File:Centripetal force vs external force vectors.svg

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Original file ‎(SVG file, nominally 623 × 327 pixels, file size: 234 KB)

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Summary

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Description
English: Shows in the upper half the position of a valve of a wheel seen from the reference frame ”Street” -> Cycloid
  • In black the position in x-y plane
  • in red the instantaneous velocity of the valve
  • in green the inst. acceleration

Lower part: The decomposition of the external force (left side of newton 2)

  • Blue tangent component (tangent to the instantaneous circle of curvature) of this force
  • Yellow centripetal component
Date
Source Own work
Author Svebert
SVG development
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Source code
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Python code

############Script to produce this figure (python)##############
from math import *
import numpy as np
import matplotlib.pyplot as plt

tarr=np.linspace(-1.0,3*pi,400)
xarr=np.zeros(len(tarr))
yarr=np.zeros(len(tarr))
vxarr=np.zeros(len(tarr))
vyarr=np.zeros(len(tarr))
axarr=np.zeros(len(tarr))
ayarr=np.zeros(len(tarr))
apxarr=np.zeros(len(tarr))
apyarr=np.zeros(len(tarr))
aZpxarr=np.zeros(len(tarr))
aZpyarr=np.zeros(len(tarr))
aFZp=np.zeros(len(tarr))
aF=np.zeros(len(tarr))
i=0
R=2.0
r=R
omega=1.0
for t in tarr:
	xarr[i]=R*omega*t-r*sin(omega*t)
	yarr[i]=R-r*cos(omega*t)
	i+=1

vxarr=np.diff(xarr)
vyarr=np.diff(yarr)

axarr=np.diff(vxarr)
ayarr=np.diff(vyarr)

#Kruemmungsradius
#vsquared
for i in range(len(tarr)-2):
	v2=vxarr[i]*vxarr[i]+vyarr[i]*vyarr[i]
	av=vxarr[i]*axarr[i]+vyarr[i]*ayarr[i]
	apxarr[i]=av*vxarr[i]/v2
	apyarr[i]=av*vyarr[i]/v2
      	aZpxarr[i]=axarr[i]-apxarr[i]
      	aZpyarr[i]=ayarr[i]-apyarr[i]
	aF[i]=sqrt(axarr[i]*axarr[i]+ayarr[i]*ayarr[i])
	aFZp[i]=sqrt(aZpxarr[i]*aZpxarr[i]+aZpyarr[i]*aZpyarr[i])


plt.figure()

plt.subplot(211,aspect='equal',adjustable='box',title='Reference frame "Street": velocity and acceleration')
plt.plot(xarr,yarr,'k')
plt.hold(1)
plt.quiver(xarr[::20],yarr[::20],vxarr[::20],vyarr[::20],color='r',label='velocity')
plt.hold(1)
plt.quiver(xarr[::20],yarr[::20],axarr[::20],ayarr[::20],color='g',label='acceleration')
plt.ylim([0,max(yarr)])
plt.xlim([0,max(xarr)])

#plt.axes().set_aspect('equal')
#pl.show()
#plt.axes().set_aspect('equal')
plt.subplot(212,aspect='equal',adjustable='box',title='Reference frame "Street": Tangent and Centripetal Force')
#plt.figure()
plt.plot(xarr,yarr,'k')
plt.hold(1)
plt.quiver(xarr[::20],yarr[::20],apxarr[::20],apyarr[::20],color='b')
plt.quiver(xarr[::20],yarr[::20],aZpxarr[::20],aZpyarr[::20],color='y')
plt.ylim([0,max(yarr)])
plt.xlim([0,max(xarr)])
plt.savefig('centripetal_force_vs_external_force_vectors.svg')
#plt.axes().set_aspect('equal')

#plt.show()
plt.figure()
plt.title("External Force, tangent and centripetal components")
plt.plot(axarr,ayarr,'g', label='External Force')
plt.hold(1)
plt.plot(aZpxarr,aZpyarr,'y.', label='Centripetal Force')
plt.plot(apxarr,apyarr,'b.',label='Tangent Force')
plt.ylim([min(min(ayarr),min(aZpyarr),min(apyarr)),max(max(ayarr),max(aZpyarr),max(apyarr))*1.6])
plt.legend()
plt.axes().set_aspect('equal')
plt.savefig('centripetal_force_vs_external_force_circles.svg')

plt.figure()
plt.title('External Force and centripetal component vs. time')
plt.hold(1)
plt.plot(tarr,aFZp,'y',label='centripetal component')
plt.plot(tarr,aF,'g',label='External Force')
plt.legend(loc='lower left')
plt.xlim([0,tarr[len(tarr)-3]])
plt.savefig('centripetal_force_vs_external_force_time_series.svg')
#plt.show()

Licensing

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I, the copyright holder of this work, hereby publish it under the following license:
Creative Commons CC-Zero This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.
The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission.

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Date/TimeThumbnailDimensionsUserComment
current17:58, 28 February 2013Thumbnail for version as of 17:58, 28 February 2013623 × 327 (234 KB)Svebert (talk | contribs)Cropped German translation
19:31, 1 February 2013Thumbnail for version as of 19:31, 1 February 2013720 × 540 (84 KB)Svebert (talk | contribs){{Information |Description ={{en|1=Shows in the upper half the position of a valve of a wheel seen from the reference frame ”Street” -> Cycloid *In black the position vs time *in red the instantaneous velocity of the valve *in green the inst. ac...

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