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
<Text-field style="Text" layout="Normal" spaceabove="8" spacebelow="2">Central Force Motion</Text-field>
<Text-field style="Text" layout="Normal">Introduction</Text-field> One may derive the equation governing all central force motion problems beginning with the total mechanical energy. i.e., 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, where 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 is the reduced mass of the system, and 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 is the potential. Solving this equation for the radial velocity, 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 and using conservation of angular momentum, 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 = const., one may then readily derive the following equation: 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. The most useful application of this equation is realized by noting that if the nature of the orbit is known (i.e., the mathematical form foir 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 ), then the mathematical form for the force may be derived. The following procedure may be used to determine the force required to execute an orbit. The necesary 'form' is a function in polar coordinates. Examples of well-known orbits are also shown. one may derive the equation governing all central force motion problems beginning with the total mechanical energy. i.e., 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, where 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 is the reduced mass of the system, and NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZHLyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYmLUYtNjlRIlVGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1JI21vR0YlNjNRMCZBcHBseUZ1bmN0aW9uO0YoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRJDBlbUYoLyUncnNwYWNlR0ZgcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlgvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkctRiQ2JS1GZG82M1EiKEYoL0Zob1EncHJlZml4RigvRltwRjtGXHAvRl9wUS50aGlubWF0aHNwYWNlRigvRmJwRmRyL0ZkcEY7RmVwRmdwRmpwRl1xRl9xRmFxRmNxRmVxRmdxRmlxLUYkNiMtRi02OVEickYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUZkbzYzUSIpRigvRmhvUShwb3N0Zml4RihGYnJGXHBGY3IvRmJwUTJ2ZXJ5dGhpbm1hdGhzcGFjZUYoRmZyRmVwRmdwRmpwRl1xRl9xRmFxRmNxRmVxRmdxRmlxRixGLDcjNiMtJSJVRzYjJSJyRw== is the potential. Solving this equation for the radial velocity, 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 and using conservation of angular momentum, NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZHLyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYmLUYtNjlRIkxGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1JI21vR0YlNjNRIj1GKC8lJWZvcm1HUSZpbmZpeEYoLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYoLyUncnNwYWNlR0ZgcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlgvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkctRiQ2KC1GLTY5USNtdUYoRjBGM0Y2L0Y6RjhGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduL0ZqblEnbm9ybWFsRihGXG8tRmRvNjNRMSZJbnZpc2libGVUaW1lcztGKEZnb0Zqb0ZccC9GX3BRJDBlbUYoL0ZicEZnckZjcEZlcEZncEZqcEZdcUZfcUZhcUZjcUZlcUZncUZpcS1GJDYlRiwtSSVtc3VwR0YlNiUtRi02OVEickYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvLUkjbW5HRiU2OVEiMkYoRjBGM0Y2RmByRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZhckZcby8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRihGLEZjci1GJDYlLUZkbzYzUSIoRigvRmhvUSdwcmVmaXhGKC9GW3BGO0ZccC9GX3BRLnRoaW5tYXRoc3BhY2VGKC9GYnBGYXQvRmRwRjtGZXBGZ3BGanBGXXFGX3FGYXFGY3FGZXFGZ3FGaXEtRiQ2JkYsLUkmbWZyYWNHRiU2Ki1GJDYjLUZkbzYzUSsmUGFydGlhbEQ7RihGXXRGam9GXHBGZnJGaHJGY3BGZXBGZ3BGanBGXXFGX3FGYXFGY3FGZXFGZ3FGaXEtRiQ2JUYsLUYkNiRGW3UtRi02OVEidEYoRjBGM0Y2RjlGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvRiwvJS5saW5ldGhpY2tuZXNzR1EiMUYoLyUrZGVub21hbGlnbkdRJ2NlbnRlckYoLyUpbnVtYWxpZ25HRmp1LyUpYmV2ZWxsZWRHRjhGZ3FGaXFGY3ItRi02OVEmdGhldGFGKEYwRjNGNkZgckY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GYXJGXG8tRmRvNjNRIilGKC9GaG9RKHBvc3RmaXhGKEZfdEZccEZgdC9GYnBRMnZlcnl0aGlubWF0aHNwYWNlRihGY3RGZXBGZ3BGanBGXXFGX3FGYXFGY3FGZXFGZ3FGaXFGLEYsRiw3IzYjLyUiTEcqKCUjbXVHIiIiKiQpJSJyRyIiI0Zfd0Zfdy0lJWRpZmZHNiQlJnRoZXRhRyUidEdGX3c= = const., one may then readily derive the following equation: 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.
<Text-field style="Text" layout="Normal">Application</Text-field> One of the most usefule application sof this equation is realized by noting that, if the nature of the orbit is known (i.e., the mathematical form foir NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USFGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjgvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGKC8lKm1hdGhjb2xvckdGRC8lL21hdGhiYWNrZ3JvdW5kR0ZHLyUrZm9udGZhbWlseUdGMi8lLG1hdGh2YXJpYW50R1EnaXRhbGljRigvJSltYXRoc2l6ZUdGNS1GJDYmLUYtNjlRInJGKEYwRjNGNkY5RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZpbkZcby1JI21vR0YlNjNRMCZBcHBseUZ1bmN0aW9uO0YoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGOC8lJ2xzcGFjZUdRJDBlbUYoLyUncnNwYWNlR0ZgcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlgvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkctRiQ2JS1GZG82M1EiKEYoL0Zob1EncHJlZml4RigvRltwRjtGXHAvRl9wUS50aGlubWF0aHNwYWNlRigvRmJwRmRyL0ZkcEY7RmVwRmdwRmpwRl1xRl9xRmFxRmNxRmVxRmdxRmlxLUYkNiMtRi02OVEmdGhldGFGKEYwRjNGNi9GOkY4RjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbi9Gam5RJ25vcm1hbEYoRlxvLUZkbzYzUSIpRigvRmhvUShwb3N0Zml4RihGYnJGXHBGY3IvRmJwUTJ2ZXJ5dGhpbm1hdGhzcGFjZUYoRmZyRmVwRmdwRmpwRl1xRl9xRmFxRmNxRmVxRmdxRmlxRixGLDcjNiMtJSJyRzYjJSZ0aGV0YUc= ), then the mathematical form for the force may be derived. The following procedure may be used to determine the force required to execute an orbit. The necesary 'form' is a function in polar coordinates. Examples of well-known orbits are also shown.
<Text-field style="Heading 4" layout="Heading 4">findForce Code</Text-field> findForce:=proc(form::algebraic) global r,F; r:=unapply(form,theta): diff(1/r(theta),theta$2)+1/r(theta) = -mu/(L^2)*r(theta)^2*f: # solve for the force as a function of the angle solve(%,f): # solve the orbit for theta solve(r=r(theta),theta): # substitute the angular form into the force solution subs(theta=%,%%): # simplify the force and "build" the force function F:=unapply(collect(simplify(%),r),r): print('r(theta)'=r(theta)); print('F(r)'=F(r)); end:
Elliptic Motion findForce(alpha/(1+epsilon*cos(theta))); 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 alpha:=1: L:=1: mu:=1: epsilon:=0.9: plot(r(theta),theta=0..2*Pi, coords=polar,scaling=constrained); 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 Linear Motion findForce(b/(sin(theta)-m*cos(theta))); 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 b:=1: m:=1: plot(r(theta),theta=0..2*Pi, coords=polar,scaling=constrained); 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 Cardiod Motion findForce(B*cos(n*theta)); 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 B:=1: n:=7: plot(r(theta),theta=0..2*Pi, coords=polar,scaling=constrained); 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<Text-field style="Text" layout="Normal">Extension - Animating Trajectories</Text-field> 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 with(plots): orbit:=proc(ecc) # declare all variables as local local r,T,N,S,j,i,a,P,L,F,eStr,tStr; global oframes; with(plots): with(plottools): if (ecc >= 1.0) then ERROR("ecc must obey 0 <= ecc <1"); fi: # define and plot the trajectory r:=(ecc,theta)->1/(1+ecc*cos(theta)): T:=plot(r(ecc,theta),theta=0..2*Pi,coords=polar,color=blue): N:=360: S:=5: j:=1: for i from 1 to N by S do a[i]:=Pi/180*i: #define an angle # plot a point on the orbit P[i]:=pointplot([r(ecc,a[i]),a[i]], coords=polar,color=red,symbol=circle): # plot a line from the origin to the point L[i]:=plot([ [0,0],[r(ecc,a[i]),a[i]] ], coords=polar,color=green): # store a picture of the system F[j]:=display(T,P[i],L[i],scaling=constrained): j:=j+1: od: # store all of the pictures as a sequence oframes:=seq(F[j],j=1..N/S): # construct the title string eStr:=convert(ecc,string): tStr:=cat("Orbital Trajectory, ecc=",eStr): # create the animation display(oframes,insequence=true,axes=none, title=tStr,scaling=constrained); end: 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 Warning, the `with' command does not work inside procedures or modules Warning, the `with' command does not work inside procedures or modules Warning, the assigned name arrow now has a global binding 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