Mass Derailed

Classical Mechanics Level 3

There are two masses of mass m m . They are connected by a spring of force constant k k and natural length L 0 L_0 .

Particle 1 is confined to move solely on the y-axis and particle 2 is confined to move solely on the x-axis.

At time t = 0 t = 0 , particle 1 starts at rest ( x 1 , y 1 ) = ( 0 , 1 ) (x_1,y_1) = (0,1) , and particle 2 starts at rest ( x 2 , y 2 ) = ( 1 , 0 ) (x_2,y_2) = (1,0) . As soon as the x-coordinate of particle 2 becomes zero for the first time, the rails confining particle 2 to the x-axis cease to exist (Turns out that the rails for particle 2 exist only along the positive x-direction). The goal of this question is to compute the y-coordinate of particle 2 one second after its x-coordinate is zero for the first time.

Note that ( x 1 , y 1 ) (x_1,y_1) and ( x 2 , y 2 ) (x_2,y_2) vary over time but their initial values are given in the problem statement.

Details and Assumptions:

  • m = 1 m = 1

  • k = 10 k = 10

  • L 0 = 2 L_0 = \sqrt{2}

  • Gravitational acceleration g = 10 g = 10 in the negative y direction.

Inspiration - Solving the mentioned problem will bring you a step closer to the answer.


The answer is -7.39966.

Karan Chatrath by Karan Chatrath , 27, Netherlands

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1 solution

Steven Chase
Nov 3, 2019

I used a Newtonian (force) approach for this. The implementation was quite general, so I just needed to "flip a switch" in the code to accommodate this case. I have attached a plot of the various positions over time. 

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import math

# Constants

m = 1.0
k = 10.0
L0 = math.sqrt(2.0)

g = 10.0

dt = 10.0**(-5.0)

########################

# Initialize pos,vel,acc
x1 = 0.0
y1 = 1.0

x2 = 1.0
y2 = 0.0

x1d = 0.0
y1d = 0.0

x2d = 0.0
y2d = 0.0

x1dd = 0.0
y1dd = 0.0

x2dd = 0.0
y2dd = 0.0

########################

print "t x1 y1 x2 y2"

# Simulation until x2 = 0

t = 0.0

count = 0

while x2 >= 0.0:

    x1 = x1 + x1d * dt  # Numerical integration
    y1 = y1 + y1d * dt

    x2 = x2 + x2d * dt
    y2 = y2 + y2d * dt

    x1d = x1d + x1dd * dt
    y1d = y1d + y1dd * dt

    x2d = x2d + x2dd * dt
    y2d = y2d + y2dd * dt

    ##########

    Lx = x2 - x1
    Ly = y2 - y1

    L = math.hypot(Lx,Ly)

    ux = Lx/L
    uy = Ly/L

    Fsmag = k*(L-L0)

    Fs1x = Fsmag * ux
    Fs1y = Fsmag * uy

    Fs2x = -Fs1x
    Fs2y = -Fs1y

    ##########

    Fgx = 0.0
    Fgy = -m*g

    ##########

    F1x = Fs1x + Fgx
    F1y = Fs1y + Fgy

    F2x = Fs2x + Fgx
    F2y = Fs2y + Fgy

    ##########

    x1dd = (F1x / m)*0.0   # Rail constraints
    y1dd = (F1y / m)        # Both rail constraints active initially

    x2dd = (F2x / m)
    y2dd = (F2y / m)*0.0

    t = t + dt
    count = count + 1

    if count % 1000 == 0:
        print t,x1,y1,x2,y2

########################

# First set of results

#print ""
#print ""
#print dt
#print t
#print x1,y1
#print x2,y2
#print ""
#print ""

#1e-05
#1.49225
#0.0 -4.32652205654
#-5.29602508791e-06 0.0

tref = t     # reference time

########################

# Simulation for another second

while t <= (tref + 1.0):

    x1 = x1 + x1d * dt
    y1 = y1 + y1d * dt

    x2 = x2 + x2d * dt
    y2 = y2 + y2d * dt

    x1d = x1d + x1dd * dt
    y1d = y1d + y1dd * dt

    x2d = x2d + x2dd * dt
    y2d = y2d + y2dd * dt

    ##########

    Lx = x2 - x1
    Ly = y2 - y1

    L = math.hypot(Lx,Ly)

    ux = Lx/L
    uy = Ly/L

    Fsmag = k*(L-L0)

    Fs1x = Fsmag * ux
    Fs1y = Fsmag * uy

    Fs2x = -Fs1x
    Fs2y = -Fs1y

    ##########

    Fgx = 0.0
    Fgy = -m*g

    ##########

    F1x = Fs1x + Fgx
    F1y = Fs1y + Fgy

    F2x = Fs2x + Fgx
    F2y = Fs2y + Fgy

    ##########

    x1dd = (F1x / m)*0.0
    y1dd = (F1y / m)

    x2dd = (F2x / m)
    y2dd = (F2y / m)*1.0   # Rail constraints removed for particle 2

    t = t + dt
    count = count + 1

    if count % 1000 == 0:
        print t,x1,y1,x2,y2

########################

# Second set of results

#print ""
#print ""
#print dt
#print t
#print x1,y1
#print x2,y2
#print ""
#print ""

#1e-05
#2.49225000001
#0.0 -3.90356699196
#-1.60860063883 -7.3993989347

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