From 2fccfdcddc97eb8c0965733b6eb429f881c8b823 Mon Sep 17 00:00:00 2001 From: CAIMEOX <1853884864@qq.com> Date: Sat, 15 Jun 2024 16:56:14 +0800 Subject: [PATCH] improve navigation harmonic motion --- trees/def.tree | 37 --------- trees/notes.tree | 4 +- trees/phy/phy-0005.tree | 164 +++++++++++++++++++++++++++++++++++++- trees/proj/proj-0005.tree | 5 ++ trees/projects.tree | 3 +- 5 files changed, 172 insertions(+), 41 deletions(-) delete mode 100644 trees/def.tree create mode 100644 trees/proj/proj-0005.tree diff --git a/trees/def.tree b/trees/def.tree deleted file mode 100644 index e3207ca7..00000000 --- a/trees/def.tree +++ /dev/null @@ -1,37 +0,0 @@ -\title{Definitions Collection} -\transclude{def-0001} -\transclude{def-0002} -\transclude{def-0003} -\transclude{def-0004} -\transclude{def-0005} -\transclude{def-0006} -\transclude{def-0007} -\transclude{def-0008} -\transclude{def-0009} -\transclude{def-000A} -\transclude{def-000B} -\transclude{def-000C} -\transclude{def-000D} -\transclude{def-000E} -\transclude{def-000F} -\transclude{def-000G} -\transclude{def-000H} -\transclude{def-000I} -\transclude{def-000J} -\transclude{def-000K} -\transclude{def-000L} -\transclude{def-000M} -\transclude{def-000N} -\transclude{def-000O} -\transclude{def-000P} -\transclude{def-000Q} -\transclude{def-000R} -\transclude{def-000S} -\transclude{def-000T} -\transclude{def-000U} -\transclude{def-000V} -\transclude{def-000W} -\transclude{def-000X} -\transclude{def-000Y} -\transclude{def-000Z} -\transclude{def-0010} \ No newline at end of file diff --git a/trees/notes.tree b/trees/notes.tree index 1ac3efb6..0c304a2f 100644 --- a/trees/notes.tree +++ b/trees/notes.tree @@ -1,6 +1,7 @@ \title{Notes} \tag{top} \put\transclude/expanded{false} +\put\transclude/toc{false} \transclude{tt-0001} \transclude{math-0003} \transclude{math-0004} @@ -17,4 +18,5 @@ \transclude{cs-0007} \transclude{math-0007} \transclude{math-0008} -\transclude{phy-0004} \ No newline at end of file +\transclude{phy-0004} +\transclude{phy-0005} \ No newline at end of file diff --git a/trees/phy/phy-0005.tree b/trees/phy/phy-0005.tree index 188ae3e9..bf7cf3b7 100644 --- a/trees/phy/phy-0005.tree +++ b/trees/phy/phy-0005.tree @@ -1,3 +1,163 @@ -\title{Period Motion} +\title{Simple Harmonic Motion} \taxon{Physics} -\p{} \ No newline at end of file +\import{macros} +\p{ + The SHM note is based on Wikipedia. +} +\p{ + In mechanics and physics, \strong{simple harmonic motion} is a special type of periodic motion an object + experiences by means of a \strong{restoring force} whose magnitude is directly proportional to the + distance of the object from an \strong{equilibrium position} and acts towards the equilibrium position. +} +\p{ + In Newtonian mechanics, for one-dimensional simple harmonic motion, the equation of motion, + which is a second-order linear ordinary differential equation with constant coefficients, + can be obtaind by Hooke's law and Newton's second law. + ##{ + F_{\mathrm {net} }=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-kx, + } + where #{m} is the inertial mass og the oscillating body, #{x} is its displacement from + the equilibrium position and #{k} is a constant. Therefore we have + ##{ + {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-{\frac {k}{m}}x, + } + solving the differential equation we get the sinusoidal function + ##{ + x(t)=C_{1}\cos \left(\omega t\right)+C_{2}\sin \left(\omega t\right), + } + where #{\omega=\sqrt{\frac{k}{m}}}. +} +\p{ + Let #{t=0} we see that #{C_1 = x(0)} so that #{C_1} is the initial position. + Taking the derivative of the equation and evaluating at #{0} we get + #{x'(0) = \omega C_2}. So #{C_2} is the initial speed of the object + diverged by the angular frequency, #{C_2 = \frac{v_0}{\omega}}. Thus + ##{ + x(t)=x_{0}\cos \left({\sqrt {\frac {k}{m}}}t\right)+{\frac {v_{0}}{\sqrt {\frac {k}{m}}}}\sin \left({\sqrt {\frac {k}{m}}}t\right). + } +} +\p{ + This equation can also be written in the form: + ##{ + x(t) = Acos(\omega t-\phi) + } + where + ##{ + \begin{align*} + A = \sqrt{C_{1}^{2}+C_{2}^{2}} \\ + \phi = \arctan\left(\frac{C_{2}}{C_{1}}\right) \\ + \sin\phi = \frac{C_{2}}{A} \\ + \cos\phi = \frac{C_{1}}{A} + \end{align*} + } + Each of these constants carries a physical meaning of the motion: + #{A} is the \strong{amplitude} and #{\omega = 2\pi f} is the \strong{angular frequency} + and #{\phi} is the initial \strong{phase}. +} +\subtree{ + \title{Energy} + \p{ + The kinetic energy of the object at time #{t} is given by + ##{ + K(t)={\tfrac {1}{2}}mv^{2}(t)={\tfrac {1}{2}}m\omega ^{2}A^{2}\sin ^{2}(\omega t-\varphi )={\tfrac {1}{2}}kA^{2}\sin ^{2}(\omega t-\varphi ) + } + Besides the kinetic energy, the potential energy of the object at time #{t} is given by + ##{ + U(t)={\tfrac {1}{2}}kx^{2}(t)={\tfrac {1}{2}}kA^{2}\cos ^{2}(\omega t-\varphi ) + } + The total energy of the object is the sum of the kinetic and potential energies + ##{ + E=K+U={\tfrac {1}{2}}kA^{2} + } + which is a constant value. + Notice that if we solve #{v} from the energy equation we get + ##{ + v = \pm\sqrt{\frac{k}{m}(A^{2}-x^{2})} = \pm\omega\sqrt{A^{2}-x^{2}} + } + which implies that the velocity is maximum when the displacement is zero and vice versa. + } +} +\subtree{ + \title{Superposition} + \p{ + According to the principle of superposition of SHM, the resultant displacement + of a number of waves in a medium at a particular point is the vector sum of the individual + displacements produced by each of the waves at that point. + Consider two waves having the same angular frequency (Suppose #{\phi_2 > \phi_1}) in the same line: + ##{ + x_{1}(t)=A_{1}\cos(\omega t+\phi_{1}) \\ + x_{2}(t)=A_{2}\cos(\omega t+\phi_{2}) + } + Use vector addition we can easily compute the resultant displacement + ##{ + A = \sqrt{A_{1}^{2}+A_{2}^{2}+2A_{1}A_{2}\cos(\phi_{2}-\phi_{1})} \\ + } + and the resultant initial phase + ##{ + \phi = \arctan\left(\frac{A_{1}\sin\phi_{1}+A_{2}\sin\phi_{2}}{A_{1}\cos\phi_{1}+A_{2}\cos\phi_{2}}\right) + } + } + \p{ + For some special case, the resultant displacement can be simplified: + \ul{ + \li{ + If #{\phi_2 - \phi_1 = 2k\pi, k \in\Z} then the resultant displacement is + ##{ + A = A_{1}+A_{2} + } + } + \li{ + If #{\phi_2 - \phi_1 = (2k+1)\pi, k \in\Z} then the resultant displacement is + ##{ + A = |A_{1}-A_{2}| + } + } + } + } + \p{ + If the angular frequencies are different, the resultant displacement changes with time. + For instance, given ##{ + x_1 = A_1\cos(\omega_1t+\phi_1) \\ + x_2 = A_2\cos(\omega_2t+\phi_2) + } + the resultant displacement is + ##{ + A = \sqrt{A_{1}^{2}+A_{2}^{2}+2A_{1}A_{2}\cos((\omega_{2}-\omega_{1})t+\phi_{2}-\phi_{1})} + } + } + \p{ + If two waves are perpendicular to each other + ##{ + x = A\cos(\omega t + \alpha) + \\ + y = B\cos(\omega t + \beta) + } + we can compute that + ##{ + \frac{x^2}{A^2} + \frac{y^2}{B^2} - + \frac{xy}{AB}\cos(\beta-\alpha) = \sin^2(\beta-\alpha) + } + which is the equation of an ellipse. + } + \ul{ + \li{ + #{\beta - \alpha = 0 \text{ or } \pi} the ellipse becomes two line: + ##{ + (\frac{x}{A}\pm\frac{y}{B})^2 = 0 \implies y = \pm \frac{B}{A}x + } + The trajectory is two straight line cross the origin. + The resultant amplitude is #{C = A^2 + B^2} + } + \li{ + #{\beta - \alpha = \pm\frac{\pi}{2}} the ellipse becomes a regular ellipse, + i.e., the ellipse that takes the coordinates axis as its major axis. + ##{ + \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1 + } + If #{\beta - \alpha > 0} the ellipse is clockwise, otherwise it is counter-clockwise. + } + } +} +\subtree{ + \title{Wave} +} \ No newline at end of file diff --git a/trees/proj/proj-0005.tree b/trees/proj/proj-0005.tree new file mode 100644 index 00000000..13ded9bd --- /dev/null +++ b/trees/proj/proj-0005.tree @@ -0,0 +1,5 @@ +\title{MoonBit Core} +\taxon{Project} +\p{ + [MoonBit Core](https://github.com/moonbitlang/core) is the standard library of the MoonBit language. +} diff --git a/trees/projects.tree b/trees/projects.tree index ec3d1053..528b5d5b 100644 --- a/trees/projects.tree +++ b/trees/projects.tree @@ -4,4 +4,5 @@ \transclude{proj-0001} \transclude{proj-0002} \transclude{proj-0003} -\transclude{proj-0004} \ No newline at end of file +\transclude{proj-0004} +\transclude{proj-0005} \ No newline at end of file