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     <!-- navigation toc: --> <li><a href="#sketcher:ex:pendulum" style="font-size: 80%;"><b>A simple pendulum</b></a></li>
     <!-- navigation toc: --> <li><a href="#sketcher:ex:pendulum:basic" style="font-size: 80%;">&nbsp;&nbsp;&nbsp;The basic physics sketch</a></li>
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<h1 id="sketcher:ex:pendulum">A simple pendulum</h1>

<h2 id="sketcher:ex:pendulum:basic">The basic physics sketch</h2>

<p>
We now want to make a sketch of simple pendulum from physics, as shown
in Figure <a href="#sketcher:ex:pendulum:fig1">8</a>. A body with mass \( m \) is attached
to a massless, stiff rod, which can rotate about a point, causing the
pendulum to oscillate.

<p>
A suggested work flow is to
first sketch the figure on a piece of paper and introduce a coordinate
system. A simple coordinate system is indicated in Figure
<a href="#sketcher:ex:pendulum:fig1wgrid">9</a>. In a code we introduce variables
<code>W</code> and <code>H</code> for the width and height of the figure (i.e., extent of
the coordinate system) and open the program like this:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pysketcher</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>

H <span style="color: #666666">=</span> <span style="color: #666666">7.</span>
W <span style="color: #666666">=</span> <span style="color: #666666">6.</span>

drawing_tool<span style="color: #666666">.</span>set_coordinate_system(xmin<span style="color: #666666">=0</span>, xmax<span style="color: #666666">=</span>W,
                                   ymin<span style="color: #666666">=0</span>, ymax<span style="color: #666666">=</span>H,
                                   axis<span style="color: #666666">=</span><span style="color: #008000">True</span>)
drawing_tool<span style="color: #666666">.</span>set_grid(<span style="color: #008000">True</span>)
drawing_tool<span style="color: #666666">.</span>set_linecolor(<span style="color: #BA2121">&#39;blue&#39;</span>)
</pre></div>
<p>
Note that when the sketch is ready for &quot;production&quot;, we will (normally)
set <code>axis=False</code> to remove the coordinate system and also remove the
grid, i.e., delete or
comment out the line <code>drawing_tool.set_grid(True)</code>.
Also note that we in this example let all lines be blue by default.

<p>
<center> <!-- figure -->
<hr class="figure">
<center><p class="caption">Figure 8:  Sketch of a simple pendulum. <div id="sketcher:ex:pendulum:fig1"></div> </p></center>
<p><img src="fig-tut/pendulum1.png" align="bottom" width=300></p>
</center>

<p>
<center> <!-- figure -->
<hr class="figure">
<center><p class="caption">Figure 9:  Sketch with assisting coordinate system. <div id="sketcher:ex:pendulum:fig1wgrid"></div> </p></center>
<p><img src="fig-tut/pendulum1_wgrid.png" align="bottom" width=400></p>
</center>

<p>
The next step is to introduce variables for key quantities in the sketch.
Let <code>L</code> be the length of the pendulum, <code>P</code> the rotation point, and let
<code>a</code> be the angle the pendulum makes with the vertical (measured in degrees).
We may set

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">L <span style="color: #666666">=</span> <span style="color: #666666">5*</span>H<span style="color: #666666">/7</span>          <span style="color: #408080; font-style: italic"># length</span>
P <span style="color: #666666">=</span> (W<span style="color: #666666">/6</span>, <span style="color: #666666">0.85*</span>H)  <span style="color: #408080; font-style: italic"># rotation point</span>
a <span style="color: #666666">=</span> <span style="color: #666666">40</span>             <span style="color: #408080; font-style: italic"># angle</span>
</pre></div>
<p>
Be careful with integer division if you use Python 2! Fortunately, we
started out with <code>float</code> objects for <code>W</code> and <code>H</code> so the expressions above
are safe.

<p>
What kind of objects do we need in this sketch? Looking at
Figure <a href="#sketcher:ex:pendulum:fig1">8</a> we see that we need

<ol>
<li> a vertical, dashed line</li>
<li> an arc with no text but dashed line to indicate the <em>path</em> of the
   mass</li>
<li> an arc with name \( \theta \) to indicate the <em>angle</em></li>
<li> a line, here called <em>rod</em>, from the rotation point to the mass</li>
<li> a blue, filled circle representing the <em>mass</em></li>
<li> a text \( m \) associated with the mass</li>
<li> an indicator of the pendulum's <em>length</em> \( L \), visualized as
   a line with two arrows tips and the text \( L \)</li>
<li> a gravity vector with the text \( g \)</li>
</ol>

Pysketcher has objects for each of these elements in our sketch.
We start with the simplest element: the vertical line, going from
<code>P</code> to <code>P</code> minus the length \( L \) in \( y \) direction:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">vertical <span style="color: #666666">=</span> Line(P, P<span style="color: #666666">-</span>point(<span style="color: #666666">0</span>,L))
</pre></div>
<p>
The class <code>point</code> is very convenient: it turns its two coordinates into
a vector with which we can compute, and is therefore one of the most
widely used Pysketcher objects.

<p>
The path of the mass is an arc that can be made by
Pysketcher's <code>Arc</code> object:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">path <span style="color: #666666">=</span> Arc(P, L, <span style="color: #666666">-90</span>, a)
</pre></div>
<p>
The first argument <code>P</code> is the center point, the second is the
radius (<code>L</code> here), the next argument is the start angle, here
it starts at -90 degrees, while the next argument is the angle of
the arc, here <code>a</code>.
For the path of the mass, we also need an arc object, but this
time with an associated text. Pysketcher has a specialized object
for this purpose, <code>Arc_wText</code>, since placing the text manually can
be somewhat cumbersome.

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">angle <span style="color: #666666">=</span> Arc_wText(<span style="color: #BA2121">r&#39;$\theta$&#39;</span>, P, L<span style="color: #666666">/4</span>, <span style="color: #666666">-90</span>, a, text_spacing<span style="color: #666666">=1/30.</span>)
</pre></div>
<p>
The arguments are as for <code>Arc</code> above, but the first one is the desired
text. Remember to use a raw string since we want a LaTeX greek letter
that contains a backslash.
The <code>text_spacing</code> argument must often be tweaked. It is recommended
to create only a few objects before rendering the sketch and then
adjust spacings as one goes along.

<p>
The rod is simply a line from <code>P</code> to the mass. We can easily
compute the position of the mass from basic geometry considerations,
but it is easier and safer to look up this point in other objects
if it is already computed. In the present case,
the <code>path</code> object stored its start and
end points, so <code>path.geometric_features()['end']</code> is the end point
of the path, which is the position of the mass. We can therefore
create the rod simply as a line from <code>P</code> to this already computed end point:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">mass_pt <span style="color: #666666">=</span> path<span style="color: #666666">.</span>geometric_features()[<span style="color: #BA2121">&#39;end&#39;</span>]
rod <span style="color: #666666">=</span> Line(P, mass_pt)
</pre></div>
<p>
The mass is a circle filled with color:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">mass <span style="color: #666666">=</span> Circle(center<span style="color: #666666">=</span>mass_pt, radius<span style="color: #666666">=</span>L<span style="color: #666666">/20.</span>)
mass<span style="color: #666666">.</span>set_filled_curves(color<span style="color: #666666">=</span><span style="color: #BA2121">&#39;blue&#39;</span>)
</pre></div>
<p>
To place the \( m \) correctly, we go a small distance in the direction of
the rod, from the center of the circle. To this end, we need to
compute the direction. This is easiest done by computing a vector
from <code>P</code> to the center of the circle and calling <code>unit_vec</code> to make
a unit vector in this direction:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">rod_vec <span style="color: #666666">=</span> rod<span style="color: #666666">.</span>geometric_features()[<span style="color: #BA2121">&#39;end&#39;</span>] <span style="color: #666666">-</span> \ 
          rod<span style="color: #666666">.</span>geometric_features()[<span style="color: #BA2121">&#39;start&#39;</span>]
unit_rod_vec <span style="color: #666666">=</span> unit_vec(rod_vec)
mass_symbol <span style="color: #666666">=</span> Text(<span style="color: #BA2121">&#39;$m$&#39;</span>, mass_pt <span style="color: #666666">+</span> L<span style="color: #666666">/10*</span>unit_rod_vec)
</pre></div>
<p>
Again, the distance <code>L/10</code> is something one has to experiment with.

<p>
The next object is the length measure with the text \( L \). Such length
measures are represented by Pysketcher's <code>Distance_wText</code> object.
An easy construction is to first place this length measure along the
rod and then translate it a little distance (<code>L/15</code>) in the
normal direction of the rod:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">length <span style="color: #666666">=</span> Distance_wText(P, mass_pt, <span style="color: #BA2121">&#39;$L$&#39;</span>)
length<span style="color: #666666">.</span>translate(L<span style="color: #666666">/15*</span>point(cos(radians(a)), sin(radians(a))))
</pre></div>
<p>
For this translation we need a unit vector in the normal direction
of the rod, which is from geometric considerations given by
\( (\cos a, \sin a) \), when \( a \) is the angle of the pendulum.
Alternatively, we could have found the normal vector as a vector that
is normal to <code>unit_rod_vec</code>: <code>point(-unit_rod_vec[1],unit_rod_vec[0])</code>.

<p>
The final object is the gravity force vector, which is so common
in physics sketches that Pysketcher has a ready-made object: <code>Gravity</code>,

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">gravity <span style="color: #666666">=</span> Gravity(start<span style="color: #666666">=</span>P<span style="color: #666666">+</span>point(<span style="color: #666666">0.8*</span>L,<span style="color: #666666">0</span>), length<span style="color: #666666">=</span>L<span style="color: #666666">/3</span>)
</pre></div>
<p>
Since blue is the default color for
lines, we want the dashed lines (for <code>vertical</code> and <code>path</code>) to be black
and with linewidth 1. These properties can be set one by one for each
object, but we can also make a little helper function:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">set_dashed_thin_blackline</span>(<span style="color: #666666">*</span>objects):
    <span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Set linestyle of objects to dashed, black, width=1.&quot;&quot;&quot;</span>
    <span style="color: #008000; font-weight: bold">for</span> obj <span style="color: #AA22FF; font-weight: bold">in</span> objects:
        obj<span style="color: #666666">.</span>set_linestyle(<span style="color: #BA2121">&#39;dashed&#39;</span>)
        obj<span style="color: #666666">.</span>set_linecolor(<span style="color: #BA2121">&#39;black&#39;</span>)
        obj<span style="color: #666666">.</span>set_linewidth(<span style="color: #666666">1</span>)

set_dashed_thin_blackline(vertical, path)
</pre></div>
<p>
Now, all objects are in place, so it remains to compose the final
figure and draw the composition:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">fig <span style="color: #666666">=</span> Composition(
    {<span style="color: #BA2121">&#39;body&#39;</span>: mass, <span style="color: #BA2121">&#39;rod&#39;</span>: rod,
     <span style="color: #BA2121">&#39;vertical&#39;</span>: vertical, <span style="color: #BA2121">&#39;theta&#39;</span>: angle, <span style="color: #BA2121">&#39;path&#39;</span>: path,
     <span style="color: #BA2121">&#39;g&#39;</span>: gravity, <span style="color: #BA2121">&#39;L&#39;</span>: length, <span style="color: #BA2121">&#39;m&#39;</span>: mass_symbol})

fig<span style="color: #666666">.</span>draw()
drawing_tool<span style="color: #666666">.</span>display()
drawing_tool<span style="color: #666666">.</span>savefig(<span style="color: #BA2121">&#39;pendulum1&#39;</span>)
</pre></div>

<h2 id="___sec10">The body diagram </h2>

<p>
Now we want to isolate the mass and draw all the forces that act on it.
Figure <a href="#sketcher:ex:pendulum:fig2wgrid">10</a> shows the desired result, but
embedded in the coordinate system.
We consider three types of forces: the gravity force, the force from the
rod, and air resistance. The body diagram is key for deriving the
equation of motion, so it is illustrative to add useful mathematical
quantities needed in the derivation, such as the unit vectors in polar
coordinates.

<p>
<center> <!-- figure -->
<hr class="figure">
<center><p class="caption">Figure 10:  Body diagram of a simple pendulum. <div id="sketcher:ex:pendulum:fig2wgrid"></div> </p></center>
<p><img src="fig-tut/pendulum5_wgrid.png" align="bottom" width=300></p>
</center>

<p>
We start by listing the objects in the sketch:

<ol>
<li> a text \( (x_0,y_0) \) representing the rotation  point <code>P</code></li>
<li> unit vector \( \boldsymbol{i}_r \) with text</li>
<li> unit vector \( \boldsymbol{i}_\theta \) with text</li>
<li> a dashed vertical line</li>
<li> a dashed line along the rod</li>
<li> an arc with text \( \theta \)</li>
<li> the gravity force with text \( mg \)</li>
<li> the force in the rod with text \( S \)</li>
<li> the air resistance force with text \( \sim |v|v \)</li>
</ol>

The first object, \( (x_0,y_0) \), is simply a plain text where we have
to experiment with its position. The unit vectors in polar coordinates
may be drawn using the Pysketcher's <code>Force</code> object since it has an
arrow with a text. The first three objects can then be made as follows:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">x0y0 <span style="color: #666666">=</span> Text(<span style="color: #BA2121">&#39;$(x_0,y_0)$&#39;</span>, P <span style="color: #666666">+</span> point(<span style="color: #666666">-0.4</span>,<span style="color: #666666">-0.1</span>))
ir <span style="color: #666666">=</span> Force(P, P <span style="color: #666666">+</span> L<span style="color: #666666">/10*</span>unit_vec(rod_vec),
           <span style="color: #BA2121">r&#39;$\boldsymbol{i}_r$&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>,
           text_spacing<span style="color: #666666">=</span>(<span style="color: #666666">0.015</span>,<span style="color: #666666">0</span>))
ith <span style="color: #666666">=</span> Force(P, P <span style="color: #666666">+</span> L<span style="color: #666666">/10*</span>unit_vec((<span style="color: #666666">-</span>rod_vec[<span style="color: #666666">1</span>], rod_vec[<span style="color: #666666">0</span>])),
           <span style="color: #BA2121">r&#39;$\boldsymbol{i}_{\theta}$&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>,
            text_spacing<span style="color: #666666">=</span>(<span style="color: #666666">0.02</span>,<span style="color: #666666">0.005</span>))
</pre></div>
<p>
Note that tweaking of the position of <code>x0y0</code> use absolute coordinates, so
if <code>W</code> or <code>H</code> is changed in the beginning of the figure, the tweaked position
will most likely not look good. A better solution would be to express
the tweaked displacement <code>point(-0.4,-0.1)</code> in terms of <code>W</code> and <code>H</code>.
The <code>text_spacing</code> values in the <code>Force</code> objects also use absolute
coordinates. Very often, this is much more convenient when adjusting
the objects, and global size parameters like <code>W</code> and <code>H</code> are in practice
seldom changed, so the solution above is quite typical.

<p>
The vertical, dashed line, the dashed rod, and the arc for \( \theta \)
are made by

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">rod_start <span style="color: #666666">=</span> rod<span style="color: #666666">.</span>geometric_features()[<span style="color: #BA2121">&#39;start&#39;</span>]  <span style="color: #408080; font-style: italic"># Point P</span>
vertical2 <span style="color: #666666">=</span> Line(rod_start, rod_start <span style="color: #666666">+</span> point(<span style="color: #666666">0</span>,<span style="color: #666666">-</span>L<span style="color: #666666">/3</span>))
set_dashed_thin_blackline(vertical2)
set_dashed_thin_blackline(rod)
angle2 <span style="color: #666666">=</span> Arc_wText(<span style="color: #BA2121">r&#39;$\theta$&#39;</span>, rod_start, L<span style="color: #666666">/6</span>, <span style="color: #666666">-90</span>, a,
                   text_spacing<span style="color: #666666">=1/30.</span>)
</pre></div>
<p>
Note how we reuse the earlier defined object <code>rod</code>.

<p>
The forces are constructed as shown below.

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">mg_force  <span style="color: #666666">=</span> Force(mass_pt, mass_pt <span style="color: #666666">+</span> L<span style="color: #666666">/5*</span>point(<span style="color: #666666">0</span>,<span style="color: #666666">-1</span>),
                  <span style="color: #BA2121">&#39;$mg$&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>)
rod_force <span style="color: #666666">=</span> Force(mass_pt, mass_pt <span style="color: #666666">-</span> L<span style="color: #666666">/3*</span>unit_vec(rod_vec),
                  <span style="color: #BA2121">&#39;$S$&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>,
                  text_spacing<span style="color: #666666">=</span>(<span style="color: #666666">0.03</span>, <span style="color: #666666">0.01</span>))
air_force <span style="color: #666666">=</span> Force(mass_pt, mass_pt <span style="color: #666666">-</span>
                  L<span style="color: #666666">/6*</span>unit_vec((rod_vec[<span style="color: #666666">1</span>], <span style="color: #666666">-</span>rod_vec[<span style="color: #666666">0</span>])),
                  <span style="color: #BA2121">&#39;$\sim|v|v$&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>,
                  text_spacing<span style="color: #666666">=</span>(<span style="color: #666666">0.04</span>,<span style="color: #666666">0.005</span>))
</pre></div>
<p>
Note that the drag force from the air is directed perpendicular to
the rod, so we construct a unit vector in this direction directly from
the <code>rod_vec</code> vector.

<p>
All objects are in place, and we can compose a figure to be drawn:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%">body_diagram <span style="color: #666666">=</span> Composition(
    {<span style="color: #BA2121">&#39;mg&#39;</span>: mg_force, <span style="color: #BA2121">&#39;S&#39;</span>: rod_force, <span style="color: #BA2121">&#39;rod&#39;</span>: rod,
     <span style="color: #BA2121">&#39;vertical&#39;</span>: vertical2, <span style="color: #BA2121">&#39;theta&#39;</span>: angle2,
     <span style="color: #BA2121">&#39;body&#39;</span>: mass, <span style="color: #BA2121">&#39;m&#39;</span>: mass_symbol})

body_diagram[<span style="color: #BA2121">&#39;air&#39;</span>] <span style="color: #666666">=</span> air_force
body_diagram[<span style="color: #BA2121">&#39;ir&#39;</span>] <span style="color: #666666">=</span> ir
body_diagram[<span style="color: #BA2121">&#39;ith&#39;</span>] <span style="color: #666666">=</span> ith
body_diagram[<span style="color: #BA2121">&#39;origin&#39;</span>] <span style="color: #666666">=</span> x0y0
</pre></div>
<p>
Here, we exemplify that we can start out with a composition as a
dictionary, but (as in ordinary Python dictionaries) add new
elements later when desired.

<p>
<!-- FIGURE: [fig-tut/pendulum1.png, width=300 frac=0.5] Sketch of a simple pendulum. <div id="sketcher:ex:pendulum:fig2"></div> -->

<h2 id="sketcher:ex:pendulum:anim">Animated body diagram</h2>

<p>
We want to make an animated body diagram so that we can see how forces
develop in time according to the motion. This means that we must
couple the sketch at each time level to a numerical solution for
the motion of the pendulum.

<h3 id="___sec12">Function for drawing the body diagram </h3>

<p>
The previous flat program for making sketches of the pendulum is not
suitable when we want to make a sketch at a lot of different points
in time, i.e., for a lot of different angles that the pendulum makes
with the vertical. We therefore need to draw the body diagram in
a function where the angle is a parameter. We also supply arrays
containing the (numerically computed) values of the angle \( \theta \) and
the forces at various time levels, plus the desired time point and level
for this particular sketch:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pysketcher</span> <span style="color: #008000; font-weight: bold">import</span> <span style="color: #666666">*</span>

H <span style="color: #666666">=</span> <span style="color: #666666">15.</span>
W <span style="color: #666666">=</span> <span style="color: #666666">17.</span>

drawing_tool<span style="color: #666666">.</span>set_coordinate_system(xmin<span style="color: #666666">=0</span>, xmax<span style="color: #666666">=</span>W,
                                   ymin<span style="color: #666666">=0</span>, ymax<span style="color: #666666">=</span>H,
                                   axis<span style="color: #666666">=</span><span style="color: #008000">False</span>)

<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">pendulum</span>(theta, S, mg, drag, t, time_level):

    drawing_tool<span style="color: #666666">.</span>set_linecolor(<span style="color: #BA2121">&#39;blue&#39;</span>)
    <span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">math</span>
    a <span style="color: #666666">=</span> math<span style="color: #666666">.</span>degrees(theta[time_level])
    L <span style="color: #666666">=</span> <span style="color: #666666">0.4*</span>H         <span style="color: #408080; font-style: italic"># length</span>
    P <span style="color: #666666">=</span> (W<span style="color: #666666">/2</span>, <span style="color: #666666">0.8*</span>H)  <span style="color: #408080; font-style: italic"># rotation point</span>

    vertical <span style="color: #666666">=</span> Line(P, P<span style="color: #666666">-</span>point(<span style="color: #666666">0</span>,L))
    path <span style="color: #666666">=</span> Arc(P, L, <span style="color: #666666">-90</span>, a)
    angle <span style="color: #666666">=</span> Arc_wText(<span style="color: #BA2121">r&#39;$\theta$&#39;</span>, P, L<span style="color: #666666">/4</span>, <span style="color: #666666">-90</span>, a, text_spacing<span style="color: #666666">=1/30.</span>)

    mass_pt <span style="color: #666666">=</span> path<span style="color: #666666">.</span>geometric_features()[<span style="color: #BA2121">&#39;end&#39;</span>]
    rod <span style="color: #666666">=</span> Line(P, mass_pt)

    mass <span style="color: #666666">=</span> Circle(center<span style="color: #666666">=</span>mass_pt, radius<span style="color: #666666">=</span>L<span style="color: #666666">/20.</span>)
    mass<span style="color: #666666">.</span>set_filled_curves(color<span style="color: #666666">=</span><span style="color: #BA2121">&#39;blue&#39;</span>)
    rod_vec <span style="color: #666666">=</span> rod<span style="color: #666666">.</span>geometric_features()[<span style="color: #BA2121">&#39;end&#39;</span>] <span style="color: #666666">-</span> \ 
              rod<span style="color: #666666">.</span>geometric_features()[<span style="color: #BA2121">&#39;start&#39;</span>]
    unit_rod_vec <span style="color: #666666">=</span> unit_vec(rod_vec)
    mass_symbol <span style="color: #666666">=</span> Text(<span style="color: #BA2121">&#39;$m$&#39;</span>, mass_pt <span style="color: #666666">+</span> L<span style="color: #666666">/10*</span>unit_rod_vec)

    length <span style="color: #666666">=</span> Distance_wText(P, mass_pt, <span style="color: #BA2121">&#39;$L$&#39;</span>)
    <span style="color: #408080; font-style: italic"># Displace length indication</span>
    length<span style="color: #666666">.</span>translate(L<span style="color: #666666">/15*</span>point(cos(radians(a)), sin(radians(a))))
    gravity <span style="color: #666666">=</span> Gravity(start<span style="color: #666666">=</span>P<span style="color: #666666">+</span>point(<span style="color: #666666">0.8*</span>L,<span style="color: #666666">0</span>), length<span style="color: #666666">=</span>L<span style="color: #666666">/3</span>)

    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">set_dashed_thin_blackline</span>(<span style="color: #666666">*</span>objects):
        <span style="color: #BA2121; font-style: italic">&quot;&quot;&quot;Set linestyle of objects to dashed, black, width=1.&quot;&quot;&quot;</span>
        <span style="color: #008000; font-weight: bold">for</span> obj <span style="color: #AA22FF; font-weight: bold">in</span> objects:
            obj<span style="color: #666666">.</span>set_linestyle(<span style="color: #BA2121">&#39;dashed&#39;</span>)
            obj<span style="color: #666666">.</span>set_linecolor(<span style="color: #BA2121">&#39;black&#39;</span>)
            obj<span style="color: #666666">.</span>set_linewidth(<span style="color: #666666">1</span>)

    set_dashed_thin_blackline(vertical, path)

    fig <span style="color: #666666">=</span> Composition(
        {<span style="color: #BA2121">&#39;body&#39;</span>: mass, <span style="color: #BA2121">&#39;rod&#39;</span>: rod,
         <span style="color: #BA2121">&#39;vertical&#39;</span>: vertical, <span style="color: #BA2121">&#39;theta&#39;</span>: angle, <span style="color: #BA2121">&#39;path&#39;</span>: path,
         <span style="color: #BA2121">&#39;g&#39;</span>: gravity, <span style="color: #BA2121">&#39;L&#39;</span>: length})

    <span style="color: #408080; font-style: italic">#fig.draw()</span>
    <span style="color: #408080; font-style: italic">#drawing_tool.display()</span>
    <span style="color: #408080; font-style: italic">#drawing_tool.savefig(&#39;tmp_pendulum1&#39;)</span>

    drawing_tool<span style="color: #666666">.</span>set_linecolor(<span style="color: #BA2121">&#39;black&#39;</span>)

    rod_start <span style="color: #666666">=</span> rod<span style="color: #666666">.</span>geometric_features()[<span style="color: #BA2121">&#39;start&#39;</span>]  <span style="color: #408080; font-style: italic"># Point P</span>
    vertical2 <span style="color: #666666">=</span> Line(rod_start, rod_start <span style="color: #666666">+</span> point(<span style="color: #666666">0</span>,<span style="color: #666666">-</span>L<span style="color: #666666">/3</span>))
    set_dashed_thin_blackline(vertical2)
    set_dashed_thin_blackline(rod)
    angle2 <span style="color: #666666">=</span> Arc_wText(<span style="color: #BA2121">r&#39;$\theta$&#39;</span>, rod_start, L<span style="color: #666666">/6</span>, <span style="color: #666666">-90</span>, a,
                       text_spacing<span style="color: #666666">=1/30.</span>)

    magnitude <span style="color: #666666">=</span> <span style="color: #666666">1.2*</span>L<span style="color: #666666">/2</span>   <span style="color: #408080; font-style: italic"># length of a unit force in figure</span>
    force <span style="color: #666666">=</span> mg[time_level]  <span style="color: #408080; font-style: italic"># constant (scaled eq: about 1)</span>
    force <span style="color: #666666">*=</span> magnitude
    mg_force  <span style="color: #666666">=</span> Force(mass_pt, mass_pt <span style="color: #666666">+</span> force<span style="color: #666666">*</span>point(<span style="color: #666666">0</span>,<span style="color: #666666">-1</span>),
                      <span style="color: #BA2121">&#39;&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>)
    force <span style="color: #666666">=</span> S[time_level]
    force <span style="color: #666666">*=</span> magnitude
    rod_force <span style="color: #666666">=</span> Force(mass_pt, mass_pt <span style="color: #666666">-</span> force<span style="color: #666666">*</span>unit_vec(rod_vec),
                      <span style="color: #BA2121">&#39;&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>,
                      text_spacing<span style="color: #666666">=</span>(<span style="color: #666666">0.03</span>, <span style="color: #666666">0.01</span>))
    force <span style="color: #666666">=</span> drag[time_level]
    force <span style="color: #666666">*=</span> magnitude
    <span style="color: #408080; font-style: italic">#print(&#39;drag(%g)=%g&#39; % (t, drag[time_level]))</span>
    air_force <span style="color: #666666">=</span> Force(mass_pt, mass_pt <span style="color: #666666">-</span>
                      force<span style="color: #666666">*</span>unit_vec((rod_vec[<span style="color: #666666">1</span>], <span style="color: #666666">-</span>rod_vec[<span style="color: #666666">0</span>])),
                      <span style="color: #BA2121">&#39;&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>,
                      text_spacing<span style="color: #666666">=</span>(<span style="color: #666666">0.04</span>,<span style="color: #666666">0.005</span>))

    body_diagram <span style="color: #666666">=</span> Composition(
        {<span style="color: #BA2121">&#39;mg&#39;</span>: mg_force, <span style="color: #BA2121">&#39;S&#39;</span>: rod_force, <span style="color: #BA2121">&#39;air&#39;</span>: air_force,
         <span style="color: #BA2121">&#39;rod&#39;</span>: rod,
         <span style="color: #BA2121">&#39;vertical&#39;</span>: vertical2, <span style="color: #BA2121">&#39;theta&#39;</span>: angle2,
         <span style="color: #BA2121">&#39;body&#39;</span>: mass})

    x0y0 <span style="color: #666666">=</span> Text(<span style="color: #BA2121">&#39;$(x_0,y_0)$&#39;</span>, P <span style="color: #666666">+</span> point(<span style="color: #666666">-0.4</span>,<span style="color: #666666">-0.1</span>))
    ir <span style="color: #666666">=</span> Force(P, P <span style="color: #666666">+</span> L<span style="color: #666666">/10*</span>unit_vec(rod_vec),
               <span style="color: #BA2121">r&#39;$\boldsymbol{i}_r$&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>,
               text_spacing<span style="color: #666666">=</span>(<span style="color: #666666">0.015</span>,<span style="color: #666666">0</span>))
    ith <span style="color: #666666">=</span> Force(P, P <span style="color: #666666">+</span> L<span style="color: #666666">/10*</span>unit_vec((<span style="color: #666666">-</span>rod_vec[<span style="color: #666666">1</span>], rod_vec[<span style="color: #666666">0</span>])),
               <span style="color: #BA2121">r&#39;$\boldsymbol{i}_{\theta}$&#39;</span>, text_pos<span style="color: #666666">=</span><span style="color: #BA2121">&#39;end&#39;</span>,
                text_spacing<span style="color: #666666">=</span>(<span style="color: #666666">0.02</span>,<span style="color: #666666">0.005</span>))

    <span style="color: #408080; font-style: italic">#body_diagram[&#39;ir&#39;] = ir</span>
    <span style="color: #408080; font-style: italic">#body_diagram[&#39;ith&#39;] = ith</span>
    <span style="color: #408080; font-style: italic">#body_diagram[&#39;origin&#39;] = x0y0</span>

    drawing_tool<span style="color: #666666">.</span>erase()
    body_diagram<span style="color: #666666">.</span>draw(verbose<span style="color: #666666">=0</span>)
    <span style="color: #408080; font-style: italic">#drawing_tool.display(&#39;Body diagram&#39;)</span>
    drawing_tool<span style="color: #666666">.</span>savefig(<span style="color: #BA2121">&#39;tmp_</span><span style="color: #BB6688; font-weight: bold">%04d</span><span style="color: #BA2121">.png&#39;</span> <span style="color: #666666">%</span> time_level, crop<span style="color: #666666">=</span><span style="color: #008000">False</span>)
    <span style="color: #408080; font-style: italic"># No cropping: otherwise movies will be very strange</span>
</pre></div>

<h3 id="___sec13">Equations for the motion and forces </h3>

<p>
The modeling of the motion of a pendulum is most conveniently done in
polar coordinates since then the unknown force in the rod is separated
from the equation determining the motion \( \theta(t) \).
The position vector for the mass is

$$ \rpos = x_0\ii + y_0\jj + L\ir\tp$$

The corresponding acceleration becomes

$$ \ddot{\rpos} = L\ddot{\theta}{\ith} - L\dot{\theta^2}{\ir}\tp$$

<!-- Note: the extra braces help to render the equation correctly in sphinx! -->

<p>
There are three forces on the mass: the gravity force
\( mg\jj = mg(-\cos\theta\,\ir + \sin\theta\,\ith) \), the force in the rod
\( -S\ir \), and the drag force because of air resistance:

$$ -\half C_D \varrho \pi R^2 |v|v\,\ith,$$

where \( C_D\approx 0.4 \) is the drag coefficient for a sphere, \( \varrho \)
is the density of air, \( R \) is the radius of the mass, and \( v \) is the
velocity (\( v=L\dot\theta \)). The drag force acts in \( -\ith \) direction
when \( v>0 \).

<p>
Newton's second law of motion for the pendulum now becomes

$$ mL\ddot\theta\ith - mL\dot\theta^2\ir = -mg(-\cos\theta\,\ir +
\sin\theta\,\ith)
-S\ir - \half C_D \varrho \pi R^2 L^2|\dot\theta|\dot\theta\ith,$$

which gives two component equations

$$
\begin{align}
mL\ddot\theta + \half C_D \varrho \pi R^2 L^2|\dot\theta|\dot\theta +
mg\sin\theta &= 0,
\tag{1}\\ 
S &= mL\dot\theta^2 + mg\cos\theta
\tag{2}\tp
\end{align}
$$

<p>
It is almost always convenient to scale such equations. Introducing
the dimensionless time

$$ \bar t = \frac{t}{t_c},\quad t_c = \sqrt{\frac{L}{g}},$$

leads to

$$
\begin{align}
\frac{d^2\theta}{d\bar t^2} +
\alpha\left\vert\frac{d\theta}{d\bar t}\right\vert\frac{d\theta}{d\bar t} +
\sin\theta &= 0,
\tag{3}\\ 
\bar S &= \left(\frac{d\theta}{d\bar t}\right)^2
+ \cos\theta,
\tag{4}
\end{align}
$$

where \( \alpha \) is a dimensionless drag coefficient

$$ \alpha = \frac{C_D\varrho\pi R^2L}{2m},$$

and \( \bar S \) is the scaled force

$$ \bar S = \frac{S}{mg}\tp$$

We see that \( \bar S = 1 \) for the equilibrium position \( \theta=0 \), so this
scaling of \( S \) seems appropriate.

<p>
The parameter \( \alpha \) is about
the ratio of the drag force and the gravity force:

$$ \frac{|\half C_D\varrho \pi R^2 |v|v|}{|mg|}\sim
\frac{C_D\varrho \pi R^2 L^2 t_c^{-2}}{mg}
\left|\frac{d\bar\theta}{d\bar t}\right|\frac{d\bar\theta}{d\bar t}
\sim \frac{C_D\varrho \pi R^2 L}{2m}\theta_0^2 = \alpha \theta_0^2\tp$$

(We have that \( \theta(t)/d\theta_0 \) is in \( [-1,1] \), so we expect
since \( \theta_0^{-1}d\bar\theta/d\bar t \) to be around unity.)

<p>
The next step is to write a numerical solver for
<a href="#mjx-eqn-3">(3)</a>-<a href="#mjx-eqn-4">(4)</a>. To
this end, we use the <a href="https://github.com/hplgit/odespy" target="_self">Odespy</a>
package. The system of second-order ODEs must be expressed as a system
of first-order ODEs. We realize that the unknown \( \bar S \) is decoupled
from \( \theta \) in the sense that we can first use
<a href="#mjx-eqn-3">(3)</a> to solve for \( \theta \) and
then compute \( \bar S \) from <a href="#mjx-eqn-4">(4)</a>.
The first-order ODEs become

$$
\begin{align}
\frac{d\omega}{d\bar t} &= -\alpha\left\vert\omega\right\vert\omega
- \sin\theta,
\tag{5}\\ 
\frac{d\theta}{d\bar t} &= \omega\tp
\tag{6}
\end{align}
$$

Then we compute

$$
\begin{equation}
\bar S = \omega^2 + \cos\theta\tp
\tag{7}
\end{equation}
$$

The dimensionless air resistance force can also be computed:

$$
\begin{equation}
-\alpha|\omega|\omega\tp
\tag{8}
\end{equation}
$$

Since we scaled the force \( S \) by \( mg \), \( mg \) is the natural force scale,
and the \( mg \) force itself is then unity.

<p>
By updating \( \omega \) in the first equation, we can use an Euler-Cromer
scheme on Odespy (all other schemes are independent of whether the
\( \theta \) or \( \omega \) equation comes first).

<h3 id="___sec14">Numerical solution </h3>

<p>
An appropriate solver is

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">simulate_pendulum</span>(alpha, theta0, dt, T):
    <span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">odespy</span>

    <span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">f</span>(u, t, alpha):
        omega, theta <span style="color: #666666">=</span> u
        <span style="color: #008000; font-weight: bold">return</span> [<span style="color: #666666">-</span>alpha<span style="color: #666666">*</span>omega<span style="color: #666666">*</span><span style="color: #008000">abs</span>(omega) <span style="color: #666666">-</span> sin(theta),
                omega]

    <span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
    Nt <span style="color: #666666">=</span> <span style="color: #008000">int</span>(<span style="color: #008000">round</span>(T<span style="color: #666666">/</span><span style="color: #008000">float</span>(dt)))
    t <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">0</span>, Nt<span style="color: #666666">*</span>dt, Nt<span style="color: #666666">+1</span>)
    solver <span style="color: #666666">=</span> odespy<span style="color: #666666">.</span>RK4(f, f_args<span style="color: #666666">=</span>[alpha])
    solver<span style="color: #666666">.</span>set_initial_condition([<span style="color: #666666">0</span>, theta0])
    u, t <span style="color: #666666">=</span> solver<span style="color: #666666">.</span>solve(t,
                        terminate<span style="color: #666666">=</span><span style="color: #008000; font-weight: bold">lambda</span> u, t, n: <span style="color: #008000">abs</span>(u[n,<span style="color: #666666">1</span>]) <span style="color: #666666">&lt;</span> <span style="color: #666666">1E-3</span>)
    omega <span style="color: #666666">=</span> u[:,<span style="color: #666666">0</span>]
    theta <span style="color: #666666">=</span> u[:,<span style="color: #666666">1</span>]
    S <span style="color: #666666">=</span> omega<span style="color: #666666">**2</span> <span style="color: #666666">+</span> np<span style="color: #666666">.</span>cos(theta)
    drag <span style="color: #666666">=</span> <span style="color: #666666">-</span>alpha<span style="color: #666666">*</span>np<span style="color: #666666">.</span>abs(omega)<span style="color: #666666">*</span>omega
    <span style="color: #008000; font-weight: bold">return</span> t, theta, omega, S, drag
</pre></div>

<h3 id="___sec15">Animation </h3>

<p>
We can finally traverse the time array and draw a body diagram
at each time level. The resulting sketches are saved to files
<code>tmp_%04d.png</code>, and these files can be combined to videos:

<p>

<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">animate</span>():
    <span style="color: #408080; font-style: italic"># Clean up old plot files</span>
    <span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span><span style="color: #666666">,</span> <span style="color: #0000FF; font-weight: bold">glob</span>
    <span style="color: #008000; font-weight: bold">for</span> filename <span style="color: #AA22FF; font-weight: bold">in</span> glob<span style="color: #666666">.</span>glob(<span style="color: #BA2121">&#39;tmp_*.png&#39;</span>) <span style="color: #666666">+</span> glob<span style="color: #666666">.</span>glob(<span style="color: #BA2121">&#39;movie.*&#39;</span>):
        os<span style="color: #666666">.</span>remove(filename)
    <span style="color: #408080; font-style: italic"># Solve problem</span>
    <span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">math</span> <span style="color: #008000; font-weight: bold">import</span> pi, radians, degrees
    <span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
    alpha <span style="color: #666666">=</span> <span style="color: #666666">0.4</span>
    period <span style="color: #666666">=</span> <span style="color: #666666">2*</span>pi
    T <span style="color: #666666">=</span> <span style="color: #666666">12*</span>period
    dt <span style="color: #666666">=</span> period<span style="color: #666666">/40</span>
    a <span style="color: #666666">=</span> <span style="color: #666666">70</span>
    theta0 <span style="color: #666666">=</span> radians(a)
    t, theta, omega, S, drag <span style="color: #666666">=</span> simulate_pendulum(alpha, theta0, dt, T)
    mg <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ones(S<span style="color: #666666">.</span>size)
    <span style="color: #408080; font-style: italic"># Visualize drag force 5 times as large</span>
    drag <span style="color: #666666">*=</span> <span style="color: #666666">5</span>
    <span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&#39;NOTE: drag force magnified 5 times!!&#39;</span>)

    <span style="color: #408080; font-style: italic"># Draw animation</span>
    <span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">time</span>
    <span style="color: #008000; font-weight: bold">for</span> time_level, t_ <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(t):
        pendulum(theta, S, mg, drag, t_, time_level)
        time<span style="color: #666666">.</span>sleep(<span style="color: #666666">0.2</span>)

    <span style="color: #408080; font-style: italic"># Make videos</span>
    prog <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;ffmpeg&#39;</span>
    filename <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;tmp_</span><span style="color: #BB6688; font-weight: bold">%04d</span><span style="color: #BA2121">.png&#39;</span>
    fps <span style="color: #666666">=</span> <span style="color: #666666">6</span>
    codecs <span style="color: #666666">=</span> {<span style="color: #BA2121">&#39;flv&#39;</span>: <span style="color: #BA2121">&#39;flv&#39;</span>, <span style="color: #BA2121">&#39;mp4&#39;</span>: <span style="color: #BA2121">&#39;libx264&#39;</span>,
              <span style="color: #BA2121">&#39;webm&#39;</span>: <span style="color: #BA2121">&#39;libvpx&#39;</span>, <span style="color: #BA2121">&#39;ogg&#39;</span>: <span style="color: #BA2121">&#39;libtheora&#39;</span>}
    <span style="color: #008000; font-weight: bold">for</span> ext <span style="color: #AA22FF; font-weight: bold">in</span> codecs:
        lib <span style="color: #666666">=</span> codecs[ext]
        cmd <span style="color: #666666">=</span> <span style="color: #BA2121">&#39;</span><span style="color: #BB6688; font-weight: bold">%(prog)s</span><span style="color: #BA2121"> -i </span><span style="color: #BB6688; font-weight: bold">%(filename)s</span><span style="color: #BA2121"> -r </span><span style="color: #BB6688; font-weight: bold">%(fps)s</span><span style="color: #BA2121"> &#39;</span> <span style="color: #666666">%</span> <span style="color: #008000">vars</span>()
        cmd <span style="color: #666666">+=</span> <span style="color: #BA2121">&#39;-vcodec </span><span style="color: #BB6688; font-weight: bold">%(lib)s</span><span style="color: #BA2121"> movie.</span><span style="color: #BB6688; font-weight: bold">%(ext)s</span><span style="color: #BA2121">&#39;</span> <span style="color: #666666">%</span> <span style="color: #008000">vars</span>()
        <span style="color: #008000; font-weight: bold">print</span>(cmd)
        os<span style="color: #666666">.</span>system(cmd)
</pre></div>
<p>

<div>
<video  loop controls width='640' height='365' preload='none'>
    <source src='https://github.com/hplgit/pysketcher/raw/master/doc/pub/tutorial/mov-tut/pendulum/movie.mp4'  type='video/mp4;  codecs="avc1.42E01E, mp4a.40.2"'>
    <source src='https://github.com/hplgit/pysketcher/raw/master/doc/pub/tutorial/mov-tut/pendulum/movie.webm' type='video/webm; codecs="vp8, vorbis"'>
    <source src='https://github.com/hplgit/pysketcher/raw/master/doc/pub/tutorial/mov-tut/pendulum/movie.ogg'  type='video/ogg;  codecs="theora, vorbis"'>
</video>
</div>
<p><em>The drag force is magnified 5 times (compared to \( mg \) and \( S \)!</em></p>

<!-- Issue warning if in a Safari browser -->
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if (!!(window.safari)) {
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</script>

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