diff --git a/TeachingExamples/SuspendedObjects-exercise.ipynb b/TeachingExamples/SuspendedObjects-exercise.ipynb index e0a1089..33a65de 100644 --- a/TeachingExamples/SuspendedObjects-exercise.ipynb +++ b/TeachingExamples/SuspendedObjects-exercise.ipynb @@ -1,302 +1,244 @@ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "

Use case scenario
\n", " This notebook is made to be used by students as an assignment or exercise, in autonomy (at home or in an exercise session).

\n", "

Features
\n", " This notebook embeds auto-corrected quizzes to engage students with the virtual demonstration and uses different types of visualisations to help students understand the phenomena.
\n", " The example chosen is voluntarily simple so that anyone can understand what is illustrated and focus the pedagogical features of the example.

\n", "

More information on using notebooks for exercises or assignements.

\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Suspended objects\n", "\n", "We consider a clothesline made of two poles and a cable.\n", "The cable is fixed on one pole. A pulley on the other pole allows to attach a counterweight to pull the cable taut. \n", "\n", "\n", "\n", "## Exercise 1\n", "Execute the cell below to activate the interactive quiz and answer the question. \n", "If you don't see the question, make sure you are logged on [moodle](https://moodle.epfl.ch/enrol/index.php?id=15917) and registered on our [Noto Community moodle page](https://moodle.epfl.ch/enrol/index.php?id=15917).\n", "\n", "" ] }, { "cell_type": "code", - "execution_count": 1, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "text/html": [ - "\n", - " \n", - " " - ], - "text/plain": [ - "" - ] - }, - "execution_count": 1, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "from IPython.display import IFrame\n", "IFrame('https://moodle.epfl.ch/mod/hvp/embed.php?id=1028285', 500, 600)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise 2\n", "\n", "The virtual lab below allows you to experiment with different counterweights to see how it affects the position of the object suspended on the clothesline. \n", "Execute the cell below to launch the virtual lab, then *answer the questions in the quiz below*." ] }, { "cell_type": "code", - "execution_count": 2, + "execution_count": null, "metadata": {}, - "outputs": [ - { - "data": { - "application/vnd.jupyter.widget-view+json": { - "model_id": "bd3abfd177574f2aa6c204469efd5739", - "version_major": 2, - "version_minor": 0 - }, - "text/plain": [ - "VBox(children=(Output(layout=Layout(margin='5px 10px')), HBox(children=(Label(value='Mass of the counterweight…" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/html": [ - "\n", - " \n", - " " - ], - "text/plain": [ - "" - ] - }, - "execution_count": 2, - "metadata": {}, - "output_type": "execute_result" - } - ], + "outputs": [], "source": [ "%matplotlib inline\n", "from lib.suspendedobjects import *\n", "SuspendedObjectsLab();\n", "IFrame('https://h5p.org/h5p/embed/584119', 1024, 350)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*If you wonder how the virtual lab works and would like to see the code, [you can have a look at it at the end of this notebook](#How-does-the-virtual-lab-work%3F).*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise 3\n", "\n", "1. Give the expression of the **mass of the counterweight $m_{cw}$ as a function of the other parameters of the problem**.\n", "2. Application: what counterweight allows to suspend wet jeans ($m = 3 kg$) on the cable so that the cable is taut at an angle of $1.5^\\circ = \\frac{\\pi}{120}$ with the horizon?\n", "3. Why is it impossible to pull the cable taut completely horizontally?\n", "\n", " \n", "\n", "## Solution\n", "\n", "### Method\n", "\n", "Given that the system is in static equilibrium, the sum of external forces exerted on the system will be zero, so using Newton's second law should be easy. The force that the counterweight exerts on the system will involve the mass of the counterweight so we should be able to rewrite Newton's second law to get an expression of the form $m_{cw} = ...$.\n", "\n", "### Hypotheses and simplifications\n", "\n", "We make the following assumptions and simplifications:\n", "* the jeans are considered as positioned exactly mid-way between the poles so the tension is equal on both sides of the cable\n", "* we represent the jeans by the point at which they are suspended\n", "* the cable is considered as rigid (not bended), with a negligible mass\n", "* the pulley is considered as perfect, without mass nor friction\n", "* we consider the static equilibrium obtained after changing the weight, once the system is stabilized\n", "\n", "### Resolution\n", "\n", "First, let's draw a diagram and represent the different forces involved.\n", "\n", "\n", "The *forces applied on the jeans* are:\n", "* the weight: $\\vec F_j = m_j \\vec g$ \n", "* the force exerted by the cable on each side of the jeans: assuming the jeans are suspended at the exact center of the cable, then the tension applied on each of the two sides is is equally distributed $\\vec T$, which combine into a vertical resulting tension $\\vec T_r = 2.\\vec T$\n", "\n", "From Newton's second law in a static equilibrium we can write: $\\sum \\vec F_j = \\vec 0$ \n", "With the forces on the jeans we get: $\\vec F_j + \\vec T_r = 0$ \n", "Using the fact that the tension is equal on both sides of the jeans we get: $\\vec F_j + 2.\\vec T = 0$\n", "\n", "If we project on $x$ and $y$ axes, we get: \n", "$\\left\\{\\begin{matrix} F_{jx} + 2.T_x = 0 \\\\ F_{jy} + 2.T_y = 0\\end{matrix}\\right. $\n", "\n", "Since the weight does not have a component on the x axis, it simplifies into: \n", "$\\left\\{\\begin{matrix} T_x = 0 \\\\ F_{jy} + 2.T_y = 0\\end{matrix}\\right. $\n", "\n", "The component of the weight on the y axis is $F_{jy} = - m_j.g$, which gives us: \n", "$\\left\\{\\begin{matrix} T_x = 0 \\\\ - m_j.g + 2.T_y = 0\\end{matrix}\\right. $\n", "\n", "Using the angle $\\alpha$ we can get the tension $T_y$ expressed as a function of T since $sin(\\alpha) = \\frac{T_y}{T}$, therefore $T_y = T.sin(\\alpha)$\n", "\n", "By replacing $T_y$ by this expression in the above equation we get: \n", "$\\left\\{\\begin{matrix} T_x = 0 \\\\ - m_j.g + 2.T.sin(\\alpha) = 0\\end{matrix}\\right. $\n", "\n", "From there we can get $T$, and this is equation number $(1)$: \n", "$T = \\frac{m_j.g}{2.sin(\\alpha)}$\n", "\n", " \n", "\n", "We now want to make the mass of the counterweight appear in this expression. \n", "So we will now look at the forces applied on the *counterweight*.\n", "\n", " \n", "\n", "The forces applied on the *counterweight* are:\n", "* the weight: $\\vec F_{cw} = m_{cw} \\vec g$ \n", "* the force exerted by the line: a simple pulley simply changes the direction of the tension so the tension applied on the counterweight is therefore $\\vec T$\n", "\n", "From Newton's second law in a static equilibrium we can write: $\\sum \\vec F_{cw} = \\vec 0$ \n", "With the forces involved in our problem : $\\vec F_{cw} + \\vec T = \\vec 0$ \n", "\n", "All forces being vertical, there is no need to project on $x$ so we get: $- F_{cw} + T = 0$ \n", "We replace the weight by its detailed expression: $-m_{cw}.g + T = 0$ \n", "Now we can express $T$ as a function of the other parameters, which is equation number $(2)$: $T = m_{cw}.g$ \n", "\n", " \n", "\n", "Let's now summarize what we have so far with equations $(1)$ and $(2)$: \n", "\n", "$\\left\\{\\begin{matrix}T = \\frac{m_j.g}{2.sin(\\alpha)} \\\\ T = m_{cw}.g\\end{matrix}\\right. $\n", "\n", "These two equations combined give us:\n", "\n", "$\\frac{m_j.g}{2.sin(\\alpha)} = m_{cw}.g$\n", "\n", "This allow us to find the mass of the counterweight as a function of the *mass of the jeans* and of the *angle that the line makes with the horizon*: \n", "\n", "$\\boxed{m_{cw} = \\frac{m_j}{2.sin(\\alpha)}}$\n", "\n", "\n", " \n", "\n", "### Application\n", "\n", "For a pair of wet jeans of $3 kg$ and an angle of $1.5^\\circ = \\frac{\\pi}{120}$, we need to put a counterweight of:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "mcw = 3 / (2 * np.sin(np.pi / 120))\n", "print(mcw)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You can **check that you get a similar result** with the virtual lab above!\n", "\n", "\n", "### Conclusion\n", "\n", "For the line to be taut completely horizontal, $\\alpha$ has to be really small i.e. really close to zero. \n", "This means that $sin(\\alpha)$ will also be close to zero, which means in turn that $m_{cw}$ will be very big.\n", "Actually, **the more we want the line to be close to the horizon, the bigger $m_{cw}$ we will need!**\n", "In fact, it is impossible to get the line taut so that it is absolutely straight...\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " \n", "\n", "---\n", "\n", "# How does the virtual lab work?\n", "\n", "If you wonder how the virtual lab works: \n", "* You can have a look at the code of the virtual lab by [opening this python file](lib/suspendedobjects.py).\n", "* You can see the documentation by executing the cell below:" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "SuspendedObjectsLab?" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.9" } }, "nbformat": 4, "nbformat_minor": 4 } diff --git a/TeachingExamples/lib/suspendedobjects.py b/TeachingExamples/lib/suspendedobjects.py index f979eb9..3373aaf 100644 --- a/TeachingExamples/lib/suspendedobjects.py +++ b/TeachingExamples/lib/suspendedobjects.py @@ -1,238 +1,239 @@ import numpy as np from ipywidgets import interact, interactive, fixed, interact_manual from ipywidgets import HBox, VBox, Label, Layout import ipywidgets as widgets from IPython.display import set_matplotlib_formats, display, Math, Markdown, Latex set_matplotlib_formats('svg') import matplotlib.pyplot as plt import matplotlib.patches as pat plt.style.use('seaborn-whitegrid') # global style for plotting +from IPython.display import IFrame class SuspendedObjectsLab: """ This class embeds all the necessary code to create a virtual lab to study the static equilibrium of an object suspended on a clothesline with a counterweight. """ def __init__(self, m_object = 3, distance = 5, height = 1, x_origin = 0, y_origin = 0): ''' Initiates and displays the virtual lab on suspended objects. :m_object: mass of the suspended object :distance: horizontal distance between the two poles :height: height of the poles (same height for both) :x_origin: x coordinate of the bottom of the left pole (origin of the coordinate system) :y_origin: y coordinate of the bottom of the left pole (origin of the coordinate system) ''' ###--- Parameters of the situation self.m_object = m_object # mass of the wet object, in kg self.distance = distance # distance between the poles, in m self.height = height # height of the poles, in m self.x_origin = x_origin # x coordinate of point of origin of the figure = x position of the left pole, in m self.y_origin = y_origin # y coordinate of point of origin of the figure = y position of the lower point (ground), in m ###--- Then we define the elements of the ihm: # parameters for sliders self.m_counterweight_min = 0 self.m_counterweight_max = 100 self.m_counterweight = self.m_counterweight_min # initial mass of the counterweight (0 by default, no counterweight at the beginning) # IHM input elements input_layout=Layout(margin='5px 10px') self.m_counterweight_label = Label('Mass of the counterweight ($kg$):', layout=input_layout) self.m_counterweight_widget = widgets.FloatSlider(min=self.m_counterweight_min,max=self.m_counterweight_max,step=1,value=self.m_counterweight, layout=input_layout) self.m_counterweight_input = HBox([self.m_counterweight_label, self.m_counterweight_widget]) # IHM output elements self.graph_output = widgets.Output(layout=input_layout) # Linking widgets to handlers self.m_counterweight_widget.observe(self.m_counterweight_event_handler, names='value') # Organize layout self.ihm = VBox([self.graph_output, self.m_counterweight_input]) ###--- Finally, we display the whole interface and we update it right away so that it plots the graph with current values display(self.ihm); self.update_lab() # Event handlers def m_counterweight_event_handler(self, change): self.m_counterweight = change.new self.update_lab() # Utility functions def get_angle(self, m_counterweight): """ Computes the angle that the cable makes with the horizon depending on the counterweight chosen: - if the counterweight is sufficient: angle = arcsin(1/2 * m_object / m_counterweight) - else (object on the ground): alpha = np.arctan(height / (distance / 2)) :m_counterweight: mass of the chosen counterweight :returns: angle that the cable makes with the horizon (in rad) """ # Default alpha value i.e. object is on the ground alpha_default = np.arctan(self.height / (self.distance / 2)) alpha = alpha_default # Let's check that there is actually a counterweight if m_counterweight > 0: # Then we compute the ratio of masses ratio = 0.5 * self.m_object / m_counterweight # Check that the ratio of masses is in the domain of validity of arcsin ([-1;1]) if abs(ratio) < 1: alpha = np.arcsin(ratio) return min(alpha_default, alpha) def get_object_coords(self, angle): """ Computes the position of the object on the cable taking into account the angle determined by the counterweight and the dimensions of the hanging system. By default: - the object is supposed to be suspended exactly in the middle of the cable - the object is on considered the ground for all values of the angle :angle: angle that the cable makes with the horizon :returns: coordinates of the point at which the object are hanged """ # the jean is midway between the poles x_object = self.x_origin + 0.5 * self.distance # default y value: the jean is on the ground y_object = self.y_origin # we check that the angle is comprised between horizontal (greater than 0) and vertical (smaller than pi/2) if angle > 0 and angle < (np.pi / 2): # we compute the delta between the horizon and the point given by the angle delta = (0.5 * self.distance * np.tan(angle)) # we check that the delta is smaller than the height of the poles (otherwise it just means the jean is on the ground) if delta <= self.height: y_object = self.y_origin + self.height - delta return [x_object, y_object] # Create visualisation def update_lab(self): # Clear outputs self.graph_output.clear_output(wait=True) # Compute new values with the counterweight selected by the user alpha = self.get_angle(self.m_counterweight) alpha_degrees = alpha*180/np.pi alpha_text = r'$\alpha$ = {:.2f} $^\circ$'.format(alpha_degrees) coord_object = self.get_object_coords(alpha) height_text = r'h = {:.2f} $m$'.format(coord_object[1]) # Create the figure fig = plt.figure(figsize=(16, 4)) ax1 = plt.subplot(131) ax2 = plt.subplot(132) ax3 = plt.subplot(133, sharex = ax2, sharey = ax1) ###--- First display the clothesline ax1.set_title('Suspended object ($m = {}$ kg)'.format(self.m_object)) # Fix graph to problem boundaries ax1.set_ylim(bottom = self.y_origin) # limit bottom of y axis to ground ax1.set_ylim(top = self.y_origin + self.height + .1) # limit top of y axis to values just above height # Customize graph style so that it doesn't look like a graph #ax1.get_xaxis().set_visible(False) # hide the x axis ax1.grid(False) # hide the grid ax1.set_ylabel("Height ($m$)") # add a label on the y axis ax1.set_xlabel("Distance ($m$)") # add a label on the x axis ax1.spines['top'].set_visible(False) # hide the frame except bottom line ax1.spines['right'].set_visible(False) ax1.spines['left'].set_visible(False) # Draw poles x_pole1 = np.array([self.x_origin, self.x_origin]) y_pole1 = np.array([self.y_origin, self.y_origin+self.height]) ax1.plot(x_pole1, y_pole1, "k-", linewidth=7) x_pole2 = np.array([self.x_origin+self.distance, self.x_origin+self.distance]) y_pole2 = np.array([self.y_origin, self.y_origin+self.height]) ax1.plot(x_pole2, y_pole2, "k-", linewidth=7) # Draw the hanging cable x = np.array([self.x_origin, coord_object[0], self.x_origin+self.distance]) y = np.array([self.y_origin+self.height, coord_object[1], self.y_origin+self.height]) ax1.plot(x, y, linewidth=3, linestyle = "-", color="green") # Draw the horizon line ax1.axhline(y=self.y_origin+self.height, color='gray', linestyle='-.', linewidth=1, zorder=1) # Draw the angle between the hanging cable and horizonline ellipse_radius = 0.2 fig_ratio = self.height / self.distance ax1.add_patch(pat.Arc(xy = (self.x_origin, self.y_origin+self.height), width = ellipse_radius/fig_ratio, height = ellipse_radius, theta1 = -1*alpha_degrees, theta2 = 0, color="gray", linestyle='-.')) ax1.annotate(alpha_text, xy=(self.x_origin, self.y_origin+self.height), xytext=(30, -15), textcoords='offset points', bbox=dict(boxstyle="round", facecolor = "white", edgecolor = "white", alpha = 0.8)) # Draw the point at which the object is suspended ax1.scatter(coord_object[0], coord_object[1], s=80, c="r", zorder=15) ax1.annotate(height_text, xy=(coord_object[0], coord_object[1]), xytext=(10, -10), textcoords='offset points', bbox=dict(boxstyle="round", facecolor = "white", edgecolor = "white", alpha = 0.8)) ###--- Then display the angle and the height as functions from the mass of the counterweight ax2.set_title(r'Angle $\alpha$ of the cable vs. horizon ($^\circ$)') ax3.set_title(r'Height of the suspension point ($m$)') # Create all possible values of the mass of the counterweight m_cw = np.linspace(self.m_counterweight_min, self.m_counterweight_max, 100) # Compute the angle (in degrees) and height for all these values angle = [] height = [] for m in m_cw: a = self.get_angle(m) #a = self.get_angle_from_masses(self.m_object, m, self.distance, self.height) angle.append(a*180/np.pi) c = self.get_object_coords(a) #c = self.get_object_coordinates(a, self.x_origin, self.y_origin, self.distance, self.height) height.append(c[1]) # Display the functions on the graphs ax2.set_xlabel('Mass of the counterweight (kg)') ax2.plot(m_cw, angle, "b") ax3.set_xlabel('Mass of the counterweight (kg)') ax3.plot(m_cw, height, "b") # Draw the horizon lines ax2.axhline(y=self.y_origin, color='gray', linestyle='-.', linewidth=1, zorder=1) ax3.axhline(y=self.y_origin+self.height, color='gray', linestyle='-.', linewidth=1, zorder=1) # Add the current angle from the counterweight selected by the user ax2.scatter(self.m_counterweight, alpha_degrees, s=80, c="r", zorder=15) ax2.annotate(alpha_text, xy=(self.m_counterweight, alpha_degrees), xytext=(10, 5), textcoords='offset points', bbox=dict(boxstyle="round", facecolor = "white", edgecolor = "white", alpha = 0.8)) # Add the current height from the counterweight selected by the user ax3.scatter(self.m_counterweight, coord_object[1], s=80, c="r", zorder=15) ax3.annotate(height_text, xy=(self.m_counterweight, coord_object[1]), xytext=(10, -10), textcoords='offset points', bbox=dict(boxstyle="round", facecolor = "white", edgecolor = "white", alpha = 0.8)) # Display graph with self.graph_output: plt.show(); # EOF