import numpy as np
import matplotlib.pyplot as plt
plt.style.use('fivethirtyeight')
plt.rcParams.update({
"text.usetex": True,
"font.family": "sans-serif"
})
#plt.rcParams['text.latex.preamble']=[r'sansmath']
from IPython.display import YouTubeVideo
YouTubeVideo('uOq4AbvWF08')
Rolling on hemisphere#
In this example, a small ball with negligible moment of inertia rolls on a hemisphere and loses contact at some point. Your goal is to determine the point where the ball loses contact with the hemisphere. The hemisphere has radius, \(R=1~m\), and the ball has mass \(m=1~kg\).
There are two states in this problem:
Ball and hemisphere are in contact with normal force \(N>0\)
Ball and hemisphere are not in contact, so normal force \(N=0\)
There cannot be a state where \(N<0\) because that would mean there is an attractive force between the ball and hemisphere.
Equation of motion and contact force#
Using the free body diagram and kinetic diagram, you can derive these two equations for state 1 (contact):
\(\ddot{\theta} = \frac{g}{R}\sin\theta\)
\(N = mg\cos\theta-mR\dot{\theta}^2\)
Integrate equation 1 by using the chain rule,
\(\frac{d\theta}{d\theta}\frac{d\dot{\theta}}{dt} = \frac{g}{R}\sin\theta\)
\(\frac{d\theta}{dt}\frac{d\dot{\theta}}{d\theta} = \frac{g}{R}\sin\theta\)
\(\int_{\dot{\theta}_0}^{\dot{\theta}}\dot{\theta}d\dot{\theta} = \int_{\theta_0}^{\theta}\frac{2g}{R}\sin\theta\)
\(\dot{\theta} = \sqrt{\dot{\theta}_0^2+\frac{g}{R}(\cos\theta_0-\cos\theta)}\)
Now, you plot the relation between \(\theta\) and \(\dot{\theta}\) for initial conditions,
\(\theta_0 = 0~rad\) and \(\dot{\theta}_0=0~rad/s\)
\(\theta_0 = 0~rad\) and \(\dot{\theta}_0=2~rad/s\)
\(\theta_0 = 0~rad\) and \(\dot{\theta}_0=4~rad/s\)
__Reflection:__When do you expect the ball to lose contact for 0, 2, and 4 rad/s angular velocities? Will it happen sooner or later if the ball is travelling faster at the top of the hemisphere?
theta = np.linspace(0,np.pi)
dtheta = lambda dt0,theta: np.sqrt(dt0**2+2*9.81/1*(1-np.cos(theta)))
plt.plot(theta*180/np.pi, dtheta(0, theta), label=r'$\dot{\theta}_0=0~rad/s$')
plt.plot(theta*180/np.pi, dtheta(2, theta), label=r'$\dot{\theta}_0=2~rad/s$')
plt.plot(theta*180/np.pi, dtheta(4, theta), label=r'$\dot{\theta}_0=4~rad/s$')
plt.title('speed vs angle along hemisphere')
plt.legend()
plt.xlabel(r'$\theta$ (deg)')
plt.ylabel(r'$\dot{\theta}$ (rad/s)');
Error in callback <function _draw_all_if_interactive at 0x7f2206dc5280> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/pyplot.py:268, in _draw_all_if_interactive()
266 def _draw_all_if_interactive() -> None:
267 if matplotlib.is_interactive():
--> 268 draw_all()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/_pylab_helpers.py:131, in Gcf.draw_all(cls, force)
129 for manager in cls.get_all_fig_managers():
130 if force or manager.canvas.figure.stale:
--> 131 manager.canvas.draw_idle()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:1905, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1903 if not self._is_idle_drawing:
1904 with self._idle_draw_cntx():
-> 1905 self.draw(*args, **kwargs)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:387, in FigureCanvasAgg.draw(self)
384 # Acquire a lock on the shared font cache.
385 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
386 else nullcontext()):
--> 387 self.figure.draw(self.renderer)
388 # A GUI class may be need to update a window using this draw, so
389 # don't forget to call the superclass.
390 super().draw()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3101, in _AxesBase.draw(self, renderer)
3098 for spine in self.spines.values():
3099 artists.remove(spine)
-> 3101 self._update_title_position(renderer)
3103 if not self.axison:
3104 for _axis in self._axis_map.values():
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3045, in _AxesBase._update_title_position(self, renderer)
3043 top = max(top, bb.ymax)
3044 if title.get_text():
-> 3045 ax.yaxis.get_tightbbox(renderer) # update offsetText
3046 if ax.yaxis.offsetText.get_text():
3047 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1372, in Axis.get_tightbbox(self, renderer, for_layout_only)
1369 renderer = self.figure._get_renderer()
1370 ticks_to_draw = self._update_ticks()
-> 1372 self._update_label_position(renderer)
1374 # go back to just this axis's tick labels
1375 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2654, in YAxis._update_label_position(self, renderer)
2650 return
2652 # get bounding boxes for this axis and any siblings
2653 # that have been set by `fig.align_ylabels()`
-> 2654 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2655 x, y = self.label.get_position()
2656 if self.label_position == 'left':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2206, in Axis._get_tick_boxes_siblings(self, renderer)
2204 axis = ax._axis_map[name]
2205 ticks_to_draw = axis._update_ticks()
-> 2206 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2207 bboxes.extend(tlb)
2208 bboxes2.extend(tlb2)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
338 pass
339 else:
--> 340 return printer(obj)
341 # Finally look for special method names
342 method = get_real_method(obj, self.print_method)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2172 # we do this instead of `self.figure.draw_without_rendering`
2173 # so that we can inject the orientation
2174 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175 self.figure.draw(renderer)
2176 if bbox_inches:
2177 if bbox_inches == "tight":
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3101, in _AxesBase.draw(self, renderer)
3098 for spine in self.spines.values():
3099 artists.remove(spine)
-> 3101 self._update_title_position(renderer)
3103 if not self.axison:
3104 for _axis in self._axis_map.values():
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3045, in _AxesBase._update_title_position(self, renderer)
3043 top = max(top, bb.ymax)
3044 if title.get_text():
-> 3045 ax.yaxis.get_tightbbox(renderer) # update offsetText
3046 if ax.yaxis.offsetText.get_text():
3047 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1372, in Axis.get_tightbbox(self, renderer, for_layout_only)
1369 renderer = self.figure._get_renderer()
1370 ticks_to_draw = self._update_ticks()
-> 1372 self._update_label_position(renderer)
1374 # go back to just this axis's tick labels
1375 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2654, in YAxis._update_label_position(self, renderer)
2650 return
2652 # get bounding boxes for this axis and any siblings
2653 # that have been set by `fig.align_ylabels()`
-> 2654 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2655 x, y = self.label.get_position()
2656 if self.label_position == 'left':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2206, in Axis._get_tick_boxes_siblings(self, renderer)
2204 axis = ax._axis_map[name]
2205 ticks_to_draw = axis._update_ticks()
-> 2206 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2207 bboxes.extend(tlb)
2208 bboxes2.extend(tlb2)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
<Figure size 640x480 with 1 Axes>
Normal force prediction#
The second equation relates the normal force, \(N\) to \(\theta\) and \(\dot{\theta}\). Now, plot the \(N-vs-\theta\).
\(N = mg\cos\theta-mR\dot{\theta}^2\)
If this equation predicts \(N<0\), then the ball has lost contact with the hemisphere. It will transition into a freefalling object with initial position and velocity,
When N=0, transition state:
\(\mathbf{r} = R\hat{e}_r=R(\sin\theta\hat{i}+\cos\theta\hat{j})\)
\(\mathbf{v} = R\dot{\theta}\hat{e}_{\theta}= R\dot{\theta}(\cos\theta\hat{i}-\sin\theta\hat{j})\)
Below, you see the plots for \(N-vs-\theta\) for \(\theta = 0-180^o\),
N0 = 9.81*np.cos(theta)-1*dtheta(0, theta)**2
N2 = 9.81*np.cos(theta)-1*dtheta(2, theta)**2
N4 = 9.81*np.cos(theta)-1*dtheta(4, theta)**2
plt.plot(theta*180/np.pi, N0, label=r'$\dot{\theta}_0=0~rad/s$')
plt.plot(theta*180/np.pi, N2, label=r'$\dot{\theta}_0=2~rad/s$')
plt.plot(theta*180/np.pi, N4, label=r'$\dot{\theta}_0=4~rad/s$')
plt.xlabel(r'$\theta$ (deg)')
plt.ylabel(r'N (N)')
plt.legend()
<matplotlib.legend.Legend at 0x7f21ec668f40>
Error in callback <function _draw_all_if_interactive at 0x7f2206dc5280> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/pyplot.py:268, in _draw_all_if_interactive()
266 def _draw_all_if_interactive() -> None:
267 if matplotlib.is_interactive():
--> 268 draw_all()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/_pylab_helpers.py:131, in Gcf.draw_all(cls, force)
129 for manager in cls.get_all_fig_managers():
130 if force or manager.canvas.figure.stale:
--> 131 manager.canvas.draw_idle()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:1905, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1903 if not self._is_idle_drawing:
1904 with self._idle_draw_cntx():
-> 1905 self.draw(*args, **kwargs)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:387, in FigureCanvasAgg.draw(self)
384 # Acquire a lock on the shared font cache.
385 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
386 else nullcontext()):
--> 387 self.figure.draw(self.renderer)
388 # A GUI class may be need to update a window using this draw, so
389 # don't forget to call the superclass.
390 super().draw()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3137, in _AxesBase.draw(self, renderer)
3134 if artists_rasterized:
3135 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3137 mimage._draw_list_compositing_images(
3138 renderer, self, artists, self.figure.suppressComposite)
3140 renderer.close_group('axes')
3141 self.stale = False
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1424, in Axis.draw(self, renderer)
1421 renderer.open_group(__name__, gid=self.get_gid())
1423 ticks_to_draw = self._update_ticks()
-> 1424 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1426 for tick in ticks_to_draw:
1427 tick.draw(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
338 pass
339 else:
--> 340 return printer(obj)
341 # Finally look for special method names
342 method = get_real_method(obj, self.print_method)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2172 # we do this instead of `self.figure.draw_without_rendering`
2173 # so that we can inject the orientation
2174 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175 self.figure.draw(renderer)
2176 if bbox_inches:
2177 if bbox_inches == "tight":
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3137, in _AxesBase.draw(self, renderer)
3134 if artists_rasterized:
3135 _draw_rasterized(self.figure, artists_rasterized, renderer)
-> 3137 mimage._draw_list_compositing_images(
3138 renderer, self, artists, self.figure.suppressComposite)
3140 renderer.close_group('axes')
3141 self.stale = False
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1424, in Axis.draw(self, renderer)
1421 renderer.open_group(__name__, gid=self.get_gid())
1423 ticks_to_draw = self._update_ticks()
-> 1424 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
1426 for tick in ticks_to_draw:
1427 tick.draw(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
<Figure size 640x480 with 1 Axes>
Solving for loss-of-contact#
The graph above reveals that at \(\dot{\theta}_0=4~rad/s\), the ball never makes contact with the hemisphere. It skims the top, and falls as
\(x(t) = 4t~[m]\)
\(y(t) = 1-\frac{g}{2}t^2~[m]\)
In the cases of \(\dot{\theta}_0=0~and~2~rad/s\), the ball loses contact between \(\theta=25^o-50^o\). You can use algebra to solve for \(N(\theta^*)=0\), where \(\theta^*\) is the location where contact is lost.
or
You have Python here that can solve nolinear problems. Import fsolve from the scipy.optimize library.
from scipy.optimize import fsolve
Defining and solving a nonlinear equation#
You have two equations,
\(\dot{\theta} = \sqrt{\dot{\theta}_0^2+\frac{g}{R}(\cos\theta_0-\cos\theta)} = g(\dot{\theta}_0,~\theta)\)
\(N = mg\cos\theta-mR\dot{\theta}^2 = f(\theta,~\dot{\theta}) = f(\theta, g(\dot{\theta}_0,~\theta))\)
For clarity, you can define equation (1) first, then use it in equation (2) as such
dtheta = lambda dt0,theta: np.sqrt(dt0**2+2*9.81/1*(1-np.cos(theta)))
N = lambda dt0, theta: 9.81*np.cos(theta)-1*dtheta(dt0, theta)**2
thetastar_0 = fsolve(lambda x: N(0, x), np.pi/3)*180/np.pi
thetastar_2 = fsolve(lambda x: N(2, x), np.pi/3)*180/np.pi
plt.plot(theta*180/np.pi, N0, label=r'$\dot{\theta}_0=0~rad/s$')
plt.plot(theta*180/np.pi, N2, label=r'$\dot{\theta}_0=2~rad/s$')
plt.plot(theta*180/np.pi, N4, label=r'$\dot{\theta}_0=4~rad/s$')
plt.plot(thetastar_0[0], N(0,thetastar_0*np.pi/180), 'b*', markersize=20)
plt.plot(thetastar_2[0], N(2,thetastar_2*np.pi/180), 'r*', markersize=20)
plt.xlabel(r'$\theta$ (deg)')
plt.ylabel(r'N (N)')
plt.title(r'Lose contact at $\theta=$ '+'{:.2f} and {:.2f}'.format(thetastar_0[0], thetastar_2[0]))
plt.ylim(-20,11)
plt.legend()
<matplotlib.legend.Legend at 0x7f21dccad670>
Error in callback <function _draw_all_if_interactive at 0x7f2206dc5280> (for post_execute), with arguments args (),kwargs {}:
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/pyplot.py:268, in _draw_all_if_interactive()
266 def _draw_all_if_interactive() -> None:
267 if matplotlib.is_interactive():
--> 268 draw_all()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/_pylab_helpers.py:131, in Gcf.draw_all(cls, force)
129 for manager in cls.get_all_fig_managers():
130 if force or manager.canvas.figure.stale:
--> 131 manager.canvas.draw_idle()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:1905, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
1903 if not self._is_idle_drawing:
1904 with self._idle_draw_cntx():
-> 1905 self.draw(*args, **kwargs)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:387, in FigureCanvasAgg.draw(self)
384 # Acquire a lock on the shared font cache.
385 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
386 else nullcontext()):
--> 387 self.figure.draw(self.renderer)
388 # A GUI class may be need to update a window using this draw, so
389 # don't forget to call the superclass.
390 super().draw()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3101, in _AxesBase.draw(self, renderer)
3098 for spine in self.spines.values():
3099 artists.remove(spine)
-> 3101 self._update_title_position(renderer)
3103 if not self.axison:
3104 for _axis in self._axis_map.values():
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3045, in _AxesBase._update_title_position(self, renderer)
3043 top = max(top, bb.ymax)
3044 if title.get_text():
-> 3045 ax.yaxis.get_tightbbox(renderer) # update offsetText
3046 if ax.yaxis.offsetText.get_text():
3047 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1372, in Axis.get_tightbbox(self, renderer, for_layout_only)
1369 renderer = self.figure._get_renderer()
1370 ticks_to_draw = self._update_ticks()
-> 1372 self._update_label_position(renderer)
1374 # go back to just this axis's tick labels
1375 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2654, in YAxis._update_label_position(self, renderer)
2650 return
2652 # get bounding boxes for this axis and any siblings
2653 # that have been set by `fig.align_ylabels()`
-> 2654 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2655 x, y = self.label.get_position()
2656 if self.label_position == 'left':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2206, in Axis._get_tick_boxes_siblings(self, renderer)
2204 axis = ax._axis_map[name]
2205 ticks_to_draw = axis._update_ticks()
-> 2206 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2207 bboxes.extend(tlb)
2208 bboxes2.extend(tlb2)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
---------------------------------------------------------------------------
FileNotFoundError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:250, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
249 try:
--> 250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:424, in check_output(timeout, *popenargs, **kwargs)
422 kwargs['input'] = empty
--> 424 return run(*popenargs, stdout=PIPE, timeout=timeout, check=True,
425 **kwargs).stdout
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:505, in run(input, capture_output, timeout, check, *popenargs, **kwargs)
503 kwargs['stderr'] = PIPE
--> 505 with Popen(*popenargs, **kwargs) as process:
506 try:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:951, in Popen.__init__(self, args, bufsize, executable, stdin, stdout, stderr, preexec_fn, close_fds, shell, cwd, env, universal_newlines, startupinfo, creationflags, restore_signals, start_new_session, pass_fds, user, group, extra_groups, encoding, errors, text, umask)
948 self.stderr = io.TextIOWrapper(self.stderr,
949 encoding=encoding, errors=errors)
--> 951 self._execute_child(args, executable, preexec_fn, close_fds,
952 pass_fds, cwd, env,
953 startupinfo, creationflags, shell,
954 p2cread, p2cwrite,
955 c2pread, c2pwrite,
956 errread, errwrite,
957 restore_signals,
958 gid, gids, uid, umask,
959 start_new_session)
960 except:
961 # Cleanup if the child failed starting.
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/subprocess.py:1837, in Popen._execute_child(self, args, executable, preexec_fn, close_fds, pass_fds, cwd, env, startupinfo, creationflags, shell, p2cread, p2cwrite, c2pread, c2pwrite, errread, errwrite, restore_signals, gid, gids, uid, umask, start_new_session)
1836 err_msg = os.strerror(errno_num)
-> 1837 raise child_exception_type(errno_num, err_msg, err_filename)
1838 raise child_exception_type(err_msg)
FileNotFoundError: [Errno 2] No such file or directory: 'latex'
The above exception was the direct cause of the following exception:
RuntimeError Traceback (most recent call last)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
338 pass
339 else:
--> 340 return printer(obj)
341 # Finally look for special method names
342 method = get_real_method(obj, self.print_method)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
149 from matplotlib.backend_bases import FigureCanvasBase
150 FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
153 data = bytes_io.getvalue()
154 if fmt == 'svg':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
2172 # we do this instead of `self.figure.draw_without_rendering`
2173 # so that we can inject the orientation
2174 with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175 self.figure.draw(renderer)
2176 if bbox_inches:
2177 if bbox_inches == "tight":
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
93 @wraps(draw)
94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95 result = draw(artist, renderer, *args, **kwargs)
96 if renderer._rasterizing:
97 renderer.stop_rasterizing()
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
3159 # ValueError can occur when resizing a window.
3161 self.patch.draw(renderer)
-> 3162 mimage._draw_list_compositing_images(
3163 renderer, self, artists, self.suppressComposite)
3165 renderer.close_group('figure')
3166 finally:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
130 if not_composite or not has_images:
131 for a in artists:
--> 132 a.draw(renderer)
133 else:
134 # Composite any adjacent images together
135 image_group = []
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
69 if artist.get_agg_filter() is not None:
70 renderer.start_filter()
---> 72 return draw(artist, renderer)
73 finally:
74 if artist.get_agg_filter() is not None:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3101, in _AxesBase.draw(self, renderer)
3098 for spine in self.spines.values():
3099 artists.remove(spine)
-> 3101 self._update_title_position(renderer)
3103 if not self.axison:
3104 for _axis in self._axis_map.values():
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axes/_base.py:3045, in _AxesBase._update_title_position(self, renderer)
3043 top = max(top, bb.ymax)
3044 if title.get_text():
-> 3045 ax.yaxis.get_tightbbox(renderer) # update offsetText
3046 if ax.yaxis.offsetText.get_text():
3047 bb = ax.yaxis.offsetText.get_tightbbox(renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1372, in Axis.get_tightbbox(self, renderer, for_layout_only)
1369 renderer = self.figure._get_renderer()
1370 ticks_to_draw = self._update_ticks()
-> 1372 self._update_label_position(renderer)
1374 # go back to just this axis's tick labels
1375 tlb1, tlb2 = self._get_ticklabel_bboxes(ticks_to_draw, renderer)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2654, in YAxis._update_label_position(self, renderer)
2650 return
2652 # get bounding boxes for this axis and any siblings
2653 # that have been set by `fig.align_ylabels()`
-> 2654 bboxes, bboxes2 = self._get_tick_boxes_siblings(renderer=renderer)
2655 x, y = self.label.get_position()
2656 if self.label_position == 'left':
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:2206, in Axis._get_tick_boxes_siblings(self, renderer)
2204 axis = ax._axis_map[name]
2205 ticks_to_draw = axis._update_ticks()
-> 2206 tlb, tlb2 = axis._get_ticklabel_bboxes(ticks_to_draw, renderer)
2207 bboxes.extend(tlb)
2208 bboxes2.extend(tlb2)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in Axis._get_ticklabel_bboxes(self, ticks, renderer)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/axis.py:1351, in <listcomp>(.0)
1349 if renderer is None:
1350 renderer = self.figure._get_renderer()
-> 1351 return ([tick.label1.get_window_extent(renderer)
1352 for tick in ticks if tick.label1.get_visible()],
1353 [tick.label2.get_window_extent(renderer)
1354 for tick in ticks if tick.label2.get_visible()])
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:959, in Text.get_window_extent(self, renderer, dpi)
954 raise RuntimeError(
955 "Cannot get window extent of text w/o renderer. You likely "
956 "want to call 'figure.draw_without_rendering()' first.")
958 with cbook._setattr_cm(self.figure, dpi=dpi):
--> 959 bbox, info, descent = self._get_layout(self._renderer)
960 x, y = self.get_unitless_position()
961 x, y = self.get_transform().transform((x, y))
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:373, in Text._get_layout(self, renderer)
370 ys = []
372 # Full vertical extent of font, including ascenders and descenders:
--> 373 _, lp_h, lp_d = _get_text_metrics_with_cache(
374 renderer, "lp", self._fontproperties,
375 ismath="TeX" if self.get_usetex() else False, dpi=self.figure.dpi)
376 min_dy = (lp_h - lp_d) * self._linespacing
378 for i, line in enumerate(lines):
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:69, in _get_text_metrics_with_cache(renderer, text, fontprop, ismath, dpi)
66 """Call ``renderer.get_text_width_height_descent``, caching the results."""
67 # Cached based on a copy of fontprop so that later in-place mutations of
68 # the passed-in argument do not mess up the cache.
---> 69 return _get_text_metrics_with_cache_impl(
70 weakref.ref(renderer), text, fontprop.copy(), ismath, dpi)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/text.py:77, in _get_text_metrics_with_cache_impl(renderer_ref, text, fontprop, ismath, dpi)
73 @functools.lru_cache(4096)
74 def _get_text_metrics_with_cache_impl(
75 renderer_ref, text, fontprop, ismath, dpi):
76 # dpi is unused, but participates in cache invalidation (via the renderer).
---> 77 return renderer_ref().get_text_width_height_descent(text, fontprop, ismath)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:212, in RendererAgg.get_text_width_height_descent(self, s, prop, ismath)
210 _api.check_in_list(["TeX", True, False], ismath=ismath)
211 if ismath == "TeX":
--> 212 return super().get_text_width_height_descent(s, prop, ismath)
214 if ismath:
215 ox, oy, width, height, descent, font_image = \
216 self.mathtext_parser.parse(s, self.dpi, prop)
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/backend_bases.py:597, in RendererBase.get_text_width_height_descent(self, s, prop, ismath)
593 fontsize = prop.get_size_in_points()
595 if ismath == 'TeX':
596 # todo: handle properties
--> 597 return self.get_texmanager().get_text_width_height_descent(
598 s, fontsize, renderer=self)
600 dpi = self.points_to_pixels(72)
601 if ismath:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:363, in TexManager.get_text_width_height_descent(cls, tex, fontsize, renderer)
361 if tex.strip() == '':
362 return 0, 0, 0
--> 363 dvifile = cls.make_dvi(tex, fontsize)
364 dpi_fraction = renderer.points_to_pixels(1.) if renderer else 1
365 with dviread.Dvi(dvifile, 72 * dpi_fraction) as dvi:
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:295, in TexManager.make_dvi(cls, tex, fontsize)
293 with TemporaryDirectory(dir=cwd) as tmpdir:
294 tmppath = Path(tmpdir)
--> 295 cls._run_checked_subprocess(
296 ["latex", "-interaction=nonstopmode", "--halt-on-error",
297 f"--output-directory={tmppath.name}",
298 f"{texfile.name}"], tex, cwd=cwd)
299 (tmppath / Path(dvifile).name).replace(dvifile)
300 return dvifile
File /opt/hostedtoolcache/Python/3.9.21/x64/lib/python3.9/site-packages/matplotlib/texmanager.py:254, in TexManager._run_checked_subprocess(cls, command, tex, cwd)
250 report = subprocess.check_output(
251 command, cwd=cwd if cwd is not None else cls._texcache,
252 stderr=subprocess.STDOUT)
253 except FileNotFoundError as exc:
--> 254 raise RuntimeError(
255 f'Failed to process string with tex because {command[0]} '
256 'could not be found') from exc
257 except subprocess.CalledProcessError as exc:
258 raise RuntimeError(
259 '{prog} was not able to process the following string:\n'
260 '{tex!r}\n\n'
(...)
267 exc=exc.output.decode('utf-8', 'backslashreplace'))
268 ) from None
RuntimeError: Failed to process string with tex because latex could not be found
<Figure size 640x480 with 1 Axes>
Wrapping up#
In this notebook, you compared two possible states for a ball rolling on a hemisphere: contact and no contact. You used free body diagrams, kinetic diagrams, and the chain rule to find the transition between contact and no contact.
You used scipy.optimize.fsolve to solve for when a function is 0.