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Scenes to show mismatch between approximated pendulum and real
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Grant Sanderson
parent
ac9c04d3b8
commit
be2ce981d9
3 changed files with 229 additions and 0 deletions
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@ -4,6 +4,12 @@ from active_projects.ode.part1.staging import *
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OUTPUT_DIRECTORY = "ode/part1"
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ALL_SCENE_CLASSES = [
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IntroducePendulum,
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FormulasAreLies,
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HighAnglePendulum,
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MediumAnglePendulum,
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LowAnglePendulum,
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ApproxWordsLowAnglePendulum,
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FailedApproxWordsHighAnglePendulum,
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# Tests
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PendulumTest,
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VectorFieldTest,
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@ -1,6 +1,15 @@
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from big_ol_pile_of_manim_imports import *
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Lg_formula_config = {
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"tex_to_color_map": {
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"\\theta_0": WHITE,
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"{L}": BLUE,
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"{g}": YELLOW,
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},
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}
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class Pendulum(VGroup):
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CONFIG = {
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"length": 3,
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@ -22,6 +31,9 @@ class Pendulum(VGroup):
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"fill_color": GREY_BROWN,
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"sheen_direction": UL,
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"sheen_factor": 0.5,
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"background_stroke_color": BLACK,
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"background_stroke_width": 3,
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"background_stroke_opacity": 0.5,
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},
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"dashed_line_config": {
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"num_dashes": 25,
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@ -514,6 +526,139 @@ class IntroducePendulum(MovingCameraScene):
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self.wait(6)
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class LowAnglePendulum(Scene):
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CONFIG = {
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"pendulum_config": {
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"initial_theta": 20 * DEGREES,
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"length": 2.0,
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"damping": 0,
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"top_point": ORIGIN,
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},
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"axes_config": {
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"y_axis_config": {"unit_size": 0.75},
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"x_axis_config": {
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"unit_size": 0.5,
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"numbers_to_show": range(2, 25, 2),
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"number_scale_val": 0.5,
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},
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"x_max": 25,
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"number_line_config": {
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"tip_length": 0.3,
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"stroke_width": 2,
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}
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},
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"axes_corner": UL,
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}
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def construct(self):
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pendulum = Pendulum(**self.pendulum_config)
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axes = ThetaVsTAxes(**self.axes_config)
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axes.center()
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axes.to_corner(self.axes_corner, buff=LARGE_BUFF)
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graph = axes.get_live_drawn_graph(pendulum)
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L = pendulum.length
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g = pendulum.gravity
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theta0 = pendulum.initial_theta
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prediction = axes.get_graph(
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lambda t: theta0 * np.cos(t * np.sqrt(g / L))
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)
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dashed_prediction = DashedVMobject(prediction, num_dashes=300)
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dashed_prediction.set_stroke(WHITE, 1)
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prediction_formula = TexMobject(
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"\\theta_0", "\\cos(\\sqrt{g / L} \\cdot t)"
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)
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prediction_formula.scale(0.75)
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prediction_formula.next_to(
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dashed_prediction, UP, SMALL_BUFF,
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)
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theta0 = prediction_formula.get_part_by_tex("\\theta_0")
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theta0_brace = Brace(theta0, UP, buff=SMALL_BUFF)
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theta0_brace.stretch(0.5, 1, about_edge=DOWN)
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theta0_label = Integer(
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pendulum.initial_theta * 180 / PI,
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unit="^\\circ"
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)
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theta0_label.scale(0.75)
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theta0_label.next_to(theta0_brace, UP, SMALL_BUFF)
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group = VGroup(theta0_brace, theta0_label, prediction_formula)
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group.shift_onto_screen(buff=MED_SMALL_BUFF)
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self.add(axes, dashed_prediction, pendulum)
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self.play(
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ShowCreation(dashed_prediction, run_time=2),
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FadeInFromDown(prediction_formula),
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FadeInFromDown(theta0_brace),
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FadeInFromDown(theta0_label),
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)
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self.play(
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ShowCreationThenFadeAround(theta0_label),
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ShowCreationThenFadeAround(pendulum.theta_label),
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)
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self.wait()
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pendulum.start_swinging()
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self.add(graph)
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self.wait(30)
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class ApproxWordsLowAnglePendulum(Scene):
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def construct(self):
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period = TexMobject(
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"\\text{Period}", "\\approx",
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"2\\pi \\sqrt{\\,{L} / {g}}",
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**Lg_formula_config
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)
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checkmark = TexMobject("\\checkmark")
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checkmark.set_color(GREEN)
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checkmark.scale(2)
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checkmark.next_to(period, RIGHT, MED_LARGE_BUFF)
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self.add(period, checkmark)
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class MediumAnglePendulum(LowAnglePendulum):
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CONFIG = {
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"pendulum_config": {
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"initial_theta": 50 * DEGREES,
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"n_steps_per_frame": 1000,
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},
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"axes_config": {
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"y_axis_config": {"unit_size": 0.75},
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"y_max": PI / 2,
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"y_min": -PI / 2,
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"number_line_config": {
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"tip_length": 0.3,
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"stroke_width": 2,
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}
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},
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"pendulum_shift_vect": 1 * RIGHT,
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}
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class HighAnglePendulum(LowAnglePendulum):
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CONFIG = {
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"pendulum_config": {
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"initial_theta": 175 * DEGREES,
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"n_steps_per_frame": 1000,
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"top_point": 1.5 * DOWN,
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"length": 2,
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},
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"axes_config": {
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"y_axis_config": {"unit_size": 0.5},
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"y_max": PI,
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"y_min": -PI,
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"number_line_config": {
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"tip_length": 0.3,
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"stroke_width": 2,
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}
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},
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"pendulum_shift_vect": 1 * RIGHT,
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}
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class PendulumTest(Scene):
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def construct(self):
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pendulum = Pendulum(
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@ -12,6 +12,7 @@ def pendulum_vector_field(point, mu=0.1, g=9.8, L=3):
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# Scenes
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class VectorFieldTest(Scene):
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def construct(self):
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plane = NumberPlane(
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@ -43,6 +44,83 @@ class VectorFieldTest(Scene):
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self.wait(10)
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class FormulasAreLies(PiCreatureScene):
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def construct(self):
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morty = self.pi_creature
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t2c = {
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"{L}": BLUE,
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"{g}": YELLOW,
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"\\theta_0": WHITE,
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"\\sqrt{\\,": WHITE,
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}
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kwargs = {"tex_to_color_map": t2c}
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period_eq = TexMobject(
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"\\text{Period} = 2\\pi \\sqrt{\\,{L} / {g}}",
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**kwargs
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)
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theta_eq = TexMobject(
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"\\theta(t) = \\theta_0 \\cos\\left("
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"\\sqrt{\\,{L} / {g}} \\cdot t"
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"\\right)",
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**kwargs
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)
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equations = VGroup(theta_eq, period_eq)
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equations.arrange(DOWN, buff=LARGE_BUFF)
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for eq in period_eq, theta_eq:
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i = eq.index_of_part_by_tex("\\sqrt")
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eq.sqrt_part = eq[i:i + 4]
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theta0 = theta_eq.get_part_by_tex("\\theta_0")
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theta0_words = TextMobject("Starting angle")
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theta0_words.next_to(theta0, UL)
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theta0_words.shift(UP + 0.5 * RIGHT)
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arrow = Arrow(
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theta0_words.get_bottom(),
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theta0,
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color=WHITE,
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tip_length=0.25,
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)
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bubble = SpeechBubble()
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bubble.pin_to(morty)
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bubble.write("Lies!")
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bubble.content.scale(2)
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bubble.resize_to_content()
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self.add(period_eq)
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morty.change("pondering", period_eq)
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self.wait()
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theta_eq.remove(*theta_eq.sqrt_part)
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self.play(
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TransformFromCopy(
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period_eq.sqrt_part,
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theta_eq.sqrt_part,
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),
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FadeIn(theta_eq)
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)
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theta_eq.add(*theta_eq.sqrt_part)
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self.play(
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FadeInFrom(theta0_words, LEFT),
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GrowArrow(arrow),
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)
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self.wait()
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self.play(morty.change, "confused")
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self.wait(0)
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self.play(
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morty.change, "angry",
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ShowCreation(bubble),
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FadeInFromPoint(bubble.content, morty.mouth),
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equations.to_edge, LEFT,
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FadeOut(arrow),
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FadeOut(theta0_words),
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)
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self.wait()
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def create_pi_creature(self):
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return Mortimer().to_corner(DR)
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class NewSceneName(Scene):
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def construct(self):
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pass
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