A little twist on Irodov (Spring-pulley mechanics)

The original problem asks for the velocity of bar $A$ when it breaks off the plane. However, an energy conservation approach can't be used since there's friction as well as the tension, which has to be computed.

The two equations of motion that can be found using Newton's 2nd law.

The normal force acting on bar $A$ can be given by:

$\bold{N} = mg - \bold{F}_s \sin(\theta)$

The equation of motion for the movement of bar $A$ is:

$m \ddot{x} = T \pm \mu \bold{N} + \bold{F}_{s} \cos(\theta)$

Since tension is acting opposite to gravity, the equation of motion for weight $B$ is:

$m \ddot{x} = mg - T$

Rearranging the equation above for $T$ and substituting into the equation on top, we can find an equation for the acceleration:

$\displaystyle \ddot{x} = \frac{mg \pm \mu \bold{N} + F_{s} \cos(\theta)}{2m}$

The code for the simulation is given below:

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import math

time = 0
deltaT = 10**-5

#constants and initial values
l = 2
k = 15
m = 5
g = 10
mu = 0.20

tension = m*g

x = 0
xDot = 0

count = 0

while tension <= 80: #Stopping when tension = 80N; when the string breaks

  if count % 10000 == 0:
    print(time)

  #spring vector angle to horizontal
  cosTheta = x/(math.sqrt(x**2+l**2))
  sinTheta = l/(math.sqrt(x**2+l**2))

  #spring force magnitude
  springForce = k*(math.sqrt(x**2+l**2)-l)

  #horizontal and vertical components of spring force
  Fsy = springForce*sinTheta
  Fsx = -springForce*cosTheta

  normalForce = m*g-Fsy

  #friction force 
  if xDot > 0:
    frictionForce = -mu*normalForce
  else:
    frictionForce = mu*normalForce

  #finding acceleration using final Newtonian equation
  xDotDot = (m*g + frictionForce + Fsx)/(2*m)
  xDot += xDotDot*deltaT
  x += xDot*deltaT

  #magnitude of tension force solved using Newton's laws
  tension = abs(m*g-m*xDotDot)

  time += deltaT
  count += 1

count = 0

while time <= 40: #Large time period to find final stopping position
  #Altered equations of motion due to absence of pulley

  if count % 10000 == 0:
    print(time)

  cosTheta = x/(math.sqrt(x**2+l**2))
  sinTheta = l/(math.sqrt(x**2+l**2))

  springForce = k*(math.sqrt(x**2+l**2)-l)

  Fsy = springForce*sinTheta
  Fsx = -springForce*cosTheta

  normalForce = m*g-Fsy

  if xDot > 0:
    frictionForce = -mu*normalForce
  else:
    frictionForce = mu*normalForce

  #New equation of motion without pulley force
  xDotDot = (frictionForce + Fsx)/m
  xDot += xDotDot*deltaT
  x += xDot*deltaT


  time += deltaT
  count += 1

count = 0