Example 4.5 from University Physics 15th edition

[Eduardo Cavazos]

Hello,

Here's example 4.5 from the book University Physics 15th edition:

# A waitress shoves a ketchup bottle with mass 0.45 kg

# to her right

# along a smooth, level lunch counter.

#

# The bottle leaves her hand moving at 2.0 m/s,

# then slows down as it slides because of a constant horizontal

# friction force exerted on it by the countertop.

#

# It slides for 1.0 m before coming to rest.

#

# What are the magnitude and direction

# of the friction force acting on the bottle?

The diagram:

A full symbolic and numeric solution in Python/sympy is shown at the end.

First I setup the equations for the bottle:

b = make_states_model('b', 2)  # bottle

b0, b1 = b.states

b01 = b.edges[0]

eqs = kinematics_fundamental(b, axes=['x'])

display_equations_(eqs)

The equations in `eqs`:

# dt_b_0_1 = -b_0_t + b_1_t

# v_av_x_b_0_1 = (-b_0_x + b_1_x)/dt_b_0_1

# a_x_b_0_1 = (-b_0_v_x + b_1_v_x)/dt_b_0_1

# v_av_x_b_0_1 = b_0_v_x/2 + b_1_v_x/2

The values we know:

values = {}

values[b0.t] = 0

values[b0.pos.x] = 0

values[b1.pos.x] = 1.0  # m

values[b0.vel.x] = 2.0  # m/s

values[b1.vel.x] = 0.0  # m/s

Setup the forces:

# forces: normal, weight, friction

n = make_point("n")  # normal force

w = make_point("w")  # weight

f = make_point("f")  # friction force

m = sp.symbols("m")  # mass

a = make_point("a")  # acceleration

eqs += newtons_second_law([n, w, f], m, a)

eqs += eq_flat(

    a.x, b01.a.x

)

values[n.x] = 0

values[w.x] = 0

values[m]   = 0.45  # kg

Now, let's display the full system of equations.

green : values we know

red : value we want

Solve the system symbolically and numerically:

Question:

I'm using some of my own libraries for this.

Are there existing libraries for stuff like this?

Would you recommend a different approach?

Ed Cavazos

Full program:

https://github.com/dharmatech/combine-equations.py/blob/master/examples/up/ch4/up-example-4.5-001-one-step.py

# A waitress shoves a ketchup bottle with mass 0.45 kg

# to her right

# along a smooth, level lunch counter.

#

# The bottle leaves her hand moving at 2.0 m/s,

# then slows down as it slides because of a constant horizontal

# friction force exerted on it by the countertop.

#

# It slides for 1.0 m before coming to rest.

#

# What are the magnitude and direction

# of the friction force acting on the bottle?

# ----------------------------------------------------------------------

from pprint import pprint

import sympy as sp

from combine_equations.kinematics_states import *

from combine_equations.solve_system import *

from combine_equations.display_equations import display_equations_

from combine_equations.solve_and_display import solve_and_display_

from combine_equations.misc import eq_flat

from combine_equations.newtons_laws import *

# ----------------------------------------------------------------------

b = make_states_model('b', 2)  # bottle

b0, b1 = b.states

b01 = b.edges[0]

eqs = kinematics_fundamental(b, axes=['x'])

display_equations_(eqs)

# dt_b_0_1 = -b_0_t + b_1_t

# v_av_x_b_0_1 = (-b_0_x + b_1_x)/dt_b_0_1

# a_x_b_0_1 = (-b_0_v_x + b_1_v_x)/dt_b_0_1

# v_av_x_b_0_1 = b_0_v_x/2 + b_1_v_x/2

# ----------------------------------------------------------------------

values = {}

values[b0.t] = 0

values[b0.pos.x] = 0

values[b1.pos.x] = 1.0  # m

values[b0.vel.x] = 2.0  # m/s

values[b1.vel.x] = 0.0  # m/s

# ----------------------------------------------------------------------

# forces: normal, weight, friction

n = make_point("n")  # normal force

w = make_point("w")  # weight

f = make_point("f")  # friction force

m = sp.symbols("m")  # mass

a = make_point("a")  # acceleration

eqs += newtons_second_law([n, w, f], m, a)

eqs += eq_flat(

    a.x, b01.a.x

)

values[n.x] = 0

values[w.x] = 0

values[m]   = 0.45  # kg

display_equations_(eqs, values, want=f.x)

# dt_b_0_1 = -b_0_t + b_1_t

# v_av_x_b_0_1 = (-b_0_x + b_1_x)/dt_b_0_1

# a_x_b_0_1 = (-b_0_v_x + b_1_v_x)/dt_b_0_1

# v_av_x_b_0_1 = b_0_v_x/2 + b_1_v_x/2

# f_x + n_x + w_x = m*a_x

# f_y + n_y + w_y = m*a_y

# a_x = a_x_b_0_1

solve_and_display_(eqs, values, want=f.x)

# f_x = (b_0_v_x**2*m/2 - b_0_x*n_x - b_0_x*w_x - b_1_v_x**2*m/2 + b_1_x*n_x + b_1_x*w_x)/(b_0_x - b_1_x)

# f_x = -0.900000000000000

[gu...@uwosh.edu]

Ed,

If I am understanding your use cases, within Sympy I think you want the sympy.physics module. See the documentation https://docs.sympy.org/latest/reference/public/physics/index.html#module-sympy.physics

Jonathan