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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Modeling and Simulation in Python
Case study
Copyright 2017 Allen Downey
License
Step1:  Yo-yo
Suppose you are holding a yo-yo with a length of string wound around its axle, and you drop it while holding the end of the string stationary.  As gravity accelerates the yo-yo downward, tension in the string exerts a force upward.  Since this force acts on a point offset from the center of mass, it exerts a torque that causes the yo-yo to spin.
This figure shows the forces on the yo-yo and the resulting torque.  The outer shaded area shows the body of the yo-yo.  The inner shaded area shows the rolled up string, the radius of which changes as the yo-yo unrolls.
In this model, we can't figure out the linear and angular acceleration independently; we have to solve a system of equations
Step2:  The results are
$T      = m g I / I^*   $
$a      = -m g r^2 / I^* $
$\alpha = m g r / I^*    $
where $I^*$ is the augmented moment of inertia, $I + m r^2$.
You can also see the derivation of these equations in this video.
To simulate the system, we don't really need $T$; we can plug $a$ and $\alpha$ directly into the slope function.
Step3:  Exercise
Step5:  Here's a make_system function that computes I and k based on the system parameters.
I estimated I by modeling the yo-yo as a solid cylinder with uniform density (see here).
In reality, the distribution of weight in a yo-yo is often designed to achieve desired effects.  But we'll keep it simple.
Step6:  Testing make_system
Step8:  Write a slope function for this system, using these results from the book
Step9:  Test your slope function with the initial paramss.
Step11:  Write an event function that will stop the simulation when y is 0.
Step12:  Test your event function
Step13:  Then run the simulation.
Step14:  Check the final state.  If things have gone according to plan, the final value of y should be close to 0.
Step15:  Plot the results.
theta should increase and accelerate.
Step16:  y should decrease and accelerate down.
Step17:  Plot velocity as a function of time; is the yo-yo accelerating?
Step18:  Use gradient to estimate the derivative of v.  How does the acceleration of the yo-yo compare to g?
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Python Code:
# Configure Jupyter so figures appear in the notebook
%matplotlib inline
# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'
# import functions from the modsim.py module
from modsim import *
Explanation: Modeling and Simulation in Python
Case study
Copyright 2017 Allen Downey
License: Creative Commons Attribution 4.0 International
End of explanation
from sympy import init_printing, symbols, Eq, solve
init_printing()
T, a, alpha, I, m, g, r = symbols('T a alpha I m g r')
eq1 = Eq(a, -r * alpha)
eq2 = Eq(T - m * g, m * a)
eq3 = Eq(T * r, I * alpha)
soln = solve([eq1, eq2, eq3], [T, a, alpha])
soln[T]
soln[a]
soln[alpha]
Explanation: Yo-yo
Suppose you are holding a yo-yo with a length of string wound around its axle, and you drop it while holding the end of the string stationary.  As gravity accelerates the yo-yo downward, tension in the string exerts a force upward.  Since this force acts on a point offset from the center of mass, it exerts a torque that causes the yo-yo to spin.
This figure shows the forces on the yo-yo and the resulting torque.  The outer shaded area shows the body of the yo-yo.  The inner shaded area shows the rolled up string, the radius of which changes as the yo-yo unrolls.
In this model, we can't figure out the linear and angular acceleration independently; we have to solve a system of equations:
$\sum F = m a $
$\sum \tau = I \alpha$
where the summations indicate that we are adding up forces and torques.
As in the previous examples, linear and angular velocity are related because of the way the string unrolls:
$\frac{dy}{dt} = -r \frac{d \theta}{dt} $
In this example, the linear and angular accelerations have opposite sign.  As the yo-yo rotates counter-clockwise, $\theta$ increases and $y$, which is the length of the rolled part of the string, decreases.
Taking the derivative of both sides yields a similar relationship between linear and angular acceleration:
$\frac{d^2 y}{dt^2} = -r \frac{d^2 \theta}{dt^2} $
Which we can write more concisely:
$ a = -r \alpha $
This relationship is not a general law of nature; it is specific to scenarios like this where there is rolling without stretching or slipping.
Because of the way we've set up the problem, $y$ actually has two meanings: it represents the length of the rolled string and the height of the yo-yo, which decreases as the yo-yo falls.  Similarly, $a$ represents acceleration in the length of the rolled string and the height of the yo-yo.
We can compute the acceleration of the yo-yo by adding up the linear forces:
$\sum F = T - mg = ma $
Where $T$ is positive because the tension force points up, and $mg$ is negative because gravity points down.
Because gravity acts on the center of mass, it creates no torque, so the only torque is due to tension:
$\sum \tau = T r = I \alpha $
Positive (upward) tension yields positive (counter-clockwise) angular acceleration.
Now we have three equations in three unknowns, $T$, $a$, and $\alpha$, with $I$, $m$, $g$, and $r$ as known parameters.  It is simple enough to solve these equations by hand, but we can also get SymPy to do it for us.
End of explanation
radian = UNITS.radian
m = UNITS.meter
s = UNITS.second
kg = UNITS.kilogram
N = UNITS.newton
Explanation: The results are
$T      = m g I / I^*   $
$a      = -m g r^2 / I^* $
$\alpha = m g r / I^*    $
where $I^*$ is the augmented moment of inertia, $I + m r^2$.
You can also see the derivation of these equations in this video.
To simulate the system, we don't really need $T$; we can plug $a$ and $\alpha$ directly into the slope function.
End of explanation
params = Params(Rmin = 8e-3 * m,
                Rmax = 16e-3 * m,
                Rout = 35e-3 * m,
                mass = 50e-3 * kg,
                L = 1 * m,
                g = 9.8 * m / s**2,
                t_end = 1 * s)
Explanation: Exercise:  Simulate the descent of a yo-yo.  How long does it take to reach the end of the string?
I provide a Params object with the system parameters:
Rmin is the radius of the axle.  Rmax is the radius of the axle plus rolled string.
Rout is the radius of the yo-yo body.  mass is the total mass of the yo-yo, ignoring the string.  
L is the length of the string.
g is the acceleration of gravity.
End of explanation
def make_system(params):
    Make a system object.
    
    params: Params with Rmin, Rmax, Rout, 
                              mass, L, g, t_end
    
    returns: System with init, k, Rmin, Rmax, mass,
                         I, g, ts
    
    L, mass = params.L, params.mass
    Rout, Rmax, Rmin = params.Rout, params.Rmax, params.Rmin 
    
    init = State(theta = 0 * radian,
                 omega = 0 * radian/s,
                 y = L,
                 v = 0 * m / s)
    
    I = mass * Rout**2 / 2
    k = (Rmax**2 - Rmin**2) / 2 / L / radian    
    
    return System(params, init=init, I=I, k=k)
Explanation: Here's a make_system function that computes I and k based on the system parameters.
I estimated I by modeling the yo-yo as a solid cylinder with uniform density (see here).
In reality, the distribution of weight in a yo-yo is often designed to achieve desired effects.  But we'll keep it simple.
End of explanation
system = make_system(params)
system.init
Explanation: Testing make_system
End of explanation
# Solution
def slope_func(state, t, system):
    Computes the derivatives of the state variables.
    
    state: State object with theta, omega, y, v
    t: time
    system: System object with Rmin, k, I, mass
    
    returns: sequence of derivatives
    
    theta, omega, y, v = state
    g, k, Rmin = system.g, system.k, system.Rmin
    I, mass = system.I, system.mass
    
    r = sqrt(2*k*y + Rmin**2)
    alpha = mass * g * r / (I + mass * r**2)
    a = -r * alpha
        
    return omega, alpha, v, a        
Explanation: Write a slope function for this system, using these results from the book:
$ r = \sqrt{2 k y + R_{min}^2} $ 
$ T      = m g I / I^*  $
$ a      = -m g r^2 / I^* $
$ \alpha  = m g r / I^*  $
where $I^*$ is the augmented moment of inertia, $I + m r^2$.
End of explanation
# Solution
slope_func(system.init, 0*s, system)
Explanation: Test your slope function with the initial paramss.
End of explanation
# Solution
def event_func(state, t, system):
    Stops when y is 0.
    
    state: State object with theta, omega, y, v
    t: time
    system: System object with Rmin, k, I, mass
    
    returns: y
    
    theta, omega, y, v = state
    return y
Explanation: Write an event function that will stop the simulation when y is 0.
End of explanation
# Solution
event_func(system.init, 0*s, system)
Explanation: Test your event function:
End of explanation
# Solution
results, details = run_ode_solver(system, slope_func, events=event_func, max_step=0.05)
details
Explanation: Then run the simulation.
End of explanation
# Solution
results.tail()
Explanation: Check the final state.  If things have gone according to plan, the final value of y should be close to 0.
End of explanation
def plot_theta(results):
    plot(results.theta, color='C0', label='theta')
    decorate(xlabel='Time (s)',
             ylabel='Angle (rad)')
plot_theta(results)
Explanation: Plot the results.
theta should increase and accelerate.
End of explanation
def plot_y(results):
    plot(results.y, color='C1', label='y')
    decorate(xlabel='Time (s)',
             ylabel='Length (m)')
    
plot_y(results)
Explanation: y should decrease and accelerate down.
End of explanation
# Solution
v = results.v # m / s
plot(v)
decorate(xlabel='Time (s)',
         ylabel='Velocity (m/s)')
Explanation: Plot velocity as a function of time; is the yo-yo accelerating?
End of explanation
# Solution
a = gradient(v)
plot(a)
decorate(xlabel='Time (s)',
         ylabel='Acceleration (m/$s^2$)')
Explanation: Use gradient to estimate the derivative of v.  How does the acceleration of the yo-yo compare to g?
End of explanation
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