Spherical Coordinates Integration Exercise
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# ## Newton's Law of Gravitation is Good for Spheres; # ## Originally Law was for Point Masses; # ## Object to Show Validity for Spheres by Integrating # ## over All Mass Differentials in the Sphere: # ## dF := -G*m*De*dV*Xvector/|Xvector|^3; # ## Assume Test Mass (m) at Position (0,0,Lm); # ## Sphere = IdealEarthModel (e); # ## Me:=MassOfSphere; RE:=RadiusOfSphere; # ## Ce:=CenterOfSphere:=(0,0,0); # ## Lm:=DistanceFromenterToTestMass; # ## G:=UniversalGravitationalConstant; # ## De:=DensityOfSphere; # ## Assume TestMass at (0,0,Lm) for Simplicity, so that # ## ForceOfGravity:=F:=(0,0,Fz), by Symmetry # ## with Fx:=0 and Fy:=0; # ## Assume Lm > RE; # ## Constant Density is Mass Me Divided by SphereVolume
< Me < De := 3/4 ------ < 3 < pi RE
# ## Total z-Component of Force of Gravity # ## in Spherical Coordinates (rho, theta, phi), # ## where z = rho*cos(phi); # ## (x,y) = rho*sin(phi)*(cos(theta),sin(theta)); # ## dV := rho^2*sin(phi)*d(theta)*d(phi)d(rho) # ## in Simplest Order of Integration:
> Fz:=-G*De*m*Int(Int(Int(rho^2*sin(phi)*(Lm-rho*cos(phi))/ \ > (Lm^2+rho^2-2*rho*cos(phi))^(3/2),theta=0..2*pi), phi=0..pi),rho=0..RE); # ## Display Math Only:
< Fz := < < RE pi 2 pi < / / / 2 < | | | rho sin(phi) (Lm - rho cos(phi)) < G Me m | | | --------------------------------- dtheta dphi drho < | | | 2 2 3/2 < / / / (Lm + rho - 2 rho cos(phi)) < 0 0 0 < - 3/4 ------------------------------------------------------------------------- < 3 < pi RE
# ## First: Partial Integration Over theta: > Fz_rho_phi:=-G*De*m*int(rho^2*sin(phi)*(Lm-rho*cos(phi))/ \ > (Lm^2+rho^2-2*rho*cos(phi))^(3/2), theta=0..2*pi);
< 2 < G Me m rho sin(phi) (- Lm + rho cos(phi)) < Fz_rho_phi := 3/2 ------------------------------------------ < 3 2 2 3/2 < RE (Lm + rho - 2 rho cos(phi)) <
# ## Second: Partial Integration Over phi: > Fz_rho:=int(Fz_rho_phi, phi=0..pi);
< 2 2 < (- Lm + Lm + rho - rho cos(pi)) G Me m rho < Fz_rho := - 3/2 -------------------------------------------- < 2 2 1/2 3 < (Lm + rho - 2 rho cos(pi)) RE < < 2 2 < (- Lm + Lm + rho - rho) G Me m rho < + 3/2 ------------------------------------ < 2 2 1/2 3 < (Lm + rho - 2 rho) RE
# ## Third: Partial Integration Over rho: > Fz:=int(Fz_rho, rho=0..RE);
< 1/2 2 3 1/2 <Fz := - 1/2 G Me m (%1 RE + 3 cos(pi) ln(RE - cos(pi) + %1 ) < < 1/2 2 1/2 < - 3 cos(pi) ln(RE - cos(pi) + %1 ) Lm + 3 cos(pi) %1 < < 1/2 1/2 2 2 1/2 < + cos(pi) %1 RE - 3 %1 Lm + 3 ln(RE - 1 + (Lm + RE - 2 RE) ) Lm < < 2 1/2 2 2 1/2 2 2 1/2 2 < + Lm %1 - (Lm + RE - 2 RE) RE - (Lm + RE - 2 RE) RE < < 2 2 1/2 2 2 2 1/2 < - 3 ln(RE - 1 + (Lm + RE - 2 RE) ) - Lm (Lm + RE - 2 RE) < < 2 2 1/2 2 2 1/2 / 3 < - 3 (Lm + RE - 2 RE) + 3 (Lm + RE - 2 RE) Lm) / RE - 3/2 G Me < / < < 3 2 1/2 2 1/2 2 2 1/2 < m (- cos(pi) ln(- cos(pi) + (Lm ) ) + (Lm ) - cos(pi) (Lm ) < < 2 1/2 2 1/2 < + ln(- 1 + (Lm ) ) - ln(- 1 + (Lm ) ) Lm < < 2 1/2 / 3 < + cos(pi) ln(- cos(pi) + (Lm ) ) Lm) / RE < / < < 2 2 <%1 := Lm + RE - 2 cos(pi) RE <
# ## Maple having Trouble with phi Integration; # ## Try Change of Variables for Singular Denominator: # ## U^2= Lm^2+rho^2-2*rho*Lm*cos(phi); # ## (L-rho*cos(phi))*rho^2*sin(phi)*d(phi)/U^3 # ## = rho*(Lm^2-rho^2+U^2)*dU/(2*Lm^2*U^2); > U1:=Lm-rho; U2:=Lm+rho;
< U1 := Lm - rho < < U2 := Lm + rho
> Fz_rho_u:=-2*pi*G*De*m/(2*Lm^2)*Int(rho*(Lm^2-rho^2+U^2)/U^2,U=U1..U2);
< Lm + rho < / 2 2 2 < | rho (Lm - rho + U ) < G Me m | --------------------- dU < | 2 < / U < Lm - rho < Fz_rho_u := - 3/4 ------------------------------------------ < 3 2 < RE Lm
> Fz_rho:=value(");
< 2 < G Me m rho < Fz_rho := - 3 ----------- < 3 2 < RE Lm
> Fz:=Int(Fz_rho, rho=0..RE);
< RE < / 2 < | G Me m rho < Fz := | - 3 ----------- drho < | 3 2 < / RE Lm < 0
> Fz:=value(");
< G Me m < Fz := - ------ < 2 < Lm
# ## Z-Component For Newton's Law Of Gravity Is As If # ## Me:=MassOfSphere Were Concentrated At Its Ce:=Center # ## For m:=TesTMass At Lm:=Distance From Ce!
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