Taylor problem 4.37 template#

In this problem the potential energy \(U(\phi)\) depends on the masses \(m\) and \(M\), with the qualitative behavior of the system starting at \(\phi=0\) depending on the ratio \(m/M\). We will determine the critical value of this ratio from the graphs of the potential energy (corresponding to taking \(\phi = 0\), meaning \(M\) at its lowest, as the zero of potential energy):

\[\begin{align} U(\phi) = M g R (1-\cos\phi) - m g R \phi \;. \end{align}\]

This notebook has a different potential energy function that has the same qualitative behavior. Your job is to implement the correct formula. Places you need to change are marked with ###.

%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt

Make a plot of the potential with simple values of the constants.

def U(phi, M_big, m_small, R=1., g=1.):
    """Potential energy for the system in Taylor, problem 4.37."""
    return M_big*g*R*(1.-np.exp(-phi**4/2.)) - m_small*g*R*phi  ### made up U(\phi)

# set non-essential constants to 1 (doesn't change shapes)
R = 1.
g = 1.
M_big = 1.

phi_pts = np.arange(0., 3.5, 0.01)

fig_U = plt.figure()
ax_U = fig_U.add_subplot(1,1,1)

m_small = 0.5  ### different values
U_of_phi_pts = U(phi_pts, M_big, m_small)
ax_U.plot(phi_pts, U_of_phi_pts, 'b-', label=fr'$m/M = {m_small:0.1f}$', lw=2)

m_small = 0.6  ### different values
U_of_phi_pts = U(phi_pts, M_big, m_small)
ax_U.plot(phi_pts, U_of_phi_pts, 'g-', label=fr'$m/M = {m_small:0.1f}$', lw=2)


ax_U.set_xlabel(r'$\phi$')
ax_U.set_ylabel(r'$U(\phi)$')

ax_U.set_xlim(0, 3.5)
#ax_U.set_ylim(-1., 6.)

# draw a black horizontal line at 0 but with alpha=0.3 so less distracting
ax_U.axhline(0., color='black', alpha=0.3 )  
ax_U.legend();

Let’s plot it with a slider to find the critical value (where the curve is tangent to the line at zero.

Ok, let’s change \(m/M\) with a slider:

from ipywidgets import interact, fixed
import ipywidgets as widgets

# set non-essential constants to 1 (doesn't change shapes)
R = 1.
g = 1.
M_big = 1.

def plot_U_given_m_over_M(m_over_M, M_big=1., R=1., g=1.):
    ""
    phi_pts = np.arange(0., 3.5, 0.01)

    fig_U = plt.figure()
    ax_U = fig_U.add_subplot(1,1,1)

    m_small = m_over_M * M_big
    U_of_phi_pts = U(phi_pts, M_big, m_small)
    ax_U.plot(phi_pts, U_of_phi_pts, 'r-', 
              label=rf'$m/M = {m_over_M:.2f}$', lw=2)

    ax_U.set_xlabel(r'$\phi$')
    ax_U.set_ylabel(r'$U(\phi)$')

    ax_U.set_xlim(0, 3.5)
    #ax_U.set_ylim(-1., 6.)

    # draw a black horizontal line at 0 but with alpha=0.3 so less distracting
    ax_U.axhline(0., color='black', alpha=0.3 )  

    ax_U.legend();

# fix the values of M_big, R, and g
interact(plot_U_given_m_over_M, 
         m_over_M = widgets.FloatSlider(min=0., max=1., step=0.01, 
                                        continuous_update=False,
                                        value=0.5, readout_format='.3f'), 
         M_big=fixed(1.), R=fixed(1.), g=fixed(1.));

Note that you can type numbers in the box to be more precise.

Theoretical and computational mechanics

Taylor problem 4.37 template

Taylor problem 4.37 template#