - [00:13.367](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=13.366999999999999): pt\_a is shown on the screen, written out. - [00:13.367](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=13.366999999999999): point is shown on the screen, grown. - [00:15.367](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=15.366999999999999): point is hidden from the screen. - [00:18.604](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=18.604): pt\_b is shown on the screen, written out. - [00:18.604](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=18.604): point\_2 is shown on the screen, grown. - [00:19.869](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=19.869): seg\_r is shown on the screen, written out. - [00:20.604](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=20.604): point\_2 is hidden from the screen. ##### [00:25.74](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=25.74) Narration: Write that fixed length as r dotted with itself, equal to the constant L squared. Board: plane — a Figure (x\_range=(0.5, 6.0), y\_range=(0.5, 4.8), aspect=(5.5, 4.3)); heading — a Heading that says "Two Points, One Angular Velocity"; body — a Polygon \[blue\] drawn in plane (vertices=((((2.1 + (0.55 \* motion)) + ((-1.2000000000000002 \* cos((0.35 …, fill\_opacity=0.12); pt\_a — a Point \[text\] labelled "A" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…); pt\_b — a Point \[text\] labelled "B" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); seg\_r — a Line \[yellow\] labelled "arrow(r)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…) Actions: - [00:26.088](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=26.087999999999997): plane moves to a new place on the board. - [00:26.088](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=26.087999999999997): work is shown on the screen, written out. - [00:28.039](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=28.038999999999998): work (the "arrow(r) dot arrow(r)" part) is emphasized. - [00:30.93](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=30.93): work (the "L^2" part) is emphasized. - [00:30.93](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=30.93): work (the "arrow(r) dot arrow(r)" part) is no longer emphasized. - [00:31.939](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=31.9395): work (the "L^2" part) is no longer emphasized. ##### [00:32.539](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=32.5395) Narration: Differentiate. The constant disappears, leaving r dotted with its own rate of change equal to zero. So that rate points square to r. Board: Unchanged from the preceding beat in this scene. Actions: - [00:32.946](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=32.946): work is shown on the screen, written out. - [00:36.697](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=36.697): work (the "arrow(r)" part) is emphasized. - [00:37.916](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=37.916): arrow\_rate is shown on the screen, written out. - [00:37.916](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=37.916): rate\_law is shown on the screen, written out. - [00:37.916](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=37.916): work (the "arrow(r)" part) is no longer emphasized. - [00:37.916](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=37.916): work (the "frac(dif arrow(r), dif t)" part) is emphasized. - [00:41.457](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=41.456999999999994): right\_rate is shown on the screen, written out. - [00:42.803](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=42.8035): work (the "frac(dif arrow(r), dif t)" part) is no longer emphasized. ##### [00:43.403](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=43.403499999999994) Narration: Here is the whole family. Change the length and even reverse the arrow; it stays perpendicular. Every member can be written as omega crossed with r. Board: rate\_law — a Math \[text\] that says "$frac(dif arrow(r), dif t)$"; plane — a Figure (x\_range=(0.5, 6.0), y\_range=(0.5, 4.8), aspect=(5.5, 4.3)); heading — a Heading that says "Two Points, One Angular Velocity"; body — a Polygon \[blue\] drawn in plane (vertices=((((2.1 + (0.55 \* motion)) + ((-1.2000000000000002 \* cos((0.35 …, fill\_opacity=0.12); pt\_a — a Point \[text\] labelled "A" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…); pt\_b — a Point \[text\] labelled "B" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); seg\_r — a Line \[yellow\] labelled "arrow(r)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); arrow\_rate — a Vector \[magenta\] labelled "frac(dif arrow(r), dif t)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…, end=((((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1…); right\_rate — an Angle \[yellow\] drawn in plane (vertex=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…, sides=((((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0…, right\_angle=True) Actions: - [00:43.403](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=43.403499999999994): right\_rate is hidden from the screen. - [00:45.574](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=45.574): arrow\_rate is redrawn as the numbers it depends on change. - [00:45.574](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=45.574): family ticks to -0.58. - [00:51.345](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=51.34499999999999): rate\_law becomes "$frac(dif arrow(r), dif t) = arrow(omega) times arrow(r)$". - [00:51.867](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=51.867): rate\_law (the "arrow(omega)" part) is emphasized. - [00:53.887](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=53.88699999999999): rate\_law (the "arrow(omega)" part) is no longer emphasized. ##### [00:54.487](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=54.486999999999995) Narration: For a rigid body, one angular velocity omega works for every pair of points at once. It belongs to the body, not to A or B. Board: rate\_law — a Math \[text\] that says "$frac(dif arrow(r), dif t)$"; plane — a Figure (x\_range=(0.5, 6.0), y\_range=(0.5, 4.8), aspect=(5.5, 4.3)); heading — a Heading that says "Two Points, One Angular Velocity"; body — a Polygon \[blue\] drawn in plane (vertices=((((2.1 + (0.55 \* motion)) + ((-1.2000000000000002 \* cos((0.35 …, fill\_opacity=0.12); pt\_a — a Point \[text\] labelled "A" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…); pt\_b — a Point \[text\] labelled "B" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); seg\_r — a Line \[yellow\] labelled "arrow(r)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); arrow\_rate — a Vector \[magenta\] labelled "frac(dif arrow(r), dif t)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…, end=((((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1…) Actions: - [00:57.831](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=57.831): rate\_law (the "arrow(omega)" part) is emphasized. - [01:2.347](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=62.347): rate\_law (the "arrow(omega)" part) is no longer emphasized. - [01:4.402](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=64.40199999999999): plane moves to a new place on the board. - [01:4.402](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=64.40199999999999): arrow\_rate is hidden from the screen. - [01:4.402](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=64.40199999999999): heading is hidden from the screen — left the board. - [01:4.402](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=64.40199999999999): rate\_law is hidden from the screen — left the board. - [01:4.402](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=64.40199999999999): work is hidden from the screen — left the board. ##### [01:5.002](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=65.002) Narration: Now compose the positions. Start at the position of A, add r, and land at the position of B. Board: plane — a Figure (x\_range=(0.5, 6.0), y\_range=(0.5, 4.8), aspect=(5.5, 4.3)); body — a Polygon \[blue\] drawn in plane (vertices=((((2.1 + (0.55 \* motion)) + ((-1.2000000000000002 \* cos((0.35 …, fill\_opacity=0.12); pt\_a — a Point \[text\] labelled "A" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…); pt\_b — a Point \[text\] labelled "B" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); seg\_r — a Line \[yellow\] labelled "arrow(r)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…) Actions: - [01:5.002](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=65.002): heading\_sum is shown on the screen, written out. - [01:5.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=65.64): position\_sum is shown on the screen, written out. ##### [01:12.759](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=72.759) Narration: Differentiate that addition. The first velocity is v A. The second term is the rate of r, which becomes omega cross r. Their sum is v B. Board: plane — a Figure (x\_range=(0.5, 6.0), y\_range=(0.5, 4.8), aspect=(5.5, 4.3)); body — a Polygon \[blue\] drawn in plane (vertices=((((2.1 + (0.55 \* motion)) + ((-1.2000000000000002 \* cos((0.35 …, fill\_opacity=0.12); pt\_a — a Point \[text\] labelled "A" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…); pt\_b — a Point \[text\] labelled "B" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); seg\_r — a Line \[yellow\] labelled "arrow(r)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); position\_sum — a Math \[text\] that says "$arrow(x)\_A + arrow(r) = arrow(x)\_B$"; heading\_sum — a Heading that says "Position Adds, So Velocity Adds" Actions: - [01:12.846](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=72.846): velocity\_sum is shown on the screen, written out. ##### [01:23.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=83.791) Narration: A translates with v A while the body turns about A with angular velocity omega. B carries both effects, so its green arrow follows the sum we just built. Board: plane — a Figure (x\_range=(0.5, 6.0), y\_range=(0.5, 4.8), aspect=(5.5, 4.3)); body — a Polygon \[blue\] drawn in plane (vertices=((((2.1 + (0.55 \* motion)) + ((-1.2000000000000002 \* cos((0.35 …, fill\_opacity=0.12); pt\_a — a Point \[text\] labelled "A" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…); pt\_b — a Point \[text\] labelled "B" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); seg\_r — a Line \[yellow\] labelled "arrow(r)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); position\_sum — a Math \[text\] that says "$arrow(x)\_A + arrow(r) = arrow(x)\_B$"; velocity\_sum — a Math \[text\] that says "$arrow(v)\_A + arrow(omega) times arrow(r) = arrow(v)\_B$"; heading\_sum — a Heading that says "Position Adds, So Velocity Adds" Actions: - [01:24.093](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=84.093): arrow\_va is shown on the screen, written out. - [01:26.554](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=86.554): curl\_omega is shown on the screen, written out. - [01:30.769](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=90.769): arrow\_vb is shown on the screen, written out. ##### [01:35.629](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=95.6295) Narration: Now let the body move through the translation and rotation we just composed. Board: plane — a Figure (x\_range=(0.5, 6.0), y\_range=(0.5, 4.8), aspect=(5.5, 4.3)); body — a Polygon \[blue\] drawn in plane (vertices=((((2.1 + (0.55 \* motion)) + ((-1.2000000000000002 \* cos((0.35 …, fill\_opacity=0.12); pt\_a — a Point \[text\] labelled "A" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…); pt\_b — a Point \[text\] labelled "B" drawn in plane (location=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); seg\_r — a Line \[yellow\] labelled "arrow(r)" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…); position\_sum — a Math \[text\] that says "$arrow(x)\_A + arrow(r) = arrow(x)\_B$"; velocity\_sum — a Math \[text\] that says "$arrow(v)\_A + arrow(omega) times arrow(r) = arrow(v)\_B$"; heading\_sum — a Heading that says "Position Adds, So Velocity Adds"; arrow\_va — an Arrow \[green\] labelled "arrow(v)\_A" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0.…, end=((((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0…); curl\_omega — a CurvedArrow \[red\] labelled "arrow(omega)" drawn in plane (start=((((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0…, end=((((2.1 + (0.55 \* motion)) + ((0.0 \* cos((0.35 \* motion))) - (0…, bend=0.45); arrow\_vb — an Arrow \[green\] labelled "arrow(v)\_B" drawn in plane (start=(((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1.…, end=((((2.1 + (0.55 \* motion)) + ((1.9 \* cos((0.35 \* motion))) - (1…) Actions: - [01:36.826](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=96.82600000000001): body is redrawn as the numbers it depends on change. - [01:36.826](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=96.82600000000001): pt\_a is redrawn as the numbers it depends on change. - [01:36.826](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=96.82600000000001): pt\_b is redrawn as the numbers it depends on change. - [01:36.826](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=96.82600000000001): seg\_r is redrawn as the numbers it depends on change. - [01:36.826](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=96.82600000000001): arrow\_va is redrawn as the numbers it depends on change. - [01:36.826](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=96.82600000000001): curl\_omega is redrawn as the numbers it depends on change. - [01:36.826](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=96.82600000000001): arrow\_vb is redrawn as the numbers it depends on change. - [01:36.826](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=96.82600000000001): motion ticks to 1.2. ##### [01:40.85](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=100.8505) Narration: Translation plus rotation is the entire instantaneous freedom of a rigid body. Every construction that follows is this one relation read in a different way. Board: Unchanged from the preceding beat in this scene. Actions: - [01:41.152](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=101.152): relation is shown on the screen, written out. - [01:49.198](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=109.19800000000001): A box is drawn around relation. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): heading\_sum is hidden from the screen — left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): plane is hidden from the screen — left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): body is hidden from the screen — plane left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): pt\_a is hidden from the screen — plane left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): pt\_b is hidden from the screen — plane left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): seg\_r is hidden from the screen — plane left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): arrow\_va is hidden from the screen — plane left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): curl\_omega is hidden from the screen — plane left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): arrow\_vb is hidden from the screen — plane left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): position\_sum is hidden from the screen — left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): relation is hidden from the screen — left the board. - [01:51.418](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=111.41816666666666): velocity\_sum is hidden from the screen — left the board. ### Scene 2: [The Instantaneous Centre](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=112.45983333333334) Span: 01:52.46–03:46.226 (112.45983333333334s–226.22579166666668s). #### Objects - arrow\_bva: an Arrow \[green\] labelled "arrow(v)\_A" drawn in build (start=(1.4, 1.2), end=(2.24, 0.86)) - arrow\_bvb: an Arrow \[green\] labelled "arrow(v)\_B" drawn in build (start=(4.6, 2.2), end=(5.16, 2.76)) - arrow\_hub: an Arrow \[green\] labelled "arrow(v)\_O" drawn in wheel (start=((3.0 + (1.4 \* roll)), 2.0), end=(((3.0 + (1.4 \* roll)) + 0.8959999999999999), 2.0)) - arrow\_q: an Arrow \[green\] labelled "arrow(v)\_Q" drawn in wheel (start=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll)…, end=((((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll…) - arrow\_top: an Arrow \[green\] labelled "2 arrow(v)\_O" drawn in wheel (start=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…, end=((((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll…) - build: a Figure (x\_range=(0.8, 5.6), y\_range=(0.5, 5.1), aspect=(4.8, 4.6)) - concept\_ic: a Panel that says "The material point that is momentarily at rest. Every other point turns about its current location, as though the body were pinned there." - formula\_ic: a Math \[text\] that says "$arrow(v)\_P = arrow(omega) times arrow(r)\_(C P)$" - ground: a Line \[gray\] drawn in wheel (start=(0.6, 0.6), end=(6.8, 0.6)) - heading\_build: a Heading that says "Finding It With a Ruler" - heading\_ic: a Heading that says "The Instantaneous Centre" - par\_a: an Arrow \[cyan\] labelled "arrow(v)\_A" drawn in build (start=(1.4, 1.2), end=(2.12, 2.16)) - par\_b: an Arrow \[cyan\] labelled "arrow(v)\_B" drawn in build (start=(4.6, 2.2), end=(5.32, 3.16)) - par\_perp\_a: a Line \[gray\] drawn in build (start=(2.2, 0.6), end=(0.6, 1.8), dashed=True) - par\_perp\_b: a Line \[gray\] drawn in build (start=(5.4, 1.6), end=(3.8, 2.8), dashed=True) - perp\_a: a Line \[gray\] drawn in build (start=(1.4, 1.2), end=(2.88, 4.91), dashed=True) - perp\_b: a Line \[gray\] drawn in build (start=(4.6, 2.2), end=(2.13, 4.68), dashed=True) - point: a Point \[yellow\] drawn in wheel (location=(3.0, 0.6)) - point\_2: a Point \[yellow\] drawn in wheel (location=(3.9899494936611664, 2.9899494936611664)) - point\_3: a Point \[yellow\] drawn in wheel (location=(4.05, 0.6)) - point\_4: a Point \[yellow\] drawn in build (location=(2.6, 4.2)) - pt\_ba: a Point \[text\] labelled "A" drawn in build (location=(1.4, 1.2)) - pt\_bb: a Point \[text\] labelled "B" drawn in build (location=(4.6, 2.2)) - pt\_bc: a Point \[red\] labelled "C" drawn in build (location=(2.6, 4.2)) - pt\_c: a Point \[red\] labelled "C\_0, thin arrow(v)\_(C\_0) = 0" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((-1.5707963267948966 - roll…) - pt\_hub: a Point \[text\] labelled "O" drawn in wheel (location=((3.0 + (1.4 \* roll)), 2.0)) - pt\_next: a Point \[green\] labelled "C\_1, thin arrow(v)\_(C\_1) = 0" drawn in wheel (location=((3.0 + (1.4 \* roll)), 0.6)) - pt\_q: a Point \[text\] labelled "Q" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll)…) - pt\_top: a Point \[text\] labelled "P" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…) - right\_a: an Angle \[yellow\] drawn in build (vertex=(1.4, 1.2), sides=((2.24, 0.86), (2.6, 4.2)), right\_angle=True) - right\_b: an Angle \[yellow\] drawn in build (vertex=(4.6, 2.2), sides=((5.16, 2.76), (2.6, 4.2)), right\_angle=True) - right\_q: an Angle \[yellow\] drawn in wheel (vertex=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll)…, sides=(((3.0 + (1.4 \* roll)), 0.6), ((((3.0 + (1.4 \* roll)) + (1.4 \* …, right\_angle=True) - rim: a Circle \[blue\] drawn in wheel (center=((3.0 + (1.4 \* roll)), 2.0), radius=1.4) - rod: a Line \[blue\] drawn in build (start=(1.4, 1.2), end=(4.6, 2.2)) - roll: a VariableNumber (format\_spec='.1f') - rules: a Block \[text\] that says "Mark the velocity direction at two points. Erect a perpendicular to each velocity. Their intersection is $C$; parallel lines place $C$ at infinity. Every point has speed $omega d$, where $d$ is its distance from $C$." - spoke\_cq: a Line \[gray\] drawn in wheel (start=((3.0 + (1.4 \* roll)), 0.6), end=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll)…, dashed=True) - spoke\_ct: a Line \[gray\] drawn in wheel (start=((3.0 + (1.4 \* roll)), 0.6), end=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…, dashed=True) - wheel: a Figure (x\_range=(0.5, 6.9), y\_range=(0.2, 4.1), aspect=(6.4, 3.9)) #### Beats ##### [01:52.46](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=112.45983333333334) Narration: In the plane, the rigid-body relation has a consequence you can watch. Here is a wheel rolling along the ground without slipping. Board: Empty. Actions: - [01:52.46](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=112.45983333333334): wheel is shown on the screen, written out. - [01:52.46](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=112.45983333333334): ground is shown on the screen, written out. - [01:52.46](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=112.45983333333334): rim is shown on the screen, written out. - [01:52.46](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=112.45983333333334): pt\_hub is shown on the screen, written out. - [01:52.762](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=112.76183333333334): heading\_ic is shown on the screen, written out. ##### [02:1.094](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=121.09433333333334) Narration: The material point touching the ground is not sliding. At this instant C has velocity zero, so it is momentarily at rest. That is the instantaneous centre. Board: wheel — a Figure (x\_range=(0.5, 6.9), y\_range=(0.2, 4.1), aspect=(6.4, 3.9)); heading\_ic — a Heading that says "The Instantaneous Centre"; ground — a Line \[gray\] drawn in wheel (start=(0.6, 0.6), end=(6.8, 0.6)); rim — a Circle \[blue\] drawn in wheel (center=((3.0 + (1.4 \* roll)), 2.0), radius=1.4); pt\_hub — a Point \[text\] labelled "O" drawn in wheel (location=((3.0 + (1.4 \* roll)), 2.0)) Actions: - [02:2.22](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=122.21983333333334): pt\_c is shown on the screen, written out. - [02:5.75](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=125.74983333333333): point is shown on the screen, grown. - [02:7.75](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=127.74983333333333): point is hidden from the screen. - [02:8.142](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=128.14183333333332): concept\_ic is shown on the screen, written out. ##### [02:12.584](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=132.58433333333335) Narration: Now take A in the relation to be C. Its velocity term is zero. What remains says that every point P moves as omega crossed with the vector from C to P. Board: concept\_ic — a Panel that says "The material point that is momentarily at rest. Every other point turns about its current location, as though the body were pinned there."; wheel — a Figure (x\_range=(0.5, 6.9), y\_range=(0.2, 4.1), aspect=(6.4, 3.9)); heading\_ic — a Heading that says "The Instantaneous Centre"; ground — a Line \[gray\] drawn in wheel (start=(0.6, 0.6), end=(6.8, 0.6)); rim — a Circle \[blue\] drawn in wheel (center=((3.0 + (1.4 \* roll)), 2.0), radius=1.4); pt\_hub — a Point \[text\] labelled "O" drawn in wheel (location=((3.0 + (1.4 \* roll)), 2.0)); pt\_c — a Point \[red\] labelled "C\_0, thin arrow(v)\_(C\_0) = 0" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((-1.5707963267948966 - roll…) Actions: - [02:13.85](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=133.84983333333332): formula\_ic is shown on the screen, written out. - [02:19.527](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=139.52683333333334): formula\_ic (the "arrow(v)\_P" part) is emphasized. - [02:20.619](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=140.61883333333333): formula\_ic (the "arrow(omega)" part) is emphasized. - [02:20.619](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=140.61883333333333): formula\_ic (the "arrow(v)\_P" part) is no longer emphasized. - [02:22.024](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=142.02383333333333): spoke\_ct is shown on the screen, written out. - [02:22.024](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=142.02383333333333): formula\_ic (the "arrow(omega)" part) is no longer emphasized. - [02:22.024](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=142.02383333333333): formula\_ic (the "arrow(r)\_(C P)" part) is emphasized. - [02:23.73](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=143.72983333333332): formula\_ic (the "arrow(r)\_(C P)" part) is no longer emphasized. ##### [02:24.33](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=144.32983333333334) Narration: The hub is one radius from C, so it moves at omega R. The top material point is two radii away, so its speed is twice the hub speed. Board: concept\_ic — a Panel that says "The material point that is momentarily at rest. Every other point turns about its current location, as though the body were pinned there."; formula\_ic — a Math \[text\] that says "$arrow(v)\_P = arrow(omega) times arrow(r)\_(C P)$"; wheel — a Figure (x\_range=(0.5, 6.9), y\_range=(0.2, 4.1), aspect=(6.4, 3.9)); heading\_ic — a Heading that says "The Instantaneous Centre"; ground — a Line \[gray\] drawn in wheel (start=(0.6, 0.6), end=(6.8, 0.6)); rim — a Circle \[blue\] drawn in wheel (center=((3.0 + (1.4 \* roll)), 2.0), radius=1.4); pt\_hub — a Point \[text\] labelled "O" drawn in wheel (location=((3.0 + (1.4 \* roll)), 2.0)); pt\_c — a Point \[red\] labelled "C\_0, thin arrow(v)\_(C\_0) = 0" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((-1.5707963267948966 - roll…); spoke\_ct — a Line \[gray\] drawn in wheel (start=((3.0 + (1.4 \* roll)), 0.6), end=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…, dashed=True) Actions: - [02:24.83](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=144.82983333333334): arrow\_hub is shown on the screen, written out. - [02:29.311](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=149.31083333333333): pt\_top is shown on the screen, written out. - [02:32.783](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=152.78283333333331): arrow\_top is shown on the screen, written out. - [02:33.305](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=153.30483333333333): arrow\_hub is indicated — a transient flash. ##### [02:34.811](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=154.81083333333333) Narration: At Q, the same rule gives a velocity perpendicular to C Q. Its length grows in direct proportion to Q's distance from C. Board: concept\_ic — a Panel that says "The material point that is momentarily at rest. Every other point turns about its current location, as though the body were pinned there."; formula\_ic — a Math \[text\] that says "$arrow(v)\_P = arrow(omega) times arrow(r)\_(C P)$"; wheel — a Figure (x\_range=(0.5, 6.9), y\_range=(0.2, 4.1), aspect=(6.4, 3.9)); heading\_ic — a Heading that says "The Instantaneous Centre"; ground — a Line \[gray\] drawn in wheel (start=(0.6, 0.6), end=(6.8, 0.6)); rim — a Circle \[blue\] drawn in wheel (center=((3.0 + (1.4 \* roll)), 2.0), radius=1.4); pt\_hub — a Point \[text\] labelled "O" drawn in wheel (location=((3.0 + (1.4 \* roll)), 2.0)); pt\_c — a Point \[red\] labelled "C\_0, thin arrow(v)\_(C\_0) = 0" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((-1.5707963267948966 - roll…); spoke\_ct — a Line \[gray\] drawn in wheel (start=((3.0 + (1.4 \* roll)), 0.6), end=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…, dashed=True); arrow\_hub — an Arrow \[green\] labelled "arrow(v)\_O" drawn in wheel (start=((3.0 + (1.4 \* roll)), 2.0), end=(((3.0 + (1.4 \* roll)) + 0.8959999999999999), 2.0)); pt\_top — a Point \[text\] labelled "P" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…); arrow\_top — an Arrow \[green\] labelled "2 arrow(v)\_O" drawn in wheel (start=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…, end=((((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll…) Actions: - [02:35.357](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=155.35683333333333): pt\_q is shown on the screen, written out. - [02:35.357](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=155.35683333333333): point\_2 is shown on the screen, grown. - [02:36.703](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=156.70283333333333): arrow\_q is shown on the screen, written out. - [02:37.295](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=157.29483333333334): right\_q is shown on the screen, written out. - [02:37.357](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=157.35683333333333): point\_2 is hidden from the screen. - [02:38.398](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=158.39783333333332): spoke\_cq is shown on the screen, written out. - [02:40.035](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=160.03483333333332): arrow\_q is indicated — a transient flash. ##### [02:44.245](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=164.24533333333335) Narration: At one instant, then, the whole velocity field looks exactly like a wheel pinned at C and rotating about it. Board: concept\_ic — a Panel that says "The material point that is momentarily at rest. Every other point turns about its current location, as though the body were pinned there."; formula\_ic — a Math \[text\] that says "$arrow(v)\_P = arrow(omega) times arrow(r)\_(C P)$"; wheel — a Figure (x\_range=(0.5, 6.9), y\_range=(0.2, 4.1), aspect=(6.4, 3.9)); heading\_ic — a Heading that says "The Instantaneous Centre"; ground — a Line \[gray\] drawn in wheel (start=(0.6, 0.6), end=(6.8, 0.6)); rim — a Circle \[blue\] drawn in wheel (center=((3.0 + (1.4 \* roll)), 2.0), radius=1.4); pt\_hub — a Point \[text\] labelled "O" drawn in wheel (location=((3.0 + (1.4 \* roll)), 2.0)); pt\_c — a Point \[red\] labelled "C\_0, thin arrow(v)\_(C\_0) = 0" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((-1.5707963267948966 - roll…); spoke\_ct — a Line \[gray\] drawn in wheel (start=((3.0 + (1.4 \* roll)), 0.6), end=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…, dashed=True); arrow\_hub — an Arrow \[green\] labelled "arrow(v)\_O" drawn in wheel (start=((3.0 + (1.4 \* roll)), 2.0), end=(((3.0 + (1.4 \* roll)) + 0.8959999999999999), 2.0)); pt\_top — a Point \[text\] labelled "P" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…); arrow\_top — an Arrow \[green\] labelled "2 arrow(v)\_O" drawn in wheel (start=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll)…, end=((((3.0 + (1.4 \* roll)) + (1.4 \* cos((1.5707963267948966 - roll…); pt\_q — a Point \[text\] labelled "Q" drawn in wheel (location=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll)…); spoke\_cq — a Line \[gray\] drawn in wheel (start=((3.0 + (1.4 \* roll)), 0.6), end=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll)…, dashed=True); arrow\_q — an Arrow \[green\] labelled "arrow(v)\_Q" drawn in wheel (start=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll)…, end=((((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll…); right\_q — an Angle \[yellow\] drawn in wheel (vertex=(((3.0 + (1.4 \* roll)) + (1.4 \* cos((0.7853981633974483 - roll)…, sides=(((3.0 + (1.4 \* roll)), 0.6), ((((3.0 + (1.4 \* roll)) + (1.4 \* …, right\_angle=True) Actions: - [02:48.46](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=168.45983333333334): concept\_ic (the "pinned there" part) is emphasized. - [02:50.666](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=170.66583333333332): concept\_ic (the "pinned there" part) is no longer emphasized. ##### [02:51.266](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=171.26583333333332) Narration: But C is not one fixed material point. Watch C zero leave the ground as the wheel rolls. A moment later, a different material point is touching down, and that new point is the one at rest. Board: Unchanged from the preceding beat in this scene. Actions: - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): rim is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): pt\_hub is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): pt\_c is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): spoke\_ct is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): arrow\_hub is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): pt\_top is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): arrow\_top is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): pt\_q is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): spoke\_cq is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): arrow\_q is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): right\_q is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): pt\_next is redrawn as the numbers it depends on change. - [02:57.64](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=177.63983333333334): roll ticks to 0.75. - [03:0.147](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=180.14683333333335): pt\_next is shown on the screen, written out. - [03:0.147](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=180.14683333333335): point\_3 is shown on the screen, grown. - [03:2.147](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=182.14683333333335): point\_3 is hidden from the screen. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): concept\_ic is hidden from the screen — left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): formula\_ic is hidden from the screen — left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): heading\_ic is hidden from the screen — left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): wheel is hidden from the screen — left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): ground is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): rim is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): pt\_hub is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): pt\_c is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): spoke\_ct is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): arrow\_hub is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): pt\_top is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): arrow\_top is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): pt\_q is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): spoke\_cq is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): arrow\_q is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): right\_q is hidden from the screen — wheel left the board. - [03:4.791](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=184.79133333333334): pt\_next is hidden from the screen — wheel left the board. ##### [03:5.991](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=185.99133333333333) Narration: The same idea gives a ruler construction. Here is a moving bar with the velocity direction already attached at A and B. Board: Empty. Actions: - [03:5.991](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=185.99133333333333): heading\_build is shown on the screen, written out. - [03:5.991](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=185.99133333333333): build is shown on the screen, written out. - [03:5.991](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=185.99133333333333): rod is shown on the screen, written out. - [03:5.991](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=185.99133333333333): pt\_ba is shown on the screen, written out. - [03:5.991](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=185.99133333333333): pt\_bb is shown on the screen, written out. - [03:5.991](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=185.99133333333333): arrow\_bva is shown on the screen, written out. - [03:5.991](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=185.99133333333333): arrow\_bvb is shown on the screen, written out. - [03:7.814](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=187.81383333333332): build moves to a new place on the board. - [03:7.814](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=187.81383333333332): rules is shown on the screen, written out. - [03:11.274](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=191.27383333333333): rules (the "velocity direction" part) is emphasized. ##### [03:14.614](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=194.61383333333333) Narration: Each point circles C, so draw a perpendicular to each velocity. The centre must lie on both lines. Board: rules — a Block \[text\] that says "Mark the velocity direction at two points. Erect a perpendicular to each velocity. Their intersection is $C$; parallel lines place $C$ at infinity. Every point has speed $omega d$, where $d$ is its distance from $C$."; build — a Figure (x\_range=(0.8, 5.6), y\_range=(0.5, 5.1), aspect=(4.8, 4.6)); heading\_build — a Heading that says "Finding It With a Ruler"; rod — a Line \[blue\] drawn in build (start=(1.4, 1.2), end=(4.6, 2.2)); pt\_ba — a Point \[text\] labelled "A" drawn in build (location=(1.4, 1.2)); pt\_bb — a Point \[text\] labelled "B" drawn in build (location=(4.6, 2.2)); arrow\_bva — an Arrow \[green\] labelled "arrow(v)\_A" drawn in build (start=(1.4, 1.2), end=(2.24, 0.86)); arrow\_bvb — an Arrow \[green\] labelled "arrow(v)\_B" drawn in build (start=(4.6, 2.2), end=(5.16, 2.76)) Actions: - [03:14.962](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=194.96183333333335): perp\_b is shown on the screen, drawn. - [03:14.962](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=194.96183333333335): right\_a is shown on the screen, written out. - [03:14.962](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=194.96183333333335): right\_b is shown on the screen, written out. - [03:17.273](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=197.27283333333332): perp\_a is shown on the screen, drawn. - [03:17.273](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=197.27283333333332): rules (the "perpendicular" part) is emphasized. - [03:17.273](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=197.27283333333332): rules (the "velocity direction" part) is no longer emphasized. ##### [03:22.865](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=202.86533333333333) Narration: The two lines meet here, at C. From that one point, every speed is omega times the distance out to the material point. Board: rules — a Block \[text\] that says "Mark the velocity direction at two points. Erect a perpendicular to each velocity. Their intersection is $C$; parallel lines place $C$ at infinity. Every point has speed $omega d$, where $d$ is its distance from $C$."; build — a Figure (x\_range=(0.8, 5.6), y\_range=(0.5, 5.1), aspect=(4.8, 4.6)); heading\_build — a Heading that says "Finding It With a Ruler"; rod — a Line \[blue\] drawn in build (start=(1.4, 1.2), end=(4.6, 2.2)); pt\_ba — a Point \[text\] labelled "A" drawn in build (location=(1.4, 1.2)); pt\_bb — a Point \[text\] labelled "B" drawn in build (location=(4.6, 2.2)); arrow\_bva — an Arrow \[green\] labelled "arrow(v)\_A" drawn in build (start=(1.4, 1.2), end=(2.24, 0.86)); arrow\_bvb — an Arrow \[green\] labelled "arrow(v)\_B" drawn in build (start=(4.6, 2.2), end=(5.16, 2.76)); perp\_a — a Line \[gray\] drawn in build (start=(1.4, 1.2), end=(2.88, 4.91), dashed=True); perp\_b — a Line \[gray\] drawn in build (start=(4.6, 2.2), end=(2.13, 4.68), dashed=True); right\_a — an Angle \[yellow\] drawn in build (vertex=(1.4, 1.2), sides=((2.24, 0.86), (2.6, 4.2)), right\_angle=True); right\_b — an Angle \[yellow\] drawn in build (vertex=(4.6, 2.2), sides=((5.16, 2.76), (2.6, 4.2)), right\_angle=True) Actions: - [03:23.91](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=203.90983333333332): rules (the "intersection" part) is emphasized. - [03:23.91](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=203.90983333333332): rules (the "perpendicular" part) is no longer emphasized. - [03:24.119](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=204.11883333333333): pt\_bc is shown on the screen, written out. - [03:24.827](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=204.8268333333333): point\_4 is shown on the screen, grown. - [03:26.827](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=206.8268333333333): point\_4 is hidden from the screen. - [03:28.113](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=208.1128333333333): rules (the "$omega d$" part) is emphasized. - [03:28.113](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=208.1128333333333): rules (the "intersection" part) is no longer emphasized. ##### [03:31.766](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=211.7663333333333) Narration: There is one limiting case. If the two velocities are equal and parallel, their perpendiculars are parallel too. They never meet, so C is at infinity and the body is purely translating. Board: rules — a Block \[text\] that says "Mark the velocity direction at two points. Erect a perpendicular to each velocity. Their intersection is $C$; parallel lines place $C$ at infinity. Every point has speed $omega d$, where $d$ is its distance from $C$."; build — a Figure (x\_range=(0.8, 5.6), y\_range=(0.5, 5.1), aspect=(4.8, 4.6)); heading\_build — a Heading that says "Finding It With a Ruler"; rod — a Line \[blue\] drawn in build (start=(1.4, 1.2), end=(4.6, 2.2)); pt\_ba — a Point \[text\] labelled "A" drawn in build (location=(1.4, 1.2)); pt\_bb — a Point \[text\] labelled "B" drawn in build (location=(4.6, 2.2)); arrow\_bva — an Arrow \[green\] labelled "arrow(v)\_A" drawn in build (start=(1.4, 1.2), end=(2.24, 0.86)); arrow\_bvb — an Arrow \[green\] labelled "arrow(v)\_B" drawn in build (start=(4.6, 2.2), end=(5.16, 2.76)); perp\_a — a Line \[gray\] drawn in build (start=(1.4, 1.2), end=(2.88, 4.91), dashed=True); perp\_b — a Line \[gray\] drawn in build (start=(4.6, 2.2), end=(2.13, 4.68), dashed=True); right\_a — an Angle \[yellow\] drawn in build (vertex=(1.4, 1.2), sides=((2.24, 0.86), (2.6, 4.2)), right\_angle=True); right\_b — an Angle \[yellow\] drawn in build (vertex=(4.6, 2.2), sides=((5.16, 2.76), (2.6, 4.2)), right\_angle=True); pt\_bc — a Point \[red\] labelled "C" drawn in build (location=(2.6, 4.2)) Actions: - [03:36.039](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=216.0388333333333): arrow\_bva is hidden from the screen. - [03:36.039](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=216.0388333333333): arrow\_bvb is hidden from the screen. - [03:36.039](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=216.0388333333333): perp\_a is hidden from the screen. - [03:36.039](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=216.0388333333333): perp\_b is hidden from the screen. - [03:36.039](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=216.0388333333333): right\_a is hidden from the screen. - [03:36.039](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=216.0388333333333): right\_b is hidden from the screen. - [03:36.039](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=216.0388333333333): pt\_bc is hidden from the screen. - [03:36.039](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=216.0388333333333): par\_a is shown on the screen, written out. - [03:36.515](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=216.51483333333329): par\_b is shown on the screen, written out. - [03:37.758](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=217.7578333333333): par\_perp\_a is shown on the screen, drawn. - [03:38.605](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=218.60483333333332): par\_perp\_b is shown on the screen, drawn. - [03:40.428](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=220.4278333333333): rules (the "$omega d$" part) is no longer emphasized. - [03:40.428](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=220.4278333333333): rules (the "parallel lines" part) is emphasized. - [03:44.934](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=224.934125): rules (the "parallel lines" part) is no longer emphasized. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): build is hidden from the screen — left the board. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): rod is hidden from the screen — build left the board. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): pt\_ba is hidden from the screen — build left the board. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): pt\_bb is hidden from the screen — build left the board. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): par\_a is hidden from the screen — build left the board. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): par\_b is hidden from the screen — build left the board. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): par\_perp\_a is hidden from the screen — build left the board. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): par\_perp\_b is hidden from the screen — build left the board. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): heading\_build is hidden from the screen — left the board. - [03:45.184](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=225.184125): rules is hidden from the screen — left the board. ### Scene 3: [The Instantaneous Axis and Screw Motion](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=226.22579166666668) Span: 03:46.226–08:1.38 (226.22579166666668s–481.38031249999995s). #### Objects - answer\_axis: a Math \[text\] that says "$arrow(r) = arrow(r)\_0 + lambda thin arrow(omega)$" - arrow\_a: a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)) - arrow\_b: a Vector \[green\] labelled "arrow(v)\_B" drawn in frame (start=(1.68, -0.56, 1.0), end=(1.96026, -0.06752000000000002, 0.6840999999999999)) - arrow\_parallel: a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157)) - arrow\_perp: a Vector \[blue\] labelled "arrow(v)\_(bot)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)) - arrow\_total: a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158)) - axis: a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True) - axis\_lambda: a VariableNumber (initial\_value=-0.75, format\_spec='.1f') - axis\_velocity: a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=((0.29999999999999993 + (axis\_lambda \* 0.6)), (-0.0800000000000…, end=(((0.29999999999999993 + (axis\_lambda \* 0.6)) + 0.4185492184628…) - body: a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6) - circle\_a: a Circle \[gray\] drawn in frame (center=(0.29999999999999993, -0.08000000000000007, 0.524), radius=1.197236818678744, normal\_vector=(0.6, 0.3, 1.0)) - cross\_a: a Vector \[magenta\] labelled "arrow(omega) times arrow(r)\_0" drawn in frame (start=(1.2, -0.8, 0.2), end=(0.91974, -1.29248, 0.5159)) - frame: an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)) - heading\_recap: a Heading that says "What to Carry Away" - heading\_screw: a Heading that says "When Nothing Is at Rest" - helix: a ParametricCurve \[yellow\] labelled "upright("screw path")" drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)) - helix\_rider: a Point \[green\] labelled "0.0" drawn in frame (location=(((0.29999999999999993 + ((0.2 + (0.28 \* helix\_t)) \* 0.49827287…) - helix\_t: a VariableNumber (format\_spec='.1f') - omega\_vec: a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)) - pitch: a Math \[text\] that says "$h = frac(v\_(parallel), omega)$" - point: a Point \[yellow\] drawn in frame (location=(1.2, -0.8, 0.2)) - pt\_a: a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)) - pt\_b: a Point \[text\] labelled "B" drawn in frame (location=(1.68, -0.56, 1.0)) - question: a Text \[text\] that says "In three dimensions, which points of the body are momentarily at rest?" - r0\_vector: a Vector \[yellow\] labelled "arrow(r)\_0" drawn in frame (start=(0.29999999999999993, -0.08000000000000007, 0.524), end=(1.2, -0.8, 0.2)) - recap: a Block \[text\] that says "Two points share one $arrow(omega)$: $arrow(v)\_B = arrow(v)\_A + arrow(omega) times arrow(r)$. Plane motion has a moving instantaneous centre. Spatial motion has an instantaneous axis; turn plus slide gives a screw." - rest\_law: a Math \[text\] that says "$arrow(v)\_(C(lambda)) = 0$" - rest\_point: a Point \[green\] labelled "-0.8" drawn in frame (location=((0.29999999999999993 + (axis\_lambda \* 0.6)), (-0.0800000000000…) - right\_cross: an Angle \[yellow\] drawn in frame (vertex=(1.2, -0.8, 0.2), sides=((1.5487910153857078, -0.6256044923071461, 0.7813183589761798),…, right\_angle=True) - slide\_only: a Line \[yellow\] drawn in frame (start=(0.3996545758244879, -0.03017271208775609, 0.69009095970748), end=(1.2762620090835795, 0.4081310045417897, 2.1511033484726325)) - spoke\_a: a Line \[gray\] drawn in frame (start=(0.29999999999999993, -0.08000000000000007, 0.524), end=(1.2, -0.8, 0.2), dashed=True) - sum\_side\_parallel: a Line \[gray\] drawn in frame (start=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True) - sum\_side\_perp: a Line \[gray\] drawn in frame (start=(1.48026, -0.30752, -0.1159), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True) - translation\_a: a Vector \[green\] labelled "arrow(v)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157)) - translation\_b: a Vector \[green\] labelled "arrow(v)" drawn in frame (start=(1.68, -0.56, 1.0), end=(2.0985492184628494, -0.35072539076857534, 1.6975820307714158)) - turn\_only: a Circle \[yellow\] drawn in frame (center=(0.8379582924540336, 0.18897914622701678, 1.4205971540900562), radius=0.72, normal\_vector=(0.6, 0.3, 1.0)) - work\_axis: a Derivation \[text\] that says "$arrow(v)\_P &= arrow(v)\_A + arrow(omega) times arrow(r) \\ 0 &= arrow(v)\_A + arrow(omega) times arrow(r) \\ arrow(r)\_0 &= frac(arrow(omega) times arrow(v)\_A, arrow(omega) dot arrow(omega))$" - work\_screw: a Derivation \[text\] that says "$arrow(v)\_A &= arrow(v)\_(bot) + arrow(v)\_(parallel) \\ arrow(v)\_(parallel) &= frac(arrow(v)\_A dot arrow(omega), arrow(omega) dot arrow(omega)) thin arrow(omega) \\ arrow(v)\_P &= arrow(v)\_(parallel) + arrow(omega) times arrow(r)$" #### Beats ##### [03:46.226](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=226.22579166666668) Narration: Lift the body into three dimensions. This solid extends through all three coordinate directions, with its material points locked into one shape. Which of those points, if any, are momentarily at rest? Board: Empty. Actions: - [03:46.226](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=226.22579166666668): question is shown on the screen, written out. - [03:46.226](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=226.22579166666668): frame is shown on the screen, written out. - [03:46.226](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=226.22579166666668): body is shown on the screen, written out. - [03:49.454](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=229.4537916666667): frame turns in its own slot. ##### [04:0.231](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=240.2312916666667) Narration: Ask the rigid-body relation. Choose A on the body, draw its velocity, and keep the body's one angular velocity omega in view. We are hunting for a point P whose velocity is zero. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); question — a Text \[text\] that says "In three dimensions, which points of the body are momentarily at rest?"; body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6) Actions: - [04:1.717](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=241.71679166666667): frame moves to a new place on the board. - [04:1.717](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=241.71679166666667): work\_axis is shown on the screen, written out. - [04:3.436](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=243.4357916666667): pt\_a is shown on the screen, written out. - [04:3.436](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=243.4357916666667): point is shown on the screen, grown. - [04:5.108](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=245.10779166666669): arrow\_a is shown on the screen, written out. - [04:5.436](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=245.4357916666667): point is hidden from the screen. - [04:8.312](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=248.3117916666667): omega\_vec is shown on the screen, written out. - [04:12.608](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=252.60779166666669): work\_axis is shown on the screen, written out. ##### [04:14.159](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=254.1592916666667) Narration: The cross-product term must cancel v A exactly. It can do that only with the part square to omega, because every omega cross r is perpendicular to omega. Here the magenta arrow is that exact cancellation. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); question — a Text \[text\] that says "In three dimensions, which points of the body are momentarily at rest?"; body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); arrow\_a — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)) Actions: - [04:14.729](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=254.72879166666667): work\_axis (the "arrow(omega) times arrow(r)" part) is emphasized. - [04:16.4](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=256.39979166666666): work\_axis (the "arrow(omega) times arrow(r)" part) is no longer emphasized. - [04:16.4](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=256.39979166666666): work\_axis (the "arrow(v)\_A" part) is emphasized. - [04:23.053](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=263.0527916666667): right\_cross is shown on the screen, written out. - [04:25.596](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=265.5957916666667): cross\_a is shown on the screen, written out. - [04:27.372](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=267.3717916666667): arrow\_a is indicated — a transient flash. - [04:28.545](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=268.5447916666667): work\_axis (the "arrow(v)\_A" part) is no longer emphasized. ##### [04:29.145](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=269.14479166666666) Narration: For the moment v A is entirely square to omega. Solving the cancellation gives this particular displacement r zero. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); question — a Text \[text\] that says "In three dimensions, which points of the body are momentarily at rest?"; body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); arrow\_a — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); cross\_a — a Vector \[magenta\] labelled "arrow(omega) times arrow(r)\_0" drawn in frame (start=(1.2, -0.8, 0.2), end=(0.91974, -1.29248, 0.5159)); right\_cross — an Angle \[yellow\] drawn in frame (vertex=(1.2, -0.8, 0.2), sides=((1.5487910153857078, -0.6256044923071461, 0.7813183589761798),…, right\_angle=True) Actions: - [04:29.145](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=269.14479166666666): right\_cross is hidden from the screen. - [04:33.278](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=273.2777916666667): work\_axis is shown on the screen, written out. - [04:35.043](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=275.0427916666667): work\_axis (the "arrow(omega) times arrow(v)\_A" part) is emphasized. - [04:36.297](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=276.29679166666665): r0\_vector is shown on the screen, written out. - [04:37.306](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=277.30629166666665): work\_axis (the "arrow(omega) times arrow(v)\_A" part) is no longer emphasized. ##### [04:37.906](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=277.9062916666667) Narration: But r zero is only one answer. Add any multiple lambda omega and the cross product does not change. Watch lambda sweep through its values: every point it reaches is another solution, so the solutions fill a line. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); question — a Text \[text\] that says "In three dimensions, which points of the body are momentarily at rest?"; body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); arrow\_a — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); cross\_a — a Vector \[magenta\] labelled "arrow(omega) times arrow(r)\_0" drawn in frame (start=(1.2, -0.8, 0.2), end=(0.91974, -1.29248, 0.5159)); r0\_vector — a Vector \[yellow\] labelled "arrow(r)\_0" drawn in frame (start=(0.29999999999999993, -0.08000000000000007, 0.524), end=(1.2, -0.8, 0.2)) Actions: - [04:37.906](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=277.9062916666667): answer\_axis is shown on the screen, written out. - [04:39.497](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=279.4967916666667): answer\_axis (the "arrow(r)\_0" part) is emphasized. - [04:41.935](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=281.9347916666667): rest\_point is shown on the screen, written out. - [04:41.935](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=281.9347916666667): answer\_axis (the "arrow(r)\_0" part) is no longer emphasized. - [04:41.935](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=281.9347916666667): answer\_axis (the "lambda thin arrow(omega)" part) is emphasized. - [04:46.661](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=286.6607916666667): rest\_point is redrawn as the numbers it depends on change. - [04:46.661](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=286.6607916666667): axis\_velocity is redrawn as the numbers it depends on change. - [04:46.661](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=286.6607916666667): axis\_lambda ticks to 1.55. - [04:52.083](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=292.08279166666665): axis is shown on the screen, drawn. - [04:52.802](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=292.8022916666667): answer\_axis (the "lambda thin arrow(omega)" part) is no longer emphasized. ##### [04:53.402](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=293.40229166666666) Narration: That tilted line is the instantaneous axis of rotation. Every green location on it has zero velocity at this instant. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); question — a Text \[text\] that says "In three dimensions, which points of the body are momentarily at rest?"; body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); answer\_axis — a Math \[text\] that says "$arrow(r) = arrow(r)\_0 + lambda thin arrow(omega)$"; pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); arrow\_a — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); cross\_a — a Vector \[magenta\] labelled "arrow(omega) times arrow(r)\_0" drawn in frame (start=(1.2, -0.8, 0.2), end=(0.91974, -1.29248, 0.5159)); r0\_vector — a Vector \[yellow\] labelled "arrow(r)\_0" drawn in frame (start=(0.29999999999999993, -0.08000000000000007, 0.524), end=(1.2, -0.8, 0.2)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); rest\_point — a Point \[green\] labelled "-0.8" drawn in frame (location=((0.29999999999999993 + (axis\_lambda \* 0.6)), (-0.0800000000000…) Actions: - [04:54.424](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=294.42379166666666): axis is indicated — a transient flash. - [04:59.37](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=299.36979166666663): rest\_law is shown on the screen, written out. - [04:59.37](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=299.36979166666663): rest\_law (the "0" part) is emphasized. - [05:1.39](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=301.39029166666666): rest\_law (the "0" part) is no longer emphasized. ##### [05:1.99](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=301.9902916666667) Narration: A point off the axis swings around it in a circle lying square to omega. Its speed is omega times its perpendicular distance from the axis. The same construction at B uses the same omega. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); question — a Text \[text\] that says "In three dimensions, which points of the body are momentarily at rest?"; body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); answer\_axis — a Math \[text\] that says "$arrow(r) = arrow(r)\_0 + lambda thin arrow(omega)$"; rest\_law — a Math \[text\] that says "$arrow(v)\_(C(lambda)) = 0$"; pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); arrow\_a — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); cross\_a — a Vector \[magenta\] labelled "arrow(omega) times arrow(r)\_0" drawn in frame (start=(1.2, -0.8, 0.2), end=(0.91974, -1.29248, 0.5159)); r0\_vector — a Vector \[yellow\] labelled "arrow(r)\_0" drawn in frame (start=(0.29999999999999993, -0.08000000000000007, 0.524), end=(1.2, -0.8, 0.2)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); rest\_point — a Point \[green\] labelled "-0.8" drawn in frame (location=((0.29999999999999993 + (axis\_lambda \* 0.6)), (-0.0800000000000…) Actions: - [05:4.672](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=304.67179166666665): circle\_a is shown on the screen, drawn. - [05:7.505](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=307.5047916666667): arrow\_a is indicated — a transient flash. - [05:10.269](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=310.2687916666667): spoke\_a is shown on the screen, written out. - [05:12.393](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=312.39279166666665): arrow\_b is shown on the screen, written out. - [05:13.403](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=313.40279166666664): pt\_b is shown on the screen, written out. ##### [05:16.011](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=316.0112916666667) Narration: Turn the view and the geometry separates cleanly: the red axis is one tilted line in space, and both green velocities stand square to it. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); question — a Text \[text\] that says "In three dimensions, which points of the body are momentarily at rest?"; body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); answer\_axis — a Math \[text\] that says "$arrow(r) = arrow(r)\_0 + lambda thin arrow(omega)$"; rest\_law — a Math \[text\] that says "$arrow(v)\_(C(lambda)) = 0$"; pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); arrow\_a — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); cross\_a — a Vector \[magenta\] labelled "arrow(omega) times arrow(r)\_0" drawn in frame (start=(1.2, -0.8, 0.2), end=(0.91974, -1.29248, 0.5159)); r0\_vector — a Vector \[yellow\] labelled "arrow(r)\_0" drawn in frame (start=(0.29999999999999993, -0.08000000000000007, 0.524), end=(1.2, -0.8, 0.2)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); rest\_point — a Point \[green\] labelled "-0.8" drawn in frame (location=((0.29999999999999993 + (axis\_lambda \* 0.6)), (-0.0800000000000…); spoke\_a — a Line \[gray\] drawn in frame (start=(0.29999999999999993, -0.08000000000000007, 0.524), end=(1.2, -0.8, 0.2), dashed=True); circle\_a — a Circle \[gray\] drawn in frame (center=(0.29999999999999993, -0.08000000000000007, 0.524), radius=1.197236818678744, normal\_vector=(0.6, 0.3, 1.0)); pt\_b — a Point \[text\] labelled "B" drawn in frame (location=(1.68, -0.56, 1.0)); arrow\_b — a Vector \[green\] labelled "arrow(v)\_B" drawn in frame (start=(1.68, -0.56, 1.0), end=(1.96026, -0.06752000000000002, 0.6840999999999999)) Actions: - [05:16.267](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=316.2667916666667): frame turns in its own slot. ##### [05:26.329](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=326.3287916666667) Narration: Omega belongs to the whole body. A and B do not get different angular velocities. What changes from point to point is r, and therefore the cross-product contribution. Board: Unchanged from the preceding beat in this scene. Actions: - [05:29.487](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=329.48679166666665): pt\_a is indicated — a transient flash. - [05:29.905](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=329.90479166666665): pt\_b is indicated — a transient flash. - [05:33.817](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=333.81679166666663): work\_axis (the "arrow(v)\_A" part) is emphasized. - [05:36.209](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=336.2087916666667): work\_axis (the "arrow(omega) times arrow(r)" part) is emphasized. - [05:36.209](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=336.2087916666667): work\_axis (the "arrow(v)\_A" part) is no longer emphasized. - [05:38.194](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=338.1942916666667): frame moves to a new place on the board. - [05:38.194](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=338.1942916666667): rest\_point is hidden from the screen. - [05:38.194](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=338.1942916666667): answer\_axis is hidden from the screen — left the board. - [05:38.194](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=338.1942916666667): question is hidden from the screen — left the board. - [05:38.194](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=338.1942916666667): rest\_law is hidden from the screen — left the board. - [05:38.194](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=338.1942916666667): work\_axis is hidden from the screen — left the board. - [05:38.194](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=338.1942916666667): work\_axis (the "arrow(omega) times arrow(r)" part) is no longer emphasized. ##### [05:39.394](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=339.3942916666667) Narration: Now remove the assumption we just used. If v A is not square to omega, split it into a perpendicular piece and a parallel piece. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); arrow\_a — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); cross\_a — a Vector \[magenta\] labelled "arrow(omega) times arrow(r)\_0" drawn in frame (start=(1.2, -0.8, 0.2), end=(0.91974, -1.29248, 0.5159)); r0\_vector — a Vector \[yellow\] labelled "arrow(r)\_0" drawn in frame (start=(0.29999999999999993, -0.08000000000000007, 0.524), end=(1.2, -0.8, 0.2)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); spoke\_a — a Line \[gray\] drawn in frame (start=(0.29999999999999993, -0.08000000000000007, 0.524), end=(1.2, -0.8, 0.2), dashed=True); circle\_a — a Circle \[gray\] drawn in frame (center=(0.29999999999999993, -0.08000000000000007, 0.524), radius=1.197236818678744, normal\_vector=(0.6, 0.3, 1.0)); pt\_b — a Point \[text\] labelled "B" drawn in frame (location=(1.68, -0.56, 1.0)); arrow\_b — a Vector \[green\] labelled "arrow(v)\_B" drawn in frame (start=(1.68, -0.56, 1.0), end=(1.96026, -0.06752000000000002, 0.6840999999999999)) Actions: - [05:39.394](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=339.3942916666667): heading\_screw is shown on the screen, written out. - [05:39.394](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=339.3942916666667): arrow\_a is hidden from the screen. - [05:39.394](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=339.3942916666667): cross\_a is hidden from the screen. - [05:39.394](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=339.3942916666667): r0\_vector is hidden from the screen. - [05:39.394](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=339.3942916666667): circle\_a is hidden from the screen. - [05:39.394](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=339.3942916666667): spoke\_a is hidden from the screen. - [05:39.394](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=339.3942916666667): arrow\_b is hidden from the screen. - [05:39.394](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=339.3942916666667): pt\_b is hidden from the screen. - [05:45.002](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=345.00179166666663): work\_screw is shown on the screen, written out. - [05:45.838](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=345.83779166666665): arrow\_perp is shown on the screen, written out. - [05:47.115](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=347.11479166666663): arrow\_parallel is shown on the screen, written out. ##### [05:48.98](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=348.9802916666667) Narration: Let those two pieces land. Complete their parallelogram, and its green diagonal is the original velocity v A. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); heading\_screw — a Heading that says "When Nothing Is at Rest"; arrow\_perp — a Vector \[blue\] labelled "arrow(v)\_(bot)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); arrow\_parallel — a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157)) Actions: - [05:51.175](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=351.17479166666664): sum\_side\_perp is shown on the screen, drawn. - [05:51.825](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=351.82479166666667): sum\_side\_parallel is shown on the screen, drawn. - [05:53.578](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=353.57779166666666): arrow\_total is shown on the screen, written out. ##### [05:57.08](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=357.08029166666665) Narration: The cross product can cancel the perpendicular piece, exactly as before. It can never touch the parallel piece, whose projection formula is this. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); heading\_screw — a Heading that says "When Nothing Is at Rest"; arrow\_perp — a Vector \[blue\] labelled "arrow(v)\_(bot)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); arrow\_parallel — a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157)); sum\_side\_perp — a Line \[gray\] drawn in frame (start=(1.48026, -0.30752, -0.1159), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True); sum\_side\_parallel — a Line \[gray\] drawn in frame (start=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True); arrow\_total — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158)) Actions: - [05:59.078](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=359.07779166666666): work\_screw (the "arrow(v)\_(bot)" part) is emphasized. - [06:3.025](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=363.02479166666666): work\_screw (the "arrow(v)\_(bot)" part) is no longer emphasized. - [06:3.025](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=363.02479166666666): work\_screw (the "arrow(v)\_(parallel)" part) is emphasized. - [06:4.268](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=364.26779166666665): work\_screw is shown on the screen, written out. - [06:6.09](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=366.09029166666664): work\_screw (the "arrow(v)\_(parallel)" part) is no longer emphasized. ##### [06:6.69](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=366.69029166666667) Narration: So the axis still exists, but its points now slide along omega instead of resting. This yellow arrow is the velocity shared by every point on it. Board: Unchanged from the preceding beat in this scene. Actions: - [06:9.964](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=369.9637916666666): axis\_velocity is shown on the screen, written out. - [06:13.401](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=373.40079166666663): work\_screw is shown on the screen, written out. ##### [06:17.391](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=377.3912916666667) Narration: Take a point away from the axis. Its velocity is the sum of a turn around the axis and that same slide along it. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); heading\_screw — a Heading that says "When Nothing Is at Rest"; arrow\_perp — a Vector \[blue\] labelled "arrow(v)\_(bot)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); arrow\_parallel — a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157)); sum\_side\_perp — a Line \[gray\] drawn in frame (start=(1.48026, -0.30752, -0.1159), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True); sum\_side\_parallel — a Line \[gray\] drawn in frame (start=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True); arrow\_total — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158)); axis\_velocity — a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=((0.29999999999999993 + (axis\_lambda \* 0.6)), (-0.0800000000000…, end=(((0.29999999999999993 + (axis\_lambda \* 0.6)) + 0.4185492184628…) Actions: - [06:21.141](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=381.14079166666664): work\_screw (the "arrow(omega) times arrow(r)" part) is emphasized. - [06:22.906](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=382.90579166666663): work\_screw (the "arrow(omega) times arrow(r)" part) is no longer emphasized. - [06:22.906](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=382.90579166666663): work\_screw (the "arrow(v)\_(parallel)" part) is emphasized. - [06:24.357](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=384.3572916666667): work\_screw (the "arrow(v)\_(parallel)" part) is no longer emphasized. ##### [06:24.957](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=384.95729166666666) Narration: Follow one material point. The combined motion traces this yellow helix, winding around the tilted axis while advancing along it. Board: Unchanged from the preceding beat in this scene. Actions: - [06:26.258](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=386.25779166666666): helix\_rider is shown on the screen, written out. - [06:29.648](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=389.64779166666665): helix is shown on the screen, drawn. ##### [06:34.207](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=394.20729166666666) Narration: This is Chasles' theorem. At every instant, rigid-body motion is a screw: a rotation about an axis together with a translation along that axis. Watch the numbered point perform both parts. The slide per unit turn is the pitch. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); heading\_screw — a Heading that says "When Nothing Is at Rest"; arrow\_perp — a Vector \[blue\] labelled "arrow(v)\_(bot)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); arrow\_parallel — a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157)); sum\_side\_perp — a Line \[gray\] drawn in frame (start=(1.48026, -0.30752, -0.1159), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True); sum\_side\_parallel — a Line \[gray\] drawn in frame (start=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True); arrow\_total — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158)); axis\_velocity — a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=((0.29999999999999993 + (axis\_lambda \* 0.6)), (-0.0800000000000…, end=(((0.29999999999999993 + (axis\_lambda \* 0.6)) + 0.4185492184628…); helix — a ParametricCurve \[yellow\] labelled "upright("screw path")" drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); helix\_rider — a Point \[green\] labelled "0.0" drawn in frame (location=(((0.29999999999999993 + ((0.2 + (0.28 \* helix\_t)) \* 0.49827287…) Actions: - [06:39.629](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=399.62879166666664): helix\_rider is redrawn as the numbers it depends on change. - [06:39.629](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=399.62879166666664): helix\_t ticks to 6.1. - [06:48.592](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=408.5917916666666): pitch is shown on the screen, written out. - [06:49.405](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=409.40479166666665): A box is drawn around pitch. ##### [06:50.005](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=410.0047916666667) Narration: First limiting case: let the slide shrink to nothing. The advancing helix closes into a circle, the axis velocity vanishes, and the motion becomes pure rotation. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); pitch — a Math \[text\] that says "$h = frac(v\_(parallel), omega)$"; heading\_screw — a Heading that says "When Nothing Is at Rest"; arrow\_perp — a Vector \[blue\] labelled "arrow(v)\_(bot)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.48026, -0.30752, -0.1159)); arrow\_parallel — a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157)); sum\_side\_perp — a Line \[gray\] drawn in frame (start=(1.48026, -0.30752, -0.1159), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True); sum\_side\_parallel — a Line \[gray\] drawn in frame (start=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158), dashed=True); arrow\_total — a Vector \[green\] labelled "arrow(v)\_A" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.8988092184628493, -0.09824539076857527, 0.5816820307714158)); axis\_velocity — a Vector \[yellow\] labelled "arrow(v)\_(parallel)" drawn in frame (start=((0.29999999999999993 + (axis\_lambda \* 0.6)), (-0.0800000000000…, end=(((0.29999999999999993 + (axis\_lambda \* 0.6)) + 0.4185492184628…); helix — a ParametricCurve \[yellow\] labelled "upright("screw path")" drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); helix\_rider — a Point \[green\] labelled "0.0" drawn in frame (location=(((0.29999999999999993 + ((0.2 + (0.28 \* helix\_t)) \* 0.49827287…) Actions: - [06:52.385](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=412.3847916666666): arrow\_perp is hidden from the screen. - [06:52.385](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=412.3847916666666): arrow\_parallel is hidden from the screen. - [06:52.385](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=412.3847916666666): sum\_side\_perp is hidden from the screen. - [06:52.385](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=412.3847916666666): sum\_side\_parallel is hidden from the screen. - [06:52.385](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=412.3847916666666): arrow\_total is hidden from the screen. - [06:54.765](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=414.7647916666666): helix is hidden from the screen. - [06:54.765](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=414.7647916666666): helix\_rider is hidden from the screen. - [06:55.485](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=415.48479166666664): turn\_only is shown on the screen, drawn. - [06:57.401](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=417.40079166666663): axis\_velocity is hidden from the screen. ##### [07:1.309](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=421.3092916666667) Narration: Second limiting case: restore the slide and let omega shrink to nothing. The winding opens into a straight path, the red omega arrow disappears, and equal green velocities show pure translation. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); pitch — a Math \[text\] that says "$h = frac(v\_(parallel), omega)$"; heading\_screw — a Heading that says "When Nothing Is at Rest"; turn\_only — a Circle \[yellow\] drawn in frame (center=(0.8379582924540336, 0.18897914622701678, 1.4205971540900562), radius=0.72, normal\_vector=(0.6, 0.3, 1.0)) Actions: - [07:7.695](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=427.6947916666666): turn\_only is hidden from the screen. - [07:8.473](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=428.47279166666664): slide\_only is shown on the screen, drawn. - [07:10.783](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=430.78279166666664): omega\_vec is hidden from the screen. - [07:10.783](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=430.78279166666664): axis is hidden from the screen. - [07:12.432](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=432.43179166666664): translation\_a is shown on the screen, written out. - [07:12.71](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=432.70979166666666): translation\_b is shown on the screen, written out. ##### [07:16.325](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=436.32479166666667) Narration: Put turn and slide back together. The tilted axis, the helix and the travelling point return as the one picture that contains the general case. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); pitch — a Math \[text\] that says "$h = frac(v\_(parallel), omega)$"; heading\_screw — a Heading that says "When Nothing Is at Rest"; slide\_only — a Line \[yellow\] drawn in frame (start=(0.3996545758244879, -0.03017271208775609, 0.69009095970748), end=(1.2762620090835795, 0.4081310045417897, 2.1511033484726325)); translation\_a — a Vector \[green\] labelled "arrow(v)" drawn in frame (start=(1.2, -0.8, 0.2), end=(1.6185492184628494, -0.5907253907685753, 0.8975820307714157)); translation\_b — a Vector \[green\] labelled "arrow(v)" drawn in frame (start=(1.68, -0.56, 1.0), end=(2.0985492184628494, -0.35072539076857534, 1.6975820307714158)) Actions: - [07:16.883](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=436.88279166666666): omega\_vec is shown on the screen, written out. - [07:17.846](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=437.8457916666666): helix\_t ticks to 0.2. - [07:18.102](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=438.10179166666666): slide\_only is hidden from the screen. - [07:18.102](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=438.10179166666666): translation\_a is hidden from the screen. - [07:18.102](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=438.10179166666666): translation\_b is hidden from the screen. - [07:19.948](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=439.9477916666666): axis is shown on the screen, drawn. - [07:20.888](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=440.88779166666666): helix is shown on the screen, drawn. - [07:22.293](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=442.29279166666663): helix\_rider is shown on the screen, written out. - [07:26.229](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=446.22879166666667): heading\_screw is hidden from the screen — left the board. - [07:26.229](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=446.22879166666667): pitch is hidden from the screen — left the board. - [07:26.229](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=446.22879166666667): work\_screw is hidden from the screen — left the board. ##### [07:27.429](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=447.42879166666665) Narration: Three ideas carry the lecture. Keep the moving screw beside them while we read the list. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); helix — a ParametricCurve \[yellow\] labelled "upright("screw path")" drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); helix\_rider — a Point \[green\] labelled "0.0" drawn in frame (location=(((0.29999999999999993 + ((0.2 + (0.28 \* helix\_t)) \* 0.49827287…) Actions: - [07:27.429](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=447.42879166666665): heading\_recap is shown on the screen, written out. - [07:27.777](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=447.77679166666667): recap is shown on the screen, written out. - [07:30.25](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=450.2497916666666): helix\_rider is redrawn as the numbers it depends on change. - [07:30.25](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=450.2497916666666): helix\_t ticks to 5.9. ##### [07:33.683](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=453.6827916666666) Narration: First, any two points share one omega, and their velocities differ by omega cross the vector between them. Board: frame — an Axes3D (x\_range=(-2.4, 2.4), y\_range=(-2.4, 2.4), z\_range=(-1.2, 3.6)); body — a Cylinder \[blue\] drawn in frame (start=(1.77, -1.72, 1.99), end=(1.11, 0.36, -0.79), radius=0.6); pt\_a — a Point \[text\] labelled "A" drawn in frame (location=(1.2, -0.8, 0.2)); omega\_vec — a Vector \[red\] labelled "arrow(omega)" drawn in frame (start=(-1.9, 0.98, 2.01), end=(-1.1775043252724622, 1.3412478373637688, 3.214159457879229)); axis — a Line \[red\] labelled "upright("instantaneous axis")" drawn in frame (start=(-0.45000000000000007, -0.45500000000000007, -0.726), end=(1.71, 0.6249999999999999, 2.874), dashed=True); helix — a ParametricCurve \[yellow\] labelled "upright("screw path")" drawn in frame (function=\, t\_range=(0.0, 6.283185307179586)); helix\_rider — a Point \[green\] labelled "0.0" drawn in frame (location=(((0.29999999999999993 + ((0.2 + (0.28 \* helix\_t)) \* 0.49827287…); recap — a Block \[text\] that says "Two points share one $arrow(omega)$: $arrow(v)\_B = arrow(v)\_A + arrow(omega) times arrow(r)$. Plane motion has a moving instantaneous centre. Spatial motion has an instantaneous axis; turn plus slide gives a screw."; heading\_recap — a Heading that says "What to Carry Away" Actions: - [07:34.031](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=454.03079166666663): recap (the "Two points" part) is emphasized. - [07:35.994](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=455.99379166666665): recap (the "Two points" part) is no longer emphasized. - [07:35.994](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=455.99379166666665): recap (the "one $arrow(omega)$" part) is emphasized. ##### [07:41.644](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=461.64379166666663) Narration: Second, plane motion has an instantaneous centre, but the material point occupying it changes as the body moves. Board: Unchanged from the preceding beat in this scene. Actions: - [07:42.12](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=462.11979166666663): recap (the "Plane motion" part) is emphasized. - [07:42.12](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=462.11979166666663): recap (the "one $arrow(omega)$" part) is no longer emphasized. ##### [07:50.057](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=470.05729166666663) Narration: Third, spatial motion has an instantaneous axis. Rotation about it plus translation along it is the general screw motion, every instant. Board: Unchanged from the preceding beat in this scene. Actions: - [07:50.406](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=470.40579166666663): recap (the "Plane motion" part) is no longer emphasized. - [07:50.406](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=470.40579166666663): recap (the "Spatial motion" part) is emphasized. - [07:54.678](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=474.6777916666666): recap (the "Spatial motion" part) is no longer emphasized. - [07:54.678](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=474.6777916666666): recap (the "turn plus slide" part) is emphasized. - [08:0.089](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.0886458333333): recap (the "turn plus slide" part) is no longer emphasized. - [08:0.339](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.3386458333333): frame is hidden from the screen — left the board. - [08:0.339](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.3386458333333): body is hidden from the screen — frame left the board. - [08:0.339](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.3386458333333): pt\_a is hidden from the screen — frame left the board. - [08:0.339](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.3386458333333): omega\_vec is hidden from the screen — frame left the board. - [08:0.339](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.3386458333333): axis is hidden from the screen — frame left the board. - [08:0.339](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.3386458333333): helix is hidden from the screen — frame left the board. - [08:0.339](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.3386458333333): helix\_rider is hidden from the screen — frame left the board. - [08:0.339](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.3386458333333): heading\_recap is hidden from the screen — left the board. - [08:0.339](https://academa.ai/lectures/instantaneous-axis-of-rotation?t=480.3386458333333): recap is hidden from the screen — left the board.