{"version":1,"lectureId":"01M19SKZ8RSJZMMJPPNCETT7MD","attempt":3,"publication":{"slug":"kinematics-in-rotating-axes","title":"Rotating Axes: Coriolis and Centrifugal","subject":"engineering","summary":"A body moves, two observers watch it, and one of them is spinning. This lecture builds the kinematics of rotating axes out of a single fact, that a basis vector painted on a turning body has derivative omega crossed with itself, and then applies that fact twice. Out come the velocity relation, the five-term acceleration relation, and with them the Euler, centripetal and Coriolis terms, each named and lit as it is spoken. We finish by carrying the extra terms across the equals sign, where they become the centrifugal and Coriolis forces a rider on the turntable genuinely feels, and by watching one launched ball fly dead straight over the ground while curving the whole way out on the disc.","metaDescription":"Rotating reference frames from scratch: the transport theorem, velocity and acceleration terms, and where Coriolis and centrifugal come from.","transcript":"Two observers watch the same particle. One uses fixed axes bolted to the floor. The other stands on a turntable with its own painted x and y axes. Their acceleration measurements do not agree. The difference is physical, and its two famous names are centrifugal and Coriolis. Both begin with the fact that the second observer's axes are spinning. A reference frame means axes together with a clock. Capital X and capital Y belong to the fixed frame, with origin O, and nothing in that frame turns. The rotating frame has origin B. Its little x and little y directions are painted on the surface, so they turn with the disc at angular velocity omega. Now let P wander outward while the turntable keeps turning. Its position from O is r sub P. The moving origin sits at r sub B. The displacement from B to P, measured in the painted coordinates, is rho. Geometry gives r P equals r B plus rho. These are the same three arrows beside the equation: first r P, then r B, then rho, joined head to tail. Differentiate that identity in the fixed frame. Rho is written in the painted basis, and the basis turns while its components change. The derivative therefore acts on both the numbers and the axes. The task is now precise: differentiate this moving triangle honestly, twice. The extra acceleration terms will come from the turning basis. Everything hard about rotating axes begins with one question: how fast does a painted axis change as its disc turns? The unit vector i hat has length one, so its tip stays on the circle. Its turned-through angle is theta. Only after that geometry is fixed do its components read cosine theta, sine theta, zero. Differentiate those components. The chain rule gives theta dot times the bracket minus sine theta, cosine theta, zero. That bracket is j hat, the second painted axis. It is exactly ninety degrees ahead of i hat on the same disc. Theta dot is the spin rate omega, so the derivative of i hat is omega times j hat. In vector form, omega points along the rotation axis. Omega crossed with i hat therefore points along j hat, which gives the same derivative. As the disc turns, i hat never changes length. Its derivative remains sideways, at right angles to i hat, at every instant. That perpendicular sweep is the geometry of the cross product. The same reasoning holds for any vector rigidly attached to the body: a painted axis, a spoke, or a robot arm. Its derivative is omega crossed with that vector. Now take an arbitrary vector A and write it in the painted basis as A x i hat, A y j hat, plus A z k hat. The product rule separates its fixed-frame derivative into two families. One differentiates the components and leaves the basis vectors alone. A rider on the disc calls that first family the relative derivative, because those are the component changes measured against the painted axes. The second family leaves the components alone and differentiates i hat, j hat, and k hat. Each basis derivative contributes omega crossed with the corresponding basis vector. Factor omega cross from that whole family. The remaining bracket is the original vector A, so the family becomes omega cross A. This is the transport theorem. A vector's fixed-frame rate equals its relative rate plus omega cross the vector. Applying this statement twice produces every acceleration term that follows. One line of geometry and one use of the transport theorem give the velocity relation. Begin with r P equals r B plus rho. Differentiate both sides in the fixed frame. The left side becomes the true velocity v P, and the first term on the right becomes the velocity v B of the moving origin. Rho needs the theorem. The red rho vector lies in the disc. The blue omega vector points along the rotation axis. Their yellow cross product is the extra velocity carried by the turning frame. Omega cross rho is at right angles to rho and omega. It lies flat in the plane of the disc, ninety degrees ahead of rho, with magnitude omega times rho. The complete velocity therefore has three pieces: motion of the origin, motion measured on the disc, and motion carried by the disc's spin. A beetle walks outward along a painted spoke while the turntable spins. The three velocity pieces form one head-to-tail chain. First is v B, the velocity of the frame's origin. A bolted turntable has v B equal to zero, while an aircraft or planet need not. Second is omega cross rho. It is tangential and square to the spoke, the velocity the beetle receives simply because the disc carries it. Third is v rel, directed outward along the spoke. Translate that vector to the tip of omega cross rho, and the two arrows now meet head to tail. The arrow from the chain's first tail to its final tip is v P, the beetle's true velocity over the ground. One more derivative produces the acceleration terms. Differentiate the velocity relation once more. Each vector on the right is measured against the turning disc, so the transport theorem remains beside the calculation. The origin contributes a B directly. The other two derivatives need the theorem because omega cross rho and v rel are both vectors written in the painted basis. Apply the product rule to omega cross rho. Omega dot becomes alpha, so the first contribution is alpha cross rho. It is tangential while the spin rate is changing. The second contribution contains the fixed derivative of rho. Substitute v rel plus omega cross rho, then distribute omega cross across both terms. The fixed derivative of v rel also uses the theorem. It contributes a rel plus a second omega cross v rel. The distributed line contains one omega cross v rel. The final derivative contains another omega cross v rel. These two identical terms add. Their sum is two omega cross v rel. The factor of two is the Coriolis coefficient, produced by two separate derivatives. Collecting the result gives five acceleration terms, all expressed in quantities available to the rotating observer. First, a B is the acceleration of the frame's origin. A floor-mounted turntable makes it zero, but a vehicle or planet need not. Second, alpha cross rho is the Euler term. It appears while the spin rate changes and vanishes at steady omega. Third, omega cross omega cross rho is centripetal. It points from P toward the rotation axis, with magnitude omega squared times the distance out. Fourth, two omega cross v rel is Coriolis. It needs a turning frame and motion relative to that frame at the same instant. Fifth, a rel is the acceleration measured by instruments fixed to the disc. It is often the unknown the rotating observer must determine. The Coriolis condition is now exact. Relative motion without rotation gives zero. Rotation without relative motion also gives zero. Only the final case has both, so only that case carries a Coriolis acceleration. Newton's second law is written in the fixed frame, but the rider reports a rel. Solving for that measured acceleration begins with the applied force F while the disc continues to turn. Subtract the origin term, the Euler term, the centripetal term, and the Coriolis term from F. The rule leaves m times a rel beneath it. Every subtracted quantity has units of force and occupies a force slot in the rider's equation. Their minus signs record the acceleration terms moved from the fixed-frame side. Bolt the turntable to the floor and hold a steady omega. Then a B is zero and alpha is zero, so those two terms disappear. The centripetal and Coriolis terms survive. The centripetal term changes sign when it crosses the equality. It becomes plus m omega squared rho, directed outward from the axis and increasing with distance from the centre. This is the centrifugal term. Centrifugal force is not an additional interaction. It is the inward centripetal acceleration moved across the equality and read from the rotating frame. The remaining correction is minus two m omega cross v rel, the Coriolis force. Its direction depends on how the body moves across the rotating disc. The painted launch spoke fixes one initial direction. A single ball leaves the centre with no sideways force, and the same flight is measured from the ground and from the disc. From the ground, the ball advances at constant speed along a straight inertial path. Its position stays on the launch direction while the blue current spoke turns underneath it. The gray launch spoke marks the disc at release. The blue current spoke is the same painted radius later in the flight, so their labels distinguish the two instants. From the disc, the painted spoke stays fixed. The identical ball traces a curved path because every later position is resolved against axes that have turned. The disc-frame path bends to the right of the ball's motion all the way outward. The Coriolis correction persists because the relative velocity persists. No sideways interaction acts on the ball. The curvature belongs to the rotating coordinates, which is why a rider assigns it a Coriolis force. The same correction matters in large weather systems, in flow across a pressure gradient, in long-range gunnery, and in the slow turn of a Foucault pendulum. The final relation has five terms, and each has a distinct physical question attached to it. The first asks how the frame origin moves. The second asks whether omega changes. The third asks how far out the point lies. Coriolis asks for relative motion, and the fifth is the rider's own acceleration. Every term follows from one fact about a painted basis vector: its derivative is omega crossed with itself. Apply that fact twice with the product rule, and the rotating-frame acceleration is complete.","watch":{"version":1,"scenes":[{"title":"Two Frames, One Motion","start":0,"end":93.28920833333333,"objects":{"axis_x_fixed":"an Arrow [gray] labelled \"X\" drawn in plane (start=(0.9, 0.8), end=(7.6, 0.8))","axis_x_rot":"an Arrow [blue] labelled \"x\" drawn in plane (start=(4.3, 3.3), end=((4.3 + (1.75 * cos(disc_angle))), (3.3 + (1.75 * sin(disc_angl…)","axis_y_fixed":"an Arrow [gray] labelled \"Y\" drawn in plane (start=(0.9, 0.8), end=(0.9, 6.4))","axis_y_rot":"an Arrow [blue] labelled \"y\" drawn in plane (start=(4.3, 3.3), end=((4.3 + (1.75 * cos((disc_angle + 1.5707963267948966)))), (3.3 …)","disc":"a Circle [gray] drawn in plane (center=(4.3, 3.3), radius=1.75)","disc_angle":"a VariableNumber (initial_value=0.42, format_spec='.2f')","frame_note":"a Panel that says \"Axes and a clock. Frames sharing one clock can still disagree about velocity and acceleration if one of them turns.\"","heading":"a Heading that says \"Two Frames, One Motion\"","origin_b":"a Point [text] labelled \"B\" drawn in plane (location=(4.3, 3.3))","origin_o":"a Point [text] labelled \"O\" drawn in plane (location=(0.9, 0.8))","particle":"a Point [magenta] labelled \"P\" drawn in plane (location=((4.3 + (particle_radius * cos(disc_angle))), (3.3 + (particle_…)","particle_radius":"a VariableNumber (initial_value=1.05, format_spec='.2f')","plane":"a Figure (x_range=(0.0, 8.2), y_range=(0.0, 7.0), aspect=(8.2, 7.0))","position_work":"a Derivation [text] that says \"$arrow(r)_P = arrow(r)_B + arrow(rho) \\ arrow(v)_P = arrow(v)_B + (frac(dif arrow(rho), dif t))_upright(\"fix\")$\"","r_b":"an Arrow [gray] labelled \"arrow(r)_B\" drawn in plane (start=(0.9, 0.8), end=(4.3, 3.3))","r_p":"an Arrow [magenta] labelled \"arrow(r)_P\" drawn in plane (start=(0.9, 0.8), end=((4.3 + (particle_radius * cos(disc_angle))), (3.3 + (particle_…)","rho":"an Arrow [red] labelled \"arrow(rho)\" drawn in plane (start=(4.3, 3.3), end=((4.3 + (particle_radius * cos(disc_angle))), (3.3 + (particle_…)","spin":"a CurvedArrow [blue] labelled \"arrow(omega)\" drawn in plane (start=(3.1, 4.83), end=(2.55, 3.64), bend=0.42)"},"beats":[{"start":0,"say":"Two observers watch the same particle. One uses fixed axes bolted to the floor. The other stands on a turntable with its own painted x and y axes.","live":[],"does":[[0,"heading is shown on the screen, written out."],[1.36,"plane is shown on the screen, written out."],[1.6800000000000002,"particle is shown on the screen, written out."],[3.3600000000000003,"axis_x_fixed is shown on the screen, written out."],[3.68,"axis_y_fixed is shown on the screen, written out."],[4.800000000000001,"origin_o is shown on the screen, written out."],[6.720000000000001,"disc is shown on the screen, written out."],[6.720000000000001,"origin_b is shown on the screen, written out."],[8.4,"axis_x_rot is shown on the screen, written out."],[8.8,"axis_y_rot is shown on the screen, written out."]]},{"start":10.395,"say":"Their acceleration measurements do not agree. The difference is physical, and its two famous names are centrifugal and Coriolis. Both begin with the fact that the second observer's axes are spinning.","live":["plane","heading","particle","axis_x_fixed","axis_y_fixed","origin_o","disc","origin_b","axis_x_rot","axis_y_rot"],"does":[[21.64,"particle is redrawn as the numbers it depends on change."],[21.64,"axis_x_rot is redrawn as the numbers it depends on change."],[21.64,"axis_y_rot is redrawn as the numbers it depends on change."],[21.64,"r_p is redrawn as the numbers it depends on change."],[21.64,"rho is redrawn as the numbers it depends on change."],[21.64,"spin is shown on the screen, written out."],[21.64,"disc_angle ticks to 0.78."]]},{"start":22.994999999999997,"say":"A reference frame means axes together with a clock. 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Its little x and little y directions are painted on the surface, so they turn with the disc at angular velocity omega.","live":["frame_note","plane","heading","particle","axis_x_fixed","axis_y_fixed","origin_o","disc","origin_b","axis_x_rot","axis_y_rot","spin"],"does":[[35.08,"origin_b is emphasized."],[36.6,"origin_b is no longer emphasized."],[36.6,"axis_x_rot is emphasized."],[37.4,"axis_y_rot is emphasized."],[40.39999999999999,"particle is redrawn as the numbers it depends on change."],[40.39999999999999,"axis_x_rot is redrawn as the numbers it depends on change."],[40.39999999999999,"axis_y_rot is redrawn as the numbers it depends on change."],[40.39999999999999,"r_p is redrawn as the numbers it depends on change."],[40.39999999999999,"rho is redrawn as the numbers it depends on change."],[40.39999999999999,"disc_angle ticks to 1.34."],[42.44,"spin is emphasized."]]},{"start":43.845,"say":"Now let P wander outward while the turntable keeps turning. Its position from O is r sub P. The moving origin sits at r sub B. The displacement from B to P, measured in the painted coordinates, is rho.","live":null,"does":[[43.845,"axis_x_rot is no longer emphasized."],[43.845,"axis_y_rot is no longer emphasized."],[43.845,"spin is no longer emphasized."],[45.12,"particle is redrawn as the numbers it depends on change."],[45.12,"axis_x_rot is redrawn as the numbers it depends on change."],[45.12,"axis_y_rot is redrawn as the numbers it depends on change."],[45.12,"particle_radius ticks to 1.62."],[47.12,"disc_angle ticks to 1.92."],[49.44,"r_p is redrawn as the numbers it depends on change."],[49.44,"r_p is shown on the screen, written out."],[52.32,"r_b is shown on the screen, written out."],[57.35999999999999,"rho is redrawn as the numbers it depends on change."],[57.35999999999999,"rho is shown on the screen, written out."]]},{"start":58.5,"say":"Geometry gives r P equals r B plus rho. These are the same three arrows beside the equation: first r P, then r B, then rho, joined head to tail.","live":["frame_note","plane","heading","particle","axis_x_fixed","axis_y_fixed","origin_o","disc","origin_b","axis_x_rot","axis_y_rot","spin","r_p","r_b","rho"],"does":[[58.76,"position_work is shown on the screen, written out."],[59.79999999999999,"r_p is emphasized."],[59.79999999999999,"position_work (the \"arrow(r)_P\" part) is emphasized."],[60.83999999999999,"r_p is no longer emphasized."],[60.83999999999999,"r_b is emphasized."],[60.83999999999999,"position_work (the \"arrow(r)_B\" part) is emphasized."],[60.83999999999999,"position_work (the \"arrow(r)_P\" part) is no longer emphasized."],[61.71999999999999,"r_b is no longer emphasized."],[61.71999999999999,"rho is emphasized."],[61.71999999999999,"position_work (the \"arrow(r)_B\" part) is no longer emphasized."],[61.71999999999999,"position_work (the \"arrow(rho)\" part) is emphasized."],[68.75999999999999,"rho is no longer emphasized."],[68.75999999999999,"position_work (the \"arrow(rho)\" part) is no longer emphasized."]]},{"start":70.315,"say":"Differentiate that identity in the fixed frame. Rho is written in the painted basis, and the basis turns while its components change. 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Its turned-through angle is theta. Only after that geometry is fixed do its components read cosine theta, sine theta, zero.","live":["dial","heading_basis","rim","hub","ref_axis","i_hat"],"does":[[103.84920833333334,"dial moves to a new place on the board."],[103.84920833333334,"unit_law is shown on the screen, written out."],[108.20920833333334,"theta_mark is shown on the screen, written out."],[111.68920833333334,"work_basis is shown on the screen, written out."]]},{"start":115.52420833333333,"say":"Differentiate those components. The chain rule gives theta dot times the bracket minus sine theta, cosine theta, zero.","live":["unit_law","dial","heading_basis","rim","hub","ref_axis","i_hat","theta_mark"],"does":[[117.84920833333334,"work_basis is shown on the screen, written out."],[119.76920833333334,"work_basis (the \"chevron.l -sin theta, thin cos theta, thin 0 chevron.r\" part) is emphasized."]]},{"start":123.80920833333333,"say":"That bracket is j hat, the second painted axis. It is exactly ninety degrees ahead of i hat on the same disc.","live":null,"does":[[124.92920833333334,"j_hat is shown on the screen, written out."],[124.92920833333334,"work_basis (the \"chevron.l -sin theta, thin cos theta, thin 0 chevron.r\" part) is no longer emphasized."],[125.80920833333333,"j_hat is emphasized."]]},{"start":131.76420833333333,"say":"Theta dot is the spin rate omega, so the derivative of i hat is omega times j hat.","live":["unit_law","dial","heading_basis","rim","hub","ref_axis","i_hat","theta_mark","j_hat"],"does":[[131.76420833333333,"j_hat is no longer emphasized."],[133.44920833333333,"work_basis (the \"omega hat(j)\" part) is emphasized."],[134.56920833333334,"work_basis is shown on the screen, written out."]]},{"start":138.12920833333334,"say":"In vector form, omega points along the rotation axis. Omega crossed with i hat therefore points along j hat, which gives the same derivative.","live":null,"does":[[138.12920833333334,"work_basis (the \"omega hat(j)\" part) is no longer emphasized."],[141.9692083333333,"work_basis is shown on the screen, written out."],[146.20920833333332,"d_i_hat is shown on the screen, written out."]]},{"start":147.54670833333336,"say":"As the disc turns, i hat never changes length. Its derivative remains sideways, at right angles to i hat, at every instant. That perpendicular sweep is the geometry of the cross product.","live":["unit_law","dial","heading_basis","rim","hub","ref_axis","i_hat","theta_mark","j_hat","d_i_hat"],"does":[[148.60920833333336,"i_hat is redrawn as the numbers it depends on change."],[148.60920833333336,"theta_mark is redrawn as the numbers it depends on change."],[148.60920833333336,"j_hat is redrawn as the numbers it depends on change."],[148.60920833333336,"d_i_hat is redrawn as the numbers it depends on change."],[148.60920833333336,"theta ticks to 2.75."],[149.76920833333332,"unit_law (the \"1\" part) is emphasized."],[151.44920833333333,"unit_law (the \"1\" part) is no longer emphasized."],[153.28920833333333,"right_mark is redrawn as the numbers it depends on change."],[153.28920833333333,"right_mark is shown on the screen, written out."]]},{"start":160.06670833333334,"say":"The same reasoning holds for any vector rigidly attached to the body: a painted axis, a spoke, or a robot arm. Its derivative is omega crossed with that vector.","live":["unit_law","dial","heading_basis","rim","hub","ref_axis","i_hat","theta_mark","j_hat","d_i_hat","right_mark"],"does":[[161.76920833333332,"fact is shown on the screen, written out."],[168.20920833333332,"fact (the \"arrow(omega) times hat(e)\" part) is emphasized."],[170.15420833333334,"dial moves to a new place on the board."],[170.15420833333334,"fact moves to a new place on the board."],[170.15420833333334,"heading_basis is hidden from the screen — left the board."],[170.15420833333334,"unit_law is hidden from the screen — left the board."],[170.15420833333334,"work_basis is hidden from the screen — left the board."]]},{"start":170.75420833333334,"say":"Now take an arbitrary vector A and write it in the painted basis as A x i hat, A y j hat, plus A z k hat.","live":["fact","dial","rim","hub","ref_axis","i_hat","theta_mark","j_hat","d_i_hat","right_mark"],"does":[[170.75420833333334,"heading_thm is shown on the screen, written out."],[170.75420833333334,"work_thm is shown on the screen, written out."],[170.75420833333334,"fact (the \"arrow(omega) times hat(e)\" part) is no longer emphasized."],[173.92920833333335,"i_hat is redrawn as the numbers it depends on change."],[173.92920833333335,"theta_mark is redrawn as the numbers it depends on change."],[173.92920833333335,"j_hat is redrawn as the numbers it depends on change."],[173.92920833333335,"d_i_hat is redrawn as the numbers it depends on change."],[173.92920833333335,"right_mark is redrawn as the numbers it depends on change."],[173.92920833333335,"theta ticks to 3.25."]]},{"start":180.08170833333332,"say":"The product rule separates its fixed-frame derivative into two families. One differentiates the components and leaves the basis vectors alone.","live":["fact","dial","rim","hub","ref_axis","i_hat","theta_mark","j_hat","d_i_hat","right_mark","heading_thm"],"does":[[180.44920833333333,"i_hat is redrawn as the numbers it depends on change."],[180.44920833333333,"theta_mark is redrawn as the numbers it depends on change."],[180.44920833333333,"j_hat is redrawn as the numbers it depends on change."],[180.44920833333333,"d_i_hat is redrawn as the numbers it depends on change."],[180.44920833333333,"right_mark is redrawn as the numbers it depends on change."],[180.44920833333333,"work_thm is shown on the screen, written out."],[180.44920833333333,"theta ticks to 3.65."],[185.88920833333333,"work_thm (the \"dot(A)_x hat(i) + dot(A)_y hat(j) + dot(A)_z hat(k)\" part) is emphasized."]]},{"start":188.75170833333334,"say":"A rider on the disc calls that first family the relative derivative, because those are the component changes measured against the painted axes.","live":null,"does":[[189.12920833333334,"i_hat is redrawn as the numbers it depends on change."],[189.12920833333334,"theta_mark is redrawn as the numbers it depends on change."],[189.12920833333334,"j_hat is redrawn as the numbers it depends on change."],[189.12920833333334,"d_i_hat is redrawn as the numbers it depends on change."],[189.12920833333334,"right_mark is redrawn as the numbers it depends on change."],[189.12920833333334,"theta ticks to 4.0."]]},{"start":196.91420833333333,"say":"The second family leaves the components alone and differentiates i hat, j hat, and k hat. Each basis derivative contributes omega crossed with the corresponding basis vector.","live":null,"does":[[197.32920833333333,"i_hat is redrawn as the numbers it depends on change."],[197.32920833333333,"theta_mark is redrawn as the numbers it depends on change."],[197.32920833333333,"j_hat is redrawn as the numbers it depends on change."],[197.32920833333333,"d_i_hat is redrawn as the numbers it depends on change."],[197.32920833333333,"right_mark is redrawn as the numbers it depends on change."],[197.32920833333333,"work_thm is shown on the screen, written out."],[197.32920833333333,"work_thm (the \"dot(A)_x hat(i) + dot(A)_y hat(j) + dot(A)_z hat(k)\" part) is no longer emphasized."],[197.32920833333333,"theta ticks to 4.45."],[199.80920833333334,"work_thm (the \"A_x frac(dif hat(i), dif t) + A_y frac(dif hat(j), dif t) + A_z frac(dif hat(k), dif t)\" part) is emphasized."],[204.92920833333332,"fact (the \"arrow(omega) times hat(e)\" part) is emphasized."]]},{"start":208.18420833333334,"say":"Factor omega cross from that whole family. The remaining bracket is the original vector A, so the family becomes omega cross A.","live":null,"does":[[208.48920833333335,"i_hat is redrawn as the numbers it depends on change."],[208.48920833333335,"theta_mark is redrawn as the numbers it depends on change."],[208.48920833333335,"j_hat is redrawn as the numbers it depends on change."],[208.48920833333335,"d_i_hat is redrawn as the numbers it depends on change."],[208.48920833333335,"right_mark is redrawn as the numbers it depends on change."],[208.48920833333335,"work_thm (the \"A_x frac(dif hat(i), dif t) + A_y frac(dif hat(j), dif t) + A_z frac(dif hat(k), dif t)\" part) is no longer emphasized."],[208.48920833333335,"work_thm (the \"A_x frac(dif hat(i), dif t) + A_y frac(dif hat(j), dif t) + A_z frac(dif hat(k), dif t)\" part) is under-braced and labelled \"arrow(omega) times arrow(A)\"."],[208.48920833333335,"theta ticks to 4.9."],[211.52920833333332,"fact (the \"arrow(omega) times hat(e)\" part) is no longer emphasized."],[212.88920833333333,"work_thm (the \"A_x hat(i) + A_y hat(j) + A_z hat(k)\" part) is emphasized."],[213.84920833333334,"The brace under work_thm (the \"A_x frac(dif hat(i), dif t) + A_y frac(dif hat(j), dif t) + A_z frac(dif hat(k), dif t)\" part) is lifted."],[214.48920833333335,"work_thm is shown on the screen, written out."],[214.48920833333335,"work_thm (the \"A_x hat(i) + A_y hat(j) + A_z hat(k)\" part) is no longer emphasized."]]},{"start":217.07420833333333,"say":"This is the transport theorem. A vector's fixed-frame rate equals its relative rate plus omega cross the vector. Applying this statement twice produces every acceleration term that follows.","live":null,"does":[[218.04920833333333,"i_hat is redrawn as the numbers it depends on change."],[218.04920833333333,"theta_mark is redrawn as the numbers it depends on change."],[218.04920833333333,"j_hat is redrawn as the numbers it depends on change."],[218.04920833333333,"d_i_hat is redrawn as the numbers it depends on change."],[218.04920833333333,"right_mark is redrawn as the numbers it depends on change."],[218.04920833333333,"theorem is shown on the screen, written out."],[218.04920833333333,"theta ticks to 5.35."],[220.20920833333332,"A box is drawn around theorem."],[221.68920833333334,"theorem (the \"(frac(dif arrow(A), dif t))_upright(\"rel\")\" part) is emphasized."],[222.68920833333334,"theorem (the \"(frac(dif arrow(A), dif t))_upright(\"rel\")\" part) is no longer emphasized."],[222.68920833333334,"theorem (the \"arrow(omega) times arrow(A)\" part) is emphasized."],[224.48920833333332,"theorem (the \"arrow(omega) times arrow(A)\" part) is no longer emphasized."],[228.81675,"dial is hidden from the screen — left the board."],[228.81675,"rim is hidden from the screen — dial left the board."],[228.81675,"hub is hidden from the screen — dial left the board."],[228.81675,"ref_axis is hidden from the screen — dial left the board."],[228.81675,"i_hat is hidden from the screen — dial left the board."],[228.81675,"theta_mark is hidden from the screen — dial left the board."],[228.81675,"j_hat is hidden from the screen — dial left the board."],[228.81675,"d_i_hat is hidden from the screen — dial left the board."],[228.81675,"right_mark is hidden from the screen — dial left the board."],[228.81675,"fact is hidden from the screen — left the board."],[228.81675,"heading_thm is hidden from the screen — left the board."],[228.81675,"theorem is hidden from the screen — left the board."],[228.81675,"work_thm is hidden from the screen — left the board."]]}]},{"title":"Velocity, Term by Term","start":229.85841666666664,"end":331.6357916666666,"objects":{"bug":"a Point [magenta] labelled \"P\" drawn in deckplan (location=((2.6 + (walk_radius * cos(disc_angle))), (2.5 + (walk_radius *…)","carry_vec":"an Arrow [yellow] labelled \"arrow(omega) times arrow(rho)\" drawn in deckplan (start=((2.6 + (walk_radius * cos(disc_angle))), (2.5 + (walk_radius *…, end=(((2.6 + (walk_radius * cos(disc_angle))) - (1.05 * sin(disc_an…)","cross_right":"an Angle [yellow] drawn in space (vertex=(0.0, 0.0, 0.0), sides=((1.6, 0.9, 0.0), (-0.9, 1.6, 0.0)), right_angle=True)","cross_vec":"a Vector [yellow] labelled \"arrow(omega) times arrow(rho)\" drawn in space (start=(0.0, 0.0, 0.0), end=(-0.9, 1.6, 0.0))","deckplan":"a Figure (x_range=(0.0, 5.6), y_range=(0.0, 5.4), aspect=(5.6, 5.4))","disc_angle":"a VariableNumber (initial_value=0.38, format_spec='.2f')","disc_plane":"a Circle [gray] drawn in space (center=(0.0, 0.0, 0.0), radius=1.65, normal_vector=(0.0, 0.0, 1.0))","heading_t":"a Heading that says \"Three Velocities, Head to Tail\"","heading_v":"a Heading that says \"Velocity, Term by Term\"","hub2":"a Point [text] labelled \"B\" drawn in deckplan (location=(2.6, 2.5))","rel_vec":"an Arrow [green] labelled \"arrow(v)_upright(\"rel\")\" drawn in deckplan (start=(((2.6 + (walk_radius * cos(disc_angle))) - (1.05 * sin(disc_an…, end=((((2.6 + (walk_radius * cos(disc_angle))) - (1.05 * sin(disc_a…)","rho_arrow":"an Arrow [red] labelled \"arrow(rho)\" drawn in deckplan (start=(2.6, 2.5), end=((2.6 + (walk_radius * cos(disc_angle))), (2.5 + (walk_radius *…)","rho_vec":"a Vector [red] labelled \"arrow(rho)\" drawn in space (start=(0.0, 0.0, 0.0), end=(1.6, 0.9, 0.0))","rim2":"a Circle [gray] drawn in deckplan (center=(2.6, 2.5), radius=2.0)","space":"an Axes3D (x_range=(-2.0, 2.0), y_range=(-2.0, 2.0), z_range=(0.0, 2.6))","spin":"a CurvedArrow [blue] labelled \"arrow(omega)\" drawn in deckplan (start=(1.25, 3.9), end=(1.05, 2.55), bend=0.45)","spoke":"a Line [gray] drawn in deckplan (start=(2.6, 2.5), end=((2.6 + (2.0 * cos(disc_angle))), (2.5 + (2.0 * sin(disc_angle)…, dashed=True)","terms":"a Block [text] that says \"$arrow(v)_B$: motion of the frame's origin $arrow(omega) times arrow(rho)$: velocity carried by the spin $arrow(v)_upright(\"rel\")$: motion measured on the disc\"","true_vec":"an Arrow [magenta] labelled \"arrow(v)_P\" drawn in deckplan (start=((2.6 + (walk_radius * cos(disc_angle))), (2.5 + (walk_radius *…, end=((((2.6 + (walk_radius * cos(disc_angle))) - (1.05 * sin(disc_a…)","w_vec":"a Vector [blue] labelled \"arrow(omega)\" drawn in space (start=(0.0, 0.0, 0.0), end=(0.0, 0.0, 2.2))","walk_radius":"a VariableNumber (initial_value=1.15, format_spec='.2f')","work_v":"a Derivation [text] that says \"$arrow(r)_P &= arrow(r)_B + arrow(rho) \\ arrow(v)_P &= arrow(v)_B + (frac(dif arrow(rho), dif t))_upright(\"fix\") \\ &= arrow(v)_B + (frac(dif arrow(rho), dif t))_upright(\"rel\") + arrow(omega) times arrow(rho) \\ &= arrow(v)_B + arrow(v)_uprig…$\""},"beats":[{"start":229.85841666666664,"say":"One line of geometry and one use of the transport theorem give the velocity relation. Begin with r P equals r B plus rho.","live":[],"does":[[229.85841666666664,"heading_v is shown on the screen, written out."],[235.69841666666665,"work_v is shown on the screen, written out."],[235.69841666666665,"work_v (the \"arrow(r)_P\" part) is emphasized."],[236.81841666666665,"work_v (the \"arrow(r)_B\" part) is emphasized."],[236.81841666666665,"work_v (the \"arrow(r)_P\" part) is no longer emphasized."],[237.69841666666665,"work_v (the \"arrow(r)_B\" part) is no longer emphasized."],[237.69841666666665,"work_v (the \"arrow(rho)\" part) is emphasized."]]},{"start":238.94091666666665,"say":"Differentiate both sides in the fixed frame. The left side becomes the true velocity v P, and the first term on the right becomes the velocity v B of the moving origin.","live":["heading_v"],"does":[[238.94091666666665,"work_v (the \"arrow(rho)\" part) is no longer emphasized."],[239.17841666666664,"work_v is shown on the screen, written out."],[240.77841666666663,"space is shown on the screen, written out."],[240.77841666666663,"disc_plane is shown on the screen, written out."],[244.05841666666663,"work_v (the \"arrow(v)_P\" part) is emphasized."],[247.41841666666664,"work_v (the \"arrow(v)_B\" part) is emphasized."],[247.41841666666664,"work_v (the \"arrow(v)_P\" part) is no longer emphasized."]]},{"start":250.07091666666665,"say":"Rho needs the theorem. The red rho vector lies in the disc. The blue omega vector points along the rotation axis. Their yellow cross product is the extra velocity carried by the turning frame.","live":["space","heading_v","disc_plane"],"does":[[250.07091666666665,"work_v (the \"arrow(v)_B\" part) is no longer emphasized."],[251.21841666666666,"work_v is shown on the screen, written out."],[251.21841666666666,"work_v (the \"(frac(dif arrow(rho), dif t))_upright(\"rel\")\" part) is emphasized."],[252.73841666666664,"rho_vec is shown on the screen, written out."],[255.13841666666664,"w_vec is shown on the screen, written out."],[258.5784166666666,"cross_vec is shown on the screen, written out."],[258.5784166666666,"work_v (the \"(frac(dif arrow(rho), dif t))_upright(\"rel\")\" part) is no longer emphasized."],[258.5784166666666,"work_v (the \"arrow(omega) times arrow(rho)\" part) is emphasized."]]},{"start":262.7559166666666,"say":"Omega cross rho is at right angles to rho and omega. It lies flat in the plane of the disc, ninety degrees ahead of rho, with magnitude omega times rho.","live":["space","heading_v","disc_plane","rho_vec","w_vec","cross_vec"],"does":[[264.61841666666663,"cross_right is shown on the screen, written out."],[267.73841666666664,"The xy plane in space is lit up."],[268.77841666666666,"cross_vec is emphasized."],[268.77841666666666,"work_v (the \"arrow(omega) times arrow(rho)\" part) is no longer emphasized."]]},{"start":273.40841666666665,"say":"The complete velocity therefore has three pieces: motion of the origin, motion measured on the disc, and motion carried by the disc's spin.","live":["space","heading_v","disc_plane","rho_vec","w_vec","cross_vec","cross_right"],"does":[[273.40841666666665,"cross_vec is no longer emphasized."],[273.40841666666665,"space: retire a lit plane (unemphasize_plane)."],[275.61841666666663,"work_v is shown on the screen, written out."],[277.3784166666666,"work_v (the \"arrow(v)_B\" part) is emphasized."],[278.5784166666666,"work_v (the \"arrow(v)_B\" part) is no longer emphasized."],[278.5784166666666,"work_v (the \"arrow(v)_upright(\"rel\")\" part) is emphasized."],[280.41841666666664,"work_v (the \"arrow(omega) times arrow(rho)\" part) is emphasized."],[280.41841666666664,"work_v (the \"arrow(v)_upright(\"rel\")\" part) is no longer emphasized."],[282.14691666666664,"heading_v is hidden from the screen — left the board."],[282.14691666666664,"space is hidden from the screen — left the board."],[282.14691666666664,"disc_plane is hidden from the screen — space left the board."],[282.14691666666664,"rho_vec is hidden from the screen — space left the board."],[282.14691666666664,"w_vec is hidden from the screen — space left the board."],[282.14691666666664,"cross_vec is hidden from the screen — space left the board."],[282.14691666666664,"cross_right is hidden from the screen — space left the board."],[282.14691666666664,"work_v is hidden from the screen — left the board."],[282.14691666666664,"work_v (the \"arrow(omega) times arrow(rho)\" part) is no longer emphasized."]]},{"start":282.7469166666666,"say":"A beetle walks outward along a painted spoke while the turntable spins. The three velocity pieces form one head-to-tail chain.","live":[],"does":[[282.7469166666666,"heading_t is shown on the screen, written out."],[282.7469166666666,"deckplan is shown on the screen, written out."],[283.3384166666666,"bug is shown on the screen, written out."],[283.8184166666666,"bug is redrawn as the numbers it depends on change."],[283.8184166666666,"carry_vec is redrawn as the numbers it depends on change."],[283.8184166666666,"rel_vec is redrawn as the numbers it depends on change."],[283.8184166666666,"true_vec is redrawn as the numbers it depends on change."],[283.8184166666666,"walk_radius ticks to 1.28."],[285.3384166666666,"spoke is redrawn as the numbers it depends on change."],[285.3384166666666,"rho_arrow is redrawn as the numbers it depends on change."],[285.3384166666666,"spoke is shown on the screen, written out."],[285.3384166666666,"rho_arrow is shown on the screen, written out."],[286.1384166666666,"rim2 is shown on the screen, written out."],[286.1384166666666,"hub2 is shown on the screen, written out."],[286.53841666666665,"spin is shown on the screen, written out."],[286.53841666666665,"disc_angle ticks to 0.85."],[287.73841666666664,"terms is shown on the screen, written out."]]},{"start":291.55091666666664,"say":"First is v B, the velocity of the frame's origin. A bolted turntable has v B equal to zero, while an aircraft or planet need not.","live":["terms","deckplan","heading_t","rim2","hub2","spoke","bug","rho_arrow","spin"],"does":[[291.55091666666664,"spoke is redrawn as the numbers it depends on change."],[291.55091666666664,"bug is redrawn as the numbers it depends on change."],[291.55091666666664,"rho_arrow is redrawn as the numbers it depends on change."],[291.55091666666664,"carry_vec is redrawn as the numbers it depends on change."],[291.55091666666664,"rel_vec is redrawn as the numbers it depends on change."],[291.55091666666664,"true_vec is redrawn as the numbers it depends on change."],[291.55091666666664,"disc_angle ticks to 1.18."],[291.55091666666664,"walk_radius ticks to 1.36."],[292.73841666666664,"terms (the \"arrow(v)_B\" part) is emphasized."]]},{"start":301.4484166666666,"say":"Second is omega cross rho. It is tangential and square to the spoke, the velocity the beetle receives simply because the disc carries it.","live":null,"does":[[301.4484166666666,"spoke is redrawn as the numbers it depends on change."],[301.4484166666666,"bug is redrawn as the numbers it depends on change."],[301.4484166666666,"rho_arrow is redrawn as the numbers it depends on change."],[301.4484166666666,"rel_vec is redrawn as the numbers it depends on change."],[301.4484166666666,"true_vec is redrawn as the numbers it depends on change."],[301.4484166666666,"disc_angle ticks to 1.55."],[301.4484166666666,"walk_radius ticks to 1.46."],[302.77841666666666,"terms (the \"arrow(omega) times arrow(rho)\" part) is emphasized."],[302.77841666666666,"terms (the \"arrow(v)_B\" part) is no longer emphasized."],[304.41841666666664,"carry_vec is redrawn as the numbers it depends on change."],[304.41841666666664,"carry_vec is shown on the screen, written out."],[309.3384166666666,"carry_vec is emphasized."]]},{"start":310.6459166666666,"say":"Third is v rel, directed outward along the spoke. Translate that vector to the tip of omega cross rho, and the two arrows now meet head to tail.","live":["terms","deckplan","heading_t","rim2","hub2","spoke","bug","rho_arrow","spin","carry_vec"],"does":[[310.6459166666666,"spoke is redrawn as the numbers it depends on change."],[310.6459166666666,"bug is redrawn as the numbers it depends on change."],[310.6459166666666,"rho_arrow is redrawn as the numbers it depends on change."],[310.6459166666666,"carry_vec is redrawn as the numbers it depends on change."],[310.6459166666666,"true_vec is redrawn as the numbers it depends on change."],[310.6459166666666,"disc_angle ticks to 1.9."],[310.6459166666666,"walk_radius ticks to 1.54."],[311.85841666666664,"terms (the \"arrow(omega) times arrow(rho)\" part) is no longer emphasized."],[311.85841666666664,"terms (the \"arrow(v)_upright(\"rel\")\" part) is emphasized."],[313.29841666666664,"rel_vec is redrawn as the numbers it depends on change."],[313.29841666666664,"rel_vec is shown on the screen, written out."],[319.3784166666667,"rel_vec is emphasized."]]},{"start":320.7484166666666,"say":"The arrow from the chain's first tail to its final tip is v P, the beetle's true velocity over the ground. One more derivative produces the acceleration terms.","live":["terms","deckplan","heading_t","rim2","hub2","spoke","bug","rho_arrow","spin","carry_vec","rel_vec"],"does":[[320.7484166666666,"spoke is redrawn as the numbers it depends on change."],[320.7484166666666,"bug is redrawn as the numbers it depends on change."],[320.7484166666666,"rho_arrow is redrawn as the numbers it depends on change."],[320.7484166666666,"carry_vec is redrawn as the numbers it depends on change."],[320.7484166666666,"rel_vec is redrawn as the numbers it depends on change."],[320.7484166666666,"carry_vec is no longer emphasized."],[320.7484166666666,"rel_vec is no longer emphasized."],[320.7484166666666,"terms (the \"arrow(v)_upright(\"rel\")\" part) is no longer emphasized."],[320.7484166666666,"disc_angle ticks to 2.28."],[320.7484166666666,"walk_radius ticks to 1.62."],[324.21841666666666,"true_vec is redrawn as the numbers it depends on change."],[324.21841666666666,"true_vec is shown on the screen, written out."],[325.65841666666665,"true_vec is emphasized."],[328.13841666666667,"true_vec is no longer emphasized."],[330.59412499999996,"deckplan is hidden from the screen — left the board."],[330.59412499999996,"rim2 is hidden from the screen — deckplan left the board."],[330.59412499999996,"hub2 is hidden from the screen — deckplan left the board."],[330.59412499999996,"spoke is hidden from the screen — deckplan left the board."],[330.59412499999996,"bug is hidden from the screen — deckplan left the board."],[330.59412499999996,"rho_arrow is hidden from the screen — deckplan left the board."],[330.59412499999996,"spin is hidden from the screen — deckplan left the board."],[330.59412499999996,"carry_vec is hidden from the screen — deckplan left the board."],[330.59412499999996,"rel_vec is hidden from the screen — deckplan left the board."],[330.59412499999996,"true_vec is hidden from the screen — deckplan left the board."],[330.59412499999996,"heading_t is hidden from the screen — left the board."],[330.59412499999996,"terms is hidden from the screen — left the board."]]}]},{"title":"Acceleration: Five Terms","start":331.6357916666666,"end":467.39520833333324,"objects":{"arrow":"an Arrow [green] labelled \"arrow(v)_upright(\"rel\")\" drawn in case (start=((1.65 + (0.82 * cos(motion_angle))), (1.5 + (0.82 * sin(motion…, end=((1.65 + (1.22 * cos(motion_angle))), (1.5 + (1.22 * sin(motion…)","arrow_2":"an Arrow [green] labelled \"arrow(v)_upright(\"rel\")\" drawn in case_3 (start=((1.65 + (0.82 * cos(both_angle))), (1.5 + (0.82 * sin(both_ang…, end=((1.65 + (1.22 * cos(both_angle))), (1.5 + (1.22 * sin(both_ang…)","arrow_3":"an Arrow [green] labelled \"2 arrow(omega) times arrow(v)_upright(\"rel\")\" drawn in case_3 (start=((1.65 + (0.82 * cos(both_angle))), (1.5 + (0.82 * sin(both_ang…, end=(((1.65 + (0.82 * cos(both_angle))) - (0.62 * sin(both_angle)))…)","both_angle":"a VariableNumber (initial_value=0.38, format_spec='.2f')","both_caption":"a Math [text] that says \"$arrow(omega) != arrow(0), thin arrow(v)_upright(\"rel\") != arrow(0)$\"","bug":"a Point [magenta] labelled \"P\" drawn in picture (location=((2.9 + (1.18 * cos(disc_angle))), (2.35 + (1.18 * sin(disc_ang…)","case":"a Figure (x_range=(0.0, 3.4), y_range=(0.0, 3.2), aspect=(3.4, 3.2))","case_2":"a Figure (x_range=(0.0, 3.4), y_range=(0.0, 3.2), aspect=(3.4, 3.2))","case_3":"a Figure (x_range=(0.0, 3.4), y_range=(0.0, 3.2), aspect=(3.4, 3.2))","case_bug":"a Point [magenta] drawn in case (location=((1.65 + (0.82 * cos(motion_angle))), (1.5 + (0.82 * sin(motion…)","case_bug_2":"a Point [magenta] drawn in case_2 (location=((1.65 + (0.82 * cos(turn_angle))), (1.5 + (0.82 * sin(turn_ang…)","case_bug_3":"a Point [magenta] drawn in case_3 (location=((1.65 + (0.82 * cos(both_angle))), (1.5 + (0.82 * sin(both_ang…)","case_rim":"a Circle [gray] drawn in case (center=(1.65, 1.5), radius=1.22)","case_rim_2":"a Circle [gray] drawn in case_2 (center=(1.65, 1.5), radius=1.22)","case_rim_3":"a Circle [gray] drawn in case_3 (center=(1.65, 1.5), radius=1.22)","case_spoke":"a Line [blue] drawn in case (start=(1.65, 1.5), end=((1.65 + (1.22 * cos(motion_angle))), (1.5 + (1.22 * sin(motion…)","case_spoke_2":"a Line [blue] drawn in case_2 (start=(1.65, 1.5), end=((1.65 + (1.22 * cos(turn_angle))), (1.5 + (1.22 * sin(turn_ang…)","case_spoke_3":"a Line [blue] drawn in case_3 (start=(1.65, 1.5), end=((1.65 + (1.22 * cos(both_angle))), (1.5 + (1.22 * sin(both_ang…)","centripetal":"an Arrow [yellow] labelled \"arrow(omega) times (arrow(omega) times arrow(rho))\" drawn in picture (start=((2.9 + (1.18 * cos(disc_angle))), (2.35 + (1.18 * sin(disc_ang…, end=(2.9, 2.35))","coriolis":"an Arrow [green] labelled \"2 arrow(omega) times arrow(v)_upright(\"rel\")\" drawn in picture (start=((2.9 + (1.18 * cos(disc_angle))), (2.35 + (1.18 * sin(disc_ang…, end=(((2.9 + (1.18 * cos(disc_angle))) - (1.2 * sin(disc_angle))), …)","coriolis_add":"an Arithmetic [text] that says \"$arrow(omega) times arrow(v)_upright(\"rel\") arrow(omega) times arrow(v)_upright(\"rel\") 2 arrow(omega) times arrow(v)_upright(\"rel\")$\" (operator='+', operands=('arrow(omega) times arrow(v)_upright(\"rel\")', 'arrow(omega) ti…, result='2 arrow(omega) times arrow(v)_upright(\"rel\")')","curved_arrow":"a CurvedArrow [blue] labelled \"arrow(omega)\" drawn in case_2 (start=(0.66, 2.38), end=(0.5, 1.38), bend=0.42)","curved_arrow_2":"a CurvedArrow [blue] labelled \"arrow(omega)\" drawn in case_3 (start=(0.66, 2.38), end=(0.5, 1.38), bend=0.42)","disc_angle":"a VariableNumber (initial_value=0.42, format_spec='.2f')","euler":"an Arrow [blue] labelled \"arrow(alpha) times arrow(rho)\" drawn in picture (start=((2.9 + (1.18 * cos(disc_angle))), (2.35 + (1.18 * sin(disc_ang…, end=(((2.9 + (1.18 * cos(disc_angle))) - (0.86 * sin(disc_angle))),…)","five_note":"a Text [text] that says \"Everything but the last term enters because the observer's frame turns or translates.\"","heading_cases":"a Heading that says \"Coriolis Needs Both Conditions\"","heading_diff":"a Heading that says \"Differentiate Once More\"","heading_five":"a Heading that says \"Five Terms\"","hub":"a Point [text] labelled \"B\" drawn in picture (location=(2.9, 2.35))","motion_caption":"a Math [text] that says \"$arrow(omega) = arrow(0) thin arrow.r thin upright(\"zero\")$\"","omega":"a CurvedArrow [blue] labelled \"arrow(omega)\" drawn in picture (start=(3.79, 0.45), end=(4.93, 1.81), bend=0.45)","picture":"a Figure (x_range=(0.0, 5.2), y_range=(0.0, 5.0), aspect=(5.2, 5.0))","rho":"an Arrow [red] labelled \"arrow(rho)\" drawn in picture (start=(2.9, 2.35), end=((2.9 + (1.18 * cos(disc_angle))), (2.35 + (1.18 * sin(disc_ang…)","rim":"a Circle [gray] drawn in picture (center=(2.9, 2.35), radius=1.8)","spoke":"a Line [blue] drawn in picture (start=(2.9, 2.35), end=((2.9 + (1.8 * cos(disc_angle))), (2.35 + (1.8 * sin(disc_angle…)","tool":"a Math [text] that says \"$(frac(dif arrow(u), dif t))_upright(\"fix\") = (frac(dif arrow(u), dif t))_upright(\"rel\") + arrow(omega) times arrow(u)$\"","turn_angle":"a VariableNumber (initial_value=0.38, format_spec='.2f')","turn_caption":"a Math [text] that says \"$arrow(v)_upright(\"rel\") = arrow(0) thin arrow.r thin upright(\"zero\")$\"","vrel":"an Arrow [green] labelled \"arrow(v)_upright(\"rel\")\" drawn in picture (start=((2.9 + (1.18 * cos(disc_angle))), (2.35 + (1.18 * sin(disc_ang…, end=((2.9 + (1.92 * cos(disc_angle))), (2.35 + (1.92 * sin(disc_ang…)","work_diff":"a Derivation [text] that says \"$arrow(a)_P &= arrow(a)_B + frac(dif, dif t)(arrow(omega) times arrow(rho)) + frac(dif, dif t)(arrow(v)_upright(\"rel\")) \\ frac(dif, dif t)(arrow(omega) times arrow(rho)) &= arrow(alpha) times arrow(rho) + arrow(omega) times frac(dif arrow(r…$\"","work_five":"a Derivation [text] that says \"$arrow(a)_P &= arrow(a)_B + arrow(alpha) times arrow(rho) + arrow(omega) times (arrow(omega) times arrow(rho)) \\ & thin + 2 arrow(omega) times arrow(v)_upright(\"rel\") + arrow(a)_upright(\"rel\")$\""},"beats":[{"start":331.6357916666666,"say":"Differentiate the velocity relation once more. Each vector on the right is measured against the turning disc, so the transport theorem remains beside the calculation.","live":[],"does":[[331.6357916666666,"heading_diff is shown on the screen, written out."],[331.6357916666666,"picture is shown on the screen, written out."],[331.6357916666666,"work_diff is shown on the screen, written out."],[334.91579166666656,"bug is shown on the screen, written out."],[334.91579166666656,"rho is shown on the screen, written out."],[336.7557916666666,"spoke is redrawn as the numbers it depends on change."],[336.7557916666666,"bug is redrawn as the numbers it depends on change."],[336.7557916666666,"rho is redrawn as the numbers it depends on change."],[336.7557916666666,"vrel is redrawn as the numbers it depends on change."],[336.7557916666666,"euler is redrawn as the numbers it depends on change."],[336.7557916666666,"centripetal is redrawn as the numbers it depends on change."],[336.7557916666666,"coriolis is redrawn as the numbers it depends on change."],[336.7557916666666,"spoke is shown on the screen, written out."],[336.7557916666666,"omega is shown on the screen, written out."],[336.7557916666666,"disc_angle ticks to 0.72."],[337.1557916666666,"rim is shown on the screen, written out."],[337.1557916666666,"hub is shown on the screen, written out."],[338.03579166666657,"tool is shown on the screen, written out."]]},{"start":341.4432916666666,"say":"The origin contributes a B directly. The other two derivatives need the theorem because omega cross rho and v rel are both vectors written in the painted basis.","live":["tool","picture","heading_diff","rim","hub","spoke","bug","rho","omega"],"does":[[341.4432916666666,"spoke is redrawn as the numbers it depends on change."],[341.4432916666666,"bug is redrawn as the numbers it depends on change."],[341.4432916666666,"rho is redrawn as the numbers it depends on change."],[341.4432916666666,"euler is redrawn as the numbers it depends on change."],[341.4432916666666,"centripetal is redrawn as the numbers it depends on change."],[341.4432916666666,"coriolis is redrawn as the numbers it depends on change."],[341.4432916666666,"disc_angle ticks to 0.92."],[342.7957916666666,"work_diff (the \"arrow(a)_B\" part) is emphasized."],[347.4357916666666,"rho is emphasized."],[347.9957916666666,"vrel is redrawn as the numbers it depends on change."],[347.9957916666666,"vrel is shown on the screen, written out."]]},{"start":352.03579166666657,"say":"Apply the product rule to omega cross rho. Omega dot becomes alpha, so the first contribution is alpha cross rho. It is tangential while the spin rate is changing.","live":["tool","picture","heading_diff","rim","hub","spoke","bug","rho","omega","vrel"],"does":[[352.03579166666657,"rho is no longer emphasized."],[352.03579166666657,"work_diff (the \"arrow(a)_B\" part) is no longer emphasized."],[352.7557916666666,"work_diff is shown on the screen, written out."],[358.59579166666657,"work_diff (the \"arrow(alpha) times arrow(rho)\" part) is emphasized."],[360.2357916666666,"euler is shown on the screen, written out."],[361.2357916666666,"euler is emphasized."],[361.9557916666666,"spoke is redrawn as the numbers it depends on change."],[361.9557916666666,"bug is redrawn as the numbers it depends on change."],[361.9557916666666,"rho is redrawn as the numbers it depends on change."],[361.9557916666666,"vrel is redrawn as the numbers it depends on change."],[361.9557916666666,"euler is redrawn as the numbers it depends on change."],[361.9557916666666,"centripetal is redrawn as the numbers it depends on change."],[361.9557916666666,"coriolis is redrawn as the numbers it depends on change."],[361.9557916666666,"disc_angle ticks to 1.2."]]},{"start":363.2582916666666,"say":"The second contribution contains the fixed derivative of rho. Substitute v rel plus omega cross rho, then distribute omega cross across both terms.","live":["tool","picture","heading_diff","rim","hub","spoke","bug","rho","omega","vrel","euler"],"does":[[363.2582916666666,"euler is no longer emphasized."],[363.2582916666666,"work_diff (the \"arrow(alpha) times arrow(rho)\" part) is no longer emphasized."],[367.1157916666666,"work_diff is shown on the screen, written out."],[367.7557916666666,"work_diff (the \"arrow(v)_upright(\"rel\")\" part) is emphasized."],[368.6357916666666,"centripetal is shown on the screen, written out."],[368.6357916666666,"work_diff (the \"arrow(omega) times arrow(rho)\" part) is emphasized."],[368.6357916666666,"work_diff (the \"arrow(v)_upright(\"rel\")\" part) is no longer emphasized."],[370.3157916666666,"work_diff is shown on the screen, written out."]]},{"start":373.54579166666656,"say":"The fixed derivative of v rel also uses the theorem. It contributes a rel plus a second omega cross v rel.","live":["tool","picture","heading_diff","rim","hub","spoke","bug","rho","omega","vrel","euler","centripetal"],"does":[[373.54579166666656,"work_diff (the \"arrow(omega) times arrow(rho)\" part) is no longer emphasized."],[376.2757916666666,"work_diff is shown on the screen, written out."],[377.7557916666666,"work_diff (the \"arrow(a)_upright(\"rel\")\" part) is emphasized."],[378.9957916666666,"work_diff (the \"arrow(a)_upright(\"rel\")\" part) is no longer emphasized."],[378.9957916666666,"work_diff (the \"arrow(omega) times arrow(v)_upright(\"rel\")\" part) is emphasized."],[379.2357916666666,"coriolis is shown on the screen, written out."]]},{"start":381.3157916666666,"say":"The distributed line contains one omega cross v rel. The final derivative contains another omega cross v rel. These two identical terms add.","live":["tool","picture","heading_diff","rim","hub","spoke","bug","rho","omega","vrel","euler","centripetal","coriolis"],"does":[[383.1157916666666,"coriolis_add is shown on the screen, written out."],[383.1157916666666,"work_diff (the \"arrow(omega) times arrow(v)_upright(\"rel\")\" part) is emphasized."],[383.1157916666666,"work_diff (the \"arrow(omega) times arrow(v)_upright(\"rel\")\" part) is no longer emphasized."],[386.3957916666666,"coriolis_add is shown on the screen, written out."],[386.3957916666666,"work_diff (the \"arrow(omega) times arrow(v)_upright(\"rel\")\" part) is no longer emphasized."],[386.3957916666666,"work_diff (the \"arrow(omega) times arrow(v)_upright(\"rel\")\" part) is emphasized."],[389.03579166666657,"coriolis is emphasized."],[390.0757916666666,"coriolis_add is shown on the screen, drawn."],[391.0885753546099,"coriolis_add is shown on the screen, drawn."]]},{"start":391.22829166666656,"say":"Their sum is two omega cross v rel. The factor of two is the Coriolis coefficient, produced by two separate derivatives.","live":null,"does":[[391.22829166666656,"coriolis is no longer emphasized."],[391.22829166666656,"work_diff (the \"arrow(omega) times arrow(v)_upright(\"rel\")\" part) is no longer emphasized."],[392.3557916666666,"coriolis_add is shown on the screen, written out."],[394.4357916666666,"coriolis_add (the \"2\" part) is emphasized."],[399.2007916666666,"coriolis_add is hidden from the screen — left the board."],[399.2007916666666,"heading_diff is hidden from the screen — left the board."],[399.2007916666666,"tool is hidden from the screen — left the board."],[399.2007916666666,"work_diff is hidden from the screen — left the board."]]},{"start":399.80079166666656,"say":"Collecting the result gives five acceleration terms, all expressed in quantities available to the rotating observer.","live":["picture","rim","hub","spoke","bug","rho","omega","vrel","euler","centripetal","coriolis"],"does":[[399.80079166666656,"heading_five is shown on the screen, written out."],[401.3557916666666,"work_five is shown on the screen, written out."],[402.3157916666666,"work_five is shown on the screen, written out."],[405.0757916666666,"spoke is redrawn as the numbers it depends on change."],[405.0757916666666,"bug is redrawn as the numbers it depends on change."],[405.0757916666666,"rho is redrawn as the numbers it depends on change."],[405.0757916666666,"vrel is redrawn as the numbers it depends on change."],[405.0757916666666,"euler is redrawn as the numbers it depends on change."],[405.0757916666666,"centripetal is redrawn as the numbers it depends on change."],[405.0757916666666,"coriolis is redrawn as the numbers it depends on change."],[405.0757916666666,"five_note is shown on the screen, written out."],[405.0757916666666,"disc_angle ticks to 1.42."]]},{"start":406.8307916666666,"say":"First, a B is the acceleration of the frame's origin. A floor-mounted turntable makes it zero, but a vehicle or planet need not.","live":["picture","rim","hub","spoke","bug","rho","omega","vrel","euler","centripetal","coriolis","five_note","heading_five"],"does":[[406.8307916666666,"spoke is redrawn as the numbers it depends on change."],[406.8307916666666,"bug is redrawn as the numbers it depends on change."],[406.8307916666666,"rho is redrawn as the numbers it depends on change."],[406.8307916666666,"vrel is redrawn as the numbers it depends on change."],[406.8307916666666,"euler is redrawn as the numbers it depends on change."],[406.8307916666666,"centripetal is redrawn as the numbers it depends on change."],[406.8307916666666,"coriolis is redrawn as the numbers it depends on change."],[406.8307916666666,"disc_angle ticks to 1.63."],[407.79579166666656,"work_five (the \"arrow(a)_B\" part) is emphasized."],[410.2757916666666,"hub is emphasized."]]},{"start":416.0482916666666,"say":"Second, alpha cross rho is the Euler term. It appears while the spin rate changes and vanishes at steady omega.","live":null,"does":[[416.0482916666666,"spoke is redrawn as the numbers it depends on change."],[416.0482916666666,"bug is redrawn as the numbers it depends on change."],[416.0482916666666,"rho is redrawn as the numbers it depends on change."],[416.0482916666666,"vrel is redrawn as the numbers it depends on change."],[416.0482916666666,"euler is redrawn as the numbers it depends on change."],[416.0482916666666,"centripetal is redrawn as the numbers it depends on change."],[416.0482916666666,"coriolis is redrawn as the numbers it depends on change."],[416.0482916666666,"hub is no longer emphasized."],[416.0482916666666,"disc_angle ticks to 1.84."],[417.1957916666666,"work_five (the \"arrow(a)_B\" part) is no longer emphasized."],[417.1957916666666,"work_five (the \"arrow(alpha) times arrow(rho)\" part) is emphasized."],[418.3957916666666,"euler is emphasized."]]},{"start":423.91329166666657,"say":"Third, omega cross omega cross rho is centripetal. It points from P toward the rotation axis, with magnitude omega squared times the distance out.","live":null,"does":[[423.91329166666657,"spoke is redrawn as the numbers it depends on change."],[423.91329166666657,"bug is redrawn as the numbers it depends on change."],[423.91329166666657,"rho is redrawn as the numbers it depends on change."],[423.91329166666657,"vrel is redrawn as the numbers it depends on change."],[423.91329166666657,"euler is redrawn as the numbers it depends on change."],[423.91329166666657,"centripetal is redrawn as the numbers it depends on change."],[423.91329166666657,"coriolis is redrawn as the numbers it depends on change."],[423.91329166666657,"euler is no longer emphasized."],[423.91329166666657,"disc_angle ticks to 2.05."],[427.1157916666666,"work_five (the \"arrow(alpha) times arrow(rho)\" part) is no longer emphasized."],[427.1157916666666,"work_five (the \"arrow(omega) times (arrow(omega) times arrow(rho))\" part) is emphasized."],[429.1557916666666,"centripetal is emphasized."]]},{"start":434.43579166666655,"say":"Fourth, two omega cross v rel is Coriolis. It needs a turning frame and motion relative to that frame at the same instant.","live":null,"does":[[434.43579166666655,"spoke is redrawn as the numbers it depends on change."],[434.43579166666655,"bug is redrawn as the numbers it depends on change."],[434.43579166666655,"rho is redrawn as the numbers it depends on change."],[434.43579166666655,"vrel is redrawn as the numbers it depends on change."],[434.43579166666655,"euler is redrawn as the numbers it depends on change."],[434.43579166666655,"centripetal is redrawn as the numbers it depends on change."],[434.43579166666655,"coriolis is redrawn as the numbers it depends on change."],[434.43579166666655,"centripetal is no longer emphasized."],[434.43579166666655,"work_five (the \"arrow(omega) times (arrow(omega) times arrow(rho))\" part) is no longer emphasized."],[434.43579166666655,"disc_angle ticks to 2.25."],[437.2757916666666,"work_five (the \"2 arrow(omega) times arrow(v)_upright(\"rel\")\" part) is emphasized."],[438.91579166666656,"omega is emphasized."],[439.9557916666666,"vrel is emphasized."],[441.6357916666666,"coriolis is emphasized."]]},{"start":443.1507916666666,"say":"Fifth, a rel is the acceleration measured by instruments fixed to the disc. It is often the unknown the rotating observer must determine.","live":null,"does":[[443.1507916666666,"spoke is redrawn as the numbers it depends on change."],[443.1507916666666,"bug is redrawn as the numbers it depends on change."],[443.1507916666666,"rho is redrawn as the numbers it depends on change."],[443.1507916666666,"vrel is redrawn as the numbers it depends on change."],[443.1507916666666,"euler is redrawn as the numbers it depends on change."],[443.1507916666666,"centripetal is redrawn as the numbers it depends on change."],[443.1507916666666,"coriolis is redrawn as the numbers it depends on change."],[443.1507916666666,"omega is no longer emphasized."],[443.1507916666666,"vrel is no longer emphasized."],[443.1507916666666,"coriolis is no longer emphasized."],[443.1507916666666,"disc_angle ticks to 2.44."],[444.3557916666666,"work_five (the \"2 arrow(omega) times arrow(v)_upright(\"rel\")\" part) is no longer emphasized."],[444.3557916666666,"work_five (the \"arrow(a)_upright(\"rel\")\" part) is emphasized."],[446.4357916666666,"bug is emphasized."],[451.49829166666655,"five_note is hidden from the screen — left the board."],[451.49829166666655,"heading_five is hidden from the screen — left the board."],[451.49829166666655,"picture is hidden from the screen — left the board."],[451.49829166666655,"rim is hidden from the screen — picture left the board."],[451.49829166666655,"hub is hidden from the screen — picture left the board."],[451.49829166666655,"spoke is hidden from the screen — picture left the board."],[451.49829166666655,"bug is hidden from the screen — picture left the board."],[451.49829166666655,"rho is hidden from the screen — picture left the board."],[451.49829166666655,"omega is hidden from the screen — picture left the board."],[451.49829166666655,"vrel is hidden from the screen — picture left the board."],[451.49829166666655,"euler is hidden from the screen — picture left the board."],[451.49829166666655,"centripetal is hidden from the screen — picture left the board."],[451.49829166666655,"coriolis is hidden from the screen — picture left the board."],[451.49829166666655,"work_five is hidden from the screen — left the board."]]},{"start":452.09829166666657,"say":"The Coriolis condition is now exact. Relative motion without rotation gives zero. Rotation without relative motion also gives zero. Only the final case has both, so only that case carries a Coriolis acceleration.","live":[],"does":[[452.09829166666657,"heading_cases is shown on the screen, written out."],[455.3557916666666,"case is shown on the screen, written out."],[455.3557916666666,"case_rim is shown on the screen, written out."],[455.3557916666666,"case_spoke is shown on the screen, written out."],[455.3557916666666,"case_bug is shown on the screen, written out."],[455.3557916666666,"arrow is shown on the screen, written out."],[456.1557916666666,"motion_caption is shown on the screen, written out."],[456.7157916666666,"case_spoke_2 is redrawn as the numbers it depends on change."],[456.7157916666666,"case_bug_2 is redrawn as the numbers it depends on change."],[456.7157916666666,"case_spoke_2 is shown on the screen, written out."],[456.7157916666666,"case_bug_2 is shown on the screen, written out."],[456.7157916666666,"case_2 is shown on the screen, written out."],[456.7157916666666,"case_rim_2 is shown on the screen, written out."],[456.7157916666666,"curved_arrow is shown on the screen, written out."],[456.7157916666666,"turn_angle ticks to 1.35."],[458.7157916666666,"turn_caption is shown on the screen, written out."],[461.7557916666666,"case_spoke_3 is redrawn as the numbers it depends on change."],[461.7557916666666,"case_bug_3 is redrawn as the numbers it depends on change."],[461.7557916666666,"arrow_2 is redrawn as the numbers it depends on change."],[461.7557916666666,"arrow_3 is redrawn as the numbers it depends on change."],[461.7557916666666,"case_spoke_3 is shown on the screen, written out."],[461.7557916666666,"case_bug_3 is shown on the screen, written out."],[461.7557916666666,"arrow_2 is shown on the screen, written out."],[461.7557916666666,"arrow_3 is shown on the screen, written out."],[461.7557916666666,"case_3 is shown on the screen, written out."],[461.7557916666666,"case_rim_3 is shown on the screen, written out."],[461.7557916666666,"curved_arrow_2 is shown on the screen, written out."],[462.7157916666666,"both_caption is shown on the screen, written out."],[462.7157916666666,"both_angle ticks to 1.35."],[466.3535416666666,"both_caption is hidden from the screen — left the board."],[466.3535416666666,"case is hidden from the screen — left the board."],[466.3535416666666,"case_rim is hidden from the screen — case left the board."],[466.3535416666666,"case_spoke is hidden from the screen — case left the board."],[466.3535416666666,"case_bug is hidden from the screen — case left the board."],[466.3535416666666,"arrow is hidden from the screen — case left the board."],[466.3535416666666,"case_2 is hidden from the screen — left the board."],[466.3535416666666,"case_rim_2 is hidden from the screen — case_2 left the board."],[466.3535416666666,"case_spoke_2 is hidden from the screen — case_2 left the board."],[466.3535416666666,"case_bug_2 is hidden from the screen — case_2 left the board."],[466.3535416666666,"curved_arrow is hidden from the screen — case_2 left the board."],[466.3535416666666,"case_3 is hidden from the screen — left the board."],[466.3535416666666,"case_rim_3 is hidden from the screen — case_3 left the board."],[466.3535416666666,"case_spoke_3 is hidden from the screen — case_3 left the board."],[466.3535416666666,"case_bug_3 is hidden from the screen — case_3 left the board."],[466.3535416666666,"arrow_2 is hidden from the screen — case_3 left the board."],[466.3535416666666,"curved_arrow_2 is hidden from the screen — case_3 left the board."],[466.3535416666666,"arrow_3 is hidden from the screen — case_3 left the board."],[466.3535416666666,"heading_cases is hidden from the screen — left the board."],[466.3535416666666,"motion_caption is hidden from the screen — left the board."],[466.3535416666666,"turn_caption is hidden from the screen — left the board."]]}]},{"title":"The Forces You Feel","start":467.39520833333324,"end":649.1276874999999,"objects":{"applications":"a Block [text] that says \"Large weather systems Flow across a pressure gradient Long-range gunnery A Foucault pendulum\"","centrifugal":"an Arrow [yellow] labelled \"m omega^2 arrow(rho)\" drawn in turntable (start=((2.7 + (1.22 * cos(disc_angle))), (3.0 + (1.22 * sin(disc_angl…, end=((2.7 + (1.94 * cos(disc_angle))), (3.0 + (1.94 * sin(disc_angl…)","closing":"a Math [text] that says \"$frac(dif hat(e), dif t) = arrow(omega) times hat(e), thin upright(\"then apply the product rule.\")$\"","deck":"a Figure (x_range=(0.0, 5.4), y_range=(0.0, 5.6), aspect=(5.4, 5.6))","deck_ball":"a Point [magenta] labelled \"P\" drawn in deck (location=((2.7 + ((1.18 * flight) * cos((0.75 * flight)))), (3.0 - ((1.1…)","deck_path":"a ParametricCurve [magenta] drawn in deck (function=<function>, t_range=(0.0, <VariableNumber flight = 1.58>))","deck_rim":"a Circle [gray] drawn in deck (center=(2.7, 3.0), radius=2.0)","deck_spoke":"a Line [blue] labelled \"upright(\"painted spoke\")\" drawn in deck (start=(2.7, 3.0), end=((2.7 + (2.0 * cos(deck_angle))), (3.0 + (2.0 * sin(deck_angle)…)","disc_angle":"a VariableNumber (initial_value=0.42, format_spec='.2f')","flight":"a VariableNumber (initial_value=0.025, format_spec='.2f')","force_balance":"an Arithmetic [text] that says \"$arrow(F) m arrow(a)_B m arrow(alpha) times arrow(rho) m arrow(omega) times (arrow(omega) times arrow(rho)) 2 m arrow(omega) times arrow(v)_upright(\"rel\") m arrow(a)_upright(\"rel\")$\" (operator='-', operands=('arrow(F)', 'm arrow(a)_B', 'm arrow(alpha) times arrow(rho)',…, result='m arrow(a)_upright(\"rel\")')","ground_ball":"a Point [magenta] labelled \"P\" drawn in turntable (location=((2.7 + ((1.18 * flight) * 0.4084874408841574)), (3.0 + ((1.18 …)","ground_path":"a ParametricCurve [magenta] drawn in turntable (function=<function>, t_range=(0.0, <VariableNumber flight = 1.58>))","heading_apps":"a Heading that says \"Where Coriolis Appears\"","heading_ball":"a Heading that says \"One Ball, Two Stories\"","heading_fict":"a Heading that says \"The Forces You Feel\"","heading_recap":"a Heading that says \"Five Terms, Once More\"","hub":"a Point [text] labelled \"B\" drawn in turntable (location=(2.7, 3.0))","label_deck":"a Tex [text] that says \"Seen from the disc\" (underline=True)","label_ground":"a Tex [text] that says \"Seen from the ground\" (underline=True)","launch_spoke":"a Line [gray] labelled \"upright(\"launch spoke\")\" drawn in turntable (start=(2.7, 3.0), end=(3.516974881768315, 4.825527880521042), dashed=True)","point":"a Point [magenta] labelled \"P\" drawn in turntable (location=((2.7 + (1.22 * cos(disc_angle))), (3.0 + (1.22 * sin(disc_angl…)","rho":"an Arrow [red] labelled \"arrow(rho)\" drawn in turntable (start=(2.7, 3.0), end=((2.7 + (1.22 * cos(disc_angle))), (3.0 + (1.22 * sin(disc_angl…)","rim":"a Circle [gray] drawn in turntable (center=(2.7, 3.0), radius=2.0)","spin":"a CurvedArrow [blue] labelled \"arrow(omega)\" drawn in turntable (start=(1.24, 4.54), end=(0.92, 3.14), bend=0.44)","spoke":"a Line [blue] labelled \"upright(\"current spoke\")\" drawn in turntable (start=(2.7, 3.0), end=((2.7 + (2.0 * cos(disc_angle))), (3.0 + (2.0 * sin(disc_angle)…)","steady_work":"a Derivation [text] that says \"$m arrow(a)_upright(\"rel\") &= arrow(F) + m omega^2 arrow(rho) \\ & thin - 2 m arrow(omega) times arrow(v)_upright(\"rel\")$\"","terms_table":"a Table [text] that says \"Term What it asks about $arrow(a)_B$ where the frame's origin is going $arrow(alpha) times arrow(rho)$ whether the spin rate is changing $arrow(omega) times (arrow(omega) times arrow(rho))$ how far out you stand $2 arrow(omega) times arrow…\" (rows=(('Term', 'What it asks about'), ('$arrow(a)_B$', \"where the fr…, header=True)","turntable":"a Figure (x_range=(0.0, 5.4), y_range=(0.0, 5.6), aspect=(5.4, 5.6))"},"beats":[{"start":467.39520833333324,"say":"Newton's second law is written in the fixed frame, but the rider reports a rel. Solving for that measured acceleration begins with the applied force F while the disc continues to turn.","live":[],"does":[[467.39520833333324,"heading_fict is shown on the screen, written out."],[467.39520833333324,"turntable is shown on the screen, written out."],[470.59520833333323,"point is shown on the screen, written out."],[470.59520833333323,"rho is shown on the screen, written out."],[475.55520833333327,"turntable moves to a new place on the board."],[475.55520833333327,"force_balance is shown on the screen, written out."],[476.75520833333326,"rim is shown on the screen, written out."],[476.75520833333326,"hub is shown on the screen, written out."],[477.15520833333323,"point is redrawn as the numbers it depends on change."],[477.15520833333323,"rho is redrawn as the numbers it depends on change."],[477.15520833333323,"centrifugal is redrawn as the numbers it depends on change."],[477.15520833333323,"disc_angle ticks to 0.72."],[477.7952083333332,"spoke is redrawn as the numbers it depends on change."],[477.7952083333332,"spoke is shown on the screen, written out."],[477.7952083333332,"spin is shown on the screen, written out."]]},{"start":479.07270833333325,"say":"Subtract the origin term, the Euler term, the centripetal term, and the Coriolis term from F. The rule leaves m times a rel beneath it.","live":["turntable","heading_fict","rim","hub","spoke","point","rho","spin"],"does":[[479.99520833333327,"force_balance is shown on the screen, written out."],[481.03520833333323,"force_balance is shown on the screen, written out."],[481.99520833333327,"force_balance is shown on the screen, written out."],[483.43520833333326,"force_balance is shown on the screen, written out."],[485.59520833333323,"force_balance is shown on the screen, drawn."],[486.6843453891708,"force_balance is shown on the screen, drawn."],[487.11520833333327,"force_balance is shown on the screen, written out."]]},{"start":489.1627083333332,"say":"Every subtracted quantity has units of force and occupies a force slot in the rider's equation. Their minus signs record the acceleration terms moved from the fixed-frame side.","live":null,"does":[[489.71520833333324,"force_balance is emphasized."],[489.71520833333324,"force_balance is emphasized."],[489.71520833333324,"force_balance is emphasized."],[489.71520833333324,"force_balance is emphasized."]]},{"start":500.08770833333324,"say":"Bolt the turntable to the floor and hold a steady omega. Then a B is zero and alpha is zero, so those two terms disappear. The centripetal and Coriolis terms survive.","live":null,"does":[[500.08770833333324,"force_balance is no longer emphasized."],[500.08770833333324,"force_balance is no longer emphasized."],[500.08770833333324,"force_balance is no longer emphasized."],[500.08770833333324,"force_balance is no longer emphasized."],[502.5552083333332,"spoke is redrawn as the numbers it depends on change."],[502.5552083333332,"point is redrawn as the numbers it depends on change."],[502.5552083333332,"rho is redrawn as the numbers it depends on change."],[502.5552083333332,"centrifugal is redrawn as the numbers it depends on change."],[502.5552083333332,"disc_angle ticks to 0.94."],[504.31520833333326,"force_balance is slashed through — it cancels."],[505.51520833333325,"force_balance is slashed through — it cancels."],[508.63520833333325,"steady_work is shown on the screen, written out."],[509.43520833333326,"steady_work is shown on the screen, written out."]]},{"start":511.7252083333332,"say":"The centripetal term changes sign when it crosses the equality. It becomes plus m omega squared rho, directed outward from the axis and increasing with distance from the centre. This is the centrifugal term.","live":null,"does":[[516.6752083333332,"steady_work (the \"m omega^2 arrow(rho)\" part) is emphasized."],[518.5952083333333,"centrifugal is shown on the screen, written out."],[520.8752083333333,"spoke is redrawn as the numbers it depends on change."],[520.8752083333333,"point is redrawn as the numbers it depends on change."],[520.8752083333333,"rho is redrawn as the numbers it depends on change."],[520.8752083333333,"centrifugal is redrawn as the numbers it depends on change."],[520.8752083333333,"disc_angle ticks to 1.04."],[522.8352083333332,"centrifugal is emphasized."]]},{"start":524.6502083333332,"say":"Centrifugal force is not an additional interaction. It is the inward centripetal acceleration moved across the equality and read from the rotating frame.","live":["turntable","heading_fict","rim","hub","spoke","point","rho","spin","centrifugal"],"does":[]},{"start":534.3577083333332,"say":"The remaining correction is minus two m omega cross v rel, the Coriolis force. Its direction depends on how the body moves across the rotating disc.","live":null,"does":[[534.3577083333332,"centrifugal is no longer emphasized."],[538.8352083333332,"steady_work (the \"m omega^2 arrow(rho)\" part) is no longer emphasized."],[538.8352083333332,"steady_work (the \"2 m arrow(omega) times arrow(v)_upright(\"rel\")\" part) is emphasized."],[542.1952083333332,"spoke is redrawn as the numbers it depends on change."],[542.1952083333332,"point is redrawn as the numbers it depends on change."],[542.1952083333332,"rho is redrawn as the numbers it depends on change."],[542.1952083333332,"centrifugal is redrawn as the numbers it depends on change."],[542.1952083333332,"disc_angle ticks to 1.15."],[542.5152083333332,"rho is hidden from the screen."],[542.5152083333332,"centrifugal is hidden from the screen."],[542.5152083333332,"point is hidden from the screen."],[544.1327083333332,"turntable moves to a new place on the board."],[544.1327083333332,"force_balance is hidden from the screen — left the board."],[544.1327083333332,"heading_fict is hidden from the screen — left the board."],[544.1327083333332,"steady_work is hidden from the screen — left the board."]]},{"start":544.7327083333332,"say":"The painted launch spoke fixes one initial direction. A single ball leaves the centre with no sideways force, and the same flight is measured from the ground and from the disc.","live":["turntable","rim","hub","spoke","spin"],"does":[[544.7327083333332,"heading_ball is shown on the screen, written out."],[544.7327083333332,"deck is shown on the screen, written out."],[545.1152083333333,"deck_spoke is shown on the screen, written out."],[545.5152083333332,"launch_spoke is shown on the screen, written out."],[548.8752083333333,"ground_path is shown on the screen, written out."],[548.8752083333333,"deck_path is shown on the screen, written out."],[548.8752083333333,"ground_ball is shown on the screen, written out."],[548.8752083333333,"deck_ball is shown on the screen, written out."],[549.1952083333332,"spoke is redrawn as the numbers it depends on change."],[549.1952083333332,"ground_path is redrawn as the numbers it depends on change."],[549.1952083333332,"deck_path is redrawn as the numbers it depends on change."],[549.1952083333332,"ground_ball is redrawn as the numbers it depends on change."],[549.1952083333332,"deck_ball is redrawn as the numbers it depends on change."],[549.1952083333332,"flight ticks to 1.58."],[549.1952083333332,"disc_angle ticks to 2.33."],[553.2752083333332,"label_ground is shown on the screen, written out."],[554.3152083333332,"label_deck is shown on the screen, written out."],[554.3152083333332,"deck_rim is shown on the screen, written out."]]},{"start":555.5377083333333,"say":"From the ground, the ball advances at constant speed along a straight inertial path. Its position stays on the launch direction while the blue current spoke turns underneath it.","live":["turntable","rim","hub","spoke","spin","label_ground","label_deck","deck","heading_ball","deck_rim","launch_spoke","deck_spoke","ground_path","deck_path","ground_ball","deck_ball"],"does":[[558.9952083333333,"ground_path is emphasized."],[562.1152083333333,"ground_path is no longer emphasized."],[562.1152083333333,"launch_spoke is emphasized."],[563.5552083333332,"spoke is emphasized."]]},{"start":566.0102083333333,"say":"The gray launch spoke marks the disc at release. The blue current spoke is the same painted radius later in the flight, so their labels distinguish the two instants.","live":null,"does":[]},{"start":576.0852083333332,"say":"From the disc, the painted spoke stays fixed. The identical ball traces a curved path because every later position is resolved against axes that have turned.","live":null,"does":[[576.0852083333332,"launch_spoke is no longer emphasized."],[576.0852083333332,"spoke is no longer emphasized."],[577.3152083333332,"deck_spoke is emphasized."],[580.9152083333332,"deck_spoke is no longer emphasized."],[580.9152083333332,"deck_path is emphasized."]]},{"start":586.3827083333332,"say":"The disc-frame path bends to the right of the ball's motion all the way outward. The Coriolis correction persists because the relative velocity persists.","live":null,"does":[[588.9952083333333,"deck_ball is emphasized."]]},{"start":595.8902083333332,"say":"No sideways interaction acts on the ball. The curvature belongs to the rotating coordinates, which is why a rider assigns it a Coriolis force.","live":null,"does":[[595.8902083333332,"deck_path is no longer emphasized."],[598.1152083333333,"ground_ball is emphasized."],[604.3727083333332,"deck is hidden from the screen — left the board."],[604.3727083333332,"deck_rim is hidden from the screen — deck left the board."],[604.3727083333332,"deck_spoke is hidden from the screen — deck left the board."],[604.3727083333332,"deck_path is hidden from the screen — deck left the board."],[604.3727083333332,"deck_ball is hidden from the screen — deck left the board."],[604.3727083333332,"heading_ball is hidden from the screen — left the board."],[604.3727083333332,"label_deck is hidden from the screen — left the board."],[604.3727083333332,"label_ground is hidden from the screen — left the board."],[604.3727083333332,"turntable is hidden from the screen — left the board."],[604.3727083333332,"rim is hidden from the screen — turntable left the board."],[604.3727083333332,"hub is hidden from the screen — turntable left the board."],[604.3727083333332,"spoke is hidden from the screen — turntable left the board."],[604.3727083333332,"spin is hidden from the screen — turntable left the board."],[604.3727083333332,"launch_spoke is hidden from the screen — turntable left the board."],[604.3727083333332,"ground_path is hidden from the screen — turntable left the board."],[604.3727083333332,"ground_ball is hidden from the screen — turntable left the board."]]},{"start":604.9727083333332,"say":"The same correction matters in large weather systems, in flow across a pressure gradient, in long-range gunnery, and in the slow turn of a Foucault pendulum.","live":[],"does":[[604.9727083333332,"heading_apps is shown on the screen, written out."],[604.9727083333332,"applications is shown on the screen, written out."],[606.8352083333333,"applications (the \"Large weather systems\" part) is emphasized."],[608.9552083333333,"applications (the \"Large weather systems\" part) is no longer emphasized."],[608.9552083333333,"applications (the \"pressure gradient\" part) is emphasized."],[610.1552083333332,"applications (the \"Long-range gunnery\" part) is emphasized."],[610.1552083333332,"applications (the \"pressure gradient\" part) is no longer emphasized."],[613.0352083333332,"applications (the \"Foucault pendulum\" part) is emphasized."],[613.0352083333332,"applications (the \"Long-range gunnery\" part) is no longer emphasized."],[614.1727083333333,"applications is hidden from the screen — left the board."],[614.1727083333333,"heading_apps is hidden from the screen — left the board."]]},{"start":614.7727083333332,"say":"The final relation has five terms, and each has a distinct physical question attached to it.","live":null,"does":[[614.7727083333332,"heading_recap is shown on the screen, written out."],[616.2752083333332,"terms_table is shown on the screen, written out."]]},{"start":620.9727083333332,"say":"The first asks how the frame origin moves. The second asks whether omega changes. The third asks how far out the point lies. Coriolis asks for relative motion, and the fifth is the rider's own acceleration.","live":["heading_recap"],"does":[[621.5152083333332,"terms_table is shown on the screen, written out."],[624.2352083333333,"terms_table is shown on the screen, written out."],[626.7952083333333,"terms_table is shown on the screen, written out."],[629.7552083333333,"terms_table is shown on the screen, written out."],[632.6352083333334,"terms_table is shown on the screen, written out."]]},{"start":635.5277083333333,"say":"Every term follows from one fact about a painted basis vector: its derivative is omega crossed with itself. Apply that fact twice with the product rule, and the rotating-frame acceleration is complete.","live":null,"does":[[637.1552083333333,"closing is shown on the screen, written out."],[640.7552083333333,"closing (the \"arrow(omega) times hat(e)\" part) is emphasized."],[642.8352083333333,"closing (the \"arrow(omega) times hat(e)\" part) is no longer emphasized."],[642.8352083333333,"A box is drawn around closing."],[648.0860208333332,"closing is hidden from the screen — left the board."],[648.0860208333332,"heading_recap is hidden from the screen — left the board."],[648.0860208333332,"terms_table is hidden from the screen — left the board."]]}]}]},"durationSeconds":649,"chapters":[{"title":"Two Frames, One Motion","startSeconds":0,"narration":"Two observers watch the same particle. One uses fixed axes bolted to the floor. The other stands on a turntable with its own painted x and y axes. Their acceleration measurements do not agree. The difference is physical, and its two famous names are centrifugal and Coriolis. Both begin with the fact that the second observer's axes are spinning. A reference frame means axes together with a clock. Capital X and capital Y belong to the fixed frame, with origin O, and nothing in that frame turns. The rotating frame has origin B. Its little x and little y directions are painted on the surface, so they turn with the disc at angular velocity omega. Now let P wander outward while the turntable keeps turning. Its position from O is r sub P. The moving origin sits at r sub B. The displacement from B to P, measured in the painted coordinates, is rho. Geometry gives r P equals r B plus rho. These are the same three arrows beside the equation: first r P, then r B, then rho, joined head to tail. Differentiate that identity in the fixed frame. Rho is written in the painted basis, and the basis turns while its components change. The derivative therefore acts on both the numbers and the axes. The task is now precise: differentiate this moving triangle honestly, twice. The extra acceleration terms will come from the turning basis."},{"title":"Differentiating in a Turning Basis","startSeconds":93.28920833333333,"narration":"Everything hard about rotating axes begins with one question: how fast does a painted axis change as its disc turns? The unit vector i hat has length one, so its tip stays on the circle. Its turned-through angle is theta. Only after that geometry is fixed do its components read cosine theta, sine theta, zero. Differentiate those components. The chain rule gives theta dot times the bracket minus sine theta, cosine theta, zero. That bracket is j hat, the second painted axis. It is exactly ninety degrees ahead of i hat on the same disc. Theta dot is the spin rate omega, so the derivative of i hat is omega times j hat. In vector form, omega points along the rotation axis. Omega crossed with i hat therefore points along j hat, which gives the same derivative. As the disc turns, i hat never changes length. Its derivative remains sideways, at right angles to i hat, at every instant. That perpendicular sweep is the geometry of the cross product. The same reasoning holds for any vector rigidly attached to the body: a painted axis, a spoke, or a robot arm. Its derivative is omega crossed with that vector. Now take an arbitrary vector A and write it in the painted basis as A x i hat, A y j hat, plus A z k hat. The product rule separates its fixed-frame derivative into two families. One differentiates the components and leaves the basis vectors alone. A rider on the disc calls that first family the relative derivative, because those are the component changes measured against the painted axes. The second family leaves the components alone and differentiates i hat, j hat, and k hat. Each basis derivative contributes omega crossed with the corresponding basis vector. Factor omega cross from that whole family. The remaining bracket is the original vector A, so the family becomes omega cross A. This is the transport theorem. A vector's fixed-frame rate equals its relative rate plus omega cross the vector. Applying this statement twice produces every acceleration term that follows."},{"title":"Velocity, Term by Term","startSeconds":229.85841666666664,"narration":"One line of geometry and one use of the transport theorem give the velocity relation. Begin with r P equals r B plus rho. Differentiate both sides in the fixed frame. The left side becomes the true velocity v P, and the first term on the right becomes the velocity v B of the moving origin. Rho needs the theorem. The red rho vector lies in the disc. The blue omega vector points along the rotation axis. Their yellow cross product is the extra velocity carried by the turning frame. Omega cross rho is at right angles to rho and omega. It lies flat in the plane of the disc, ninety degrees ahead of rho, with magnitude omega times rho. The complete velocity therefore has three pieces: motion of the origin, motion measured on the disc, and motion carried by the disc's spin. A beetle walks outward along a painted spoke while the turntable spins. The three velocity pieces form one head-to-tail chain. First is v B, the velocity of the frame's origin. A bolted turntable has v B equal to zero, while an aircraft or planet need not. Second is omega cross rho. It is tangential and square to the spoke, the velocity the beetle receives simply because the disc carries it. Third is v rel, directed outward along the spoke. Translate that vector to the tip of omega cross rho, and the two arrows now meet head to tail. The arrow from the chain's first tail to its final tip is v P, the beetle's true velocity over the ground. One more derivative produces the acceleration terms."},{"title":"Acceleration: Five Terms","startSeconds":331.6357916666666,"narration":"Differentiate the velocity relation once more. Each vector on the right is measured against the turning disc, so the transport theorem remains beside the calculation. The origin contributes a B directly. The other two derivatives need the theorem because omega cross rho and v rel are both vectors written in the painted basis. Apply the product rule to omega cross rho. Omega dot becomes alpha, so the first contribution is alpha cross rho. It is tangential while the spin rate is changing. The second contribution contains the fixed derivative of rho. Substitute v rel plus omega cross rho, then distribute omega cross across both terms. The fixed derivative of v rel also uses the theorem. It contributes a rel plus a second omega cross v rel. The distributed line contains one omega cross v rel. The final derivative contains another omega cross v rel. These two identical terms add. Their sum is two omega cross v rel. The factor of two is the Coriolis coefficient, produced by two separate derivatives. Collecting the result gives five acceleration terms, all expressed in quantities available to the rotating observer. First, a B is the acceleration of the frame's origin. A floor-mounted turntable makes it zero, but a vehicle or planet need not. Second, alpha cross rho is the Euler term. It appears while the spin rate changes and vanishes at steady omega. Third, omega cross omega cross rho is centripetal. It points from P toward the rotation axis, with magnitude omega squared times the distance out. Fourth, two omega cross v rel is Coriolis. It needs a turning frame and motion relative to that frame at the same instant. Fifth, a rel is the acceleration measured by instruments fixed to the disc. It is often the unknown the rotating observer must determine. The Coriolis condition is now exact. Relative motion without rotation gives zero. Rotation without relative motion also gives zero. Only the final case has both, so only that case carries a Coriolis acceleration."},{"title":"The Forces You Feel","startSeconds":467.39520833333324,"narration":"Newton's second law is written in the fixed frame, but the rider reports a rel. Solving for that measured acceleration begins with the applied force F while the disc continues to turn. Subtract the origin term, the Euler term, the centripetal term, and the Coriolis term from F. The rule leaves m times a rel beneath it. Every subtracted quantity has units of force and occupies a force slot in the rider's equation. Their minus signs record the acceleration terms moved from the fixed-frame side. Bolt the turntable to the floor and hold a steady omega. Then a B is zero and alpha is zero, so those two terms disappear. The centripetal and Coriolis terms survive. The centripetal term changes sign when it crosses the equality. It becomes plus m omega squared rho, directed outward from the axis and increasing with distance from the centre. This is the centrifugal term. Centrifugal force is not an additional interaction. It is the inward centripetal acceleration moved across the equality and read from the rotating frame. The remaining correction is minus two m omega cross v rel, the Coriolis force. Its direction depends on how the body moves across the rotating disc. The painted launch spoke fixes one initial direction. A single ball leaves the centre with no sideways force, and the same flight is measured from the ground and from the disc. From the ground, the ball advances at constant speed along a straight inertial path. Its position stays on the launch direction while the blue current spoke turns underneath it. The gray launch spoke marks the disc at release. The blue current spoke is the same painted radius later in the flight, so their labels distinguish the two instants. From the disc, the painted spoke stays fixed. The identical ball traces a curved path because every later position is resolved against axes that have turned. The disc-frame path bends to the right of the ball's motion all the way outward. The Coriolis correction persists because the relative velocity persists. No sideways interaction acts on the ball. The curvature belongs to the rotating coordinates, which is why a rider assigns it a Coriolis force. The same correction matters in large weather systems, in flow across a pressure gradient, in long-range gunnery, and in the slow turn of a Foucault pendulum. The final relation has five terms, and each has a distinct physical question attached to it. The first asks how the frame origin moves. The second asks whether omega changes. The third asks how far out the point lies. Coriolis asks for relative motion, and the fifth is the rider's own acceleration. Every term follows from one fact about a painted basis vector: its derivative is omega crossed with itself. Apply that fact twice with the product rule, and the rotating-frame acceleration is complete."}]}}