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April 29

Feynman Lectures. Exercises. Exercise 21-8 PNG

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21-8. The gravitational force felt by a particle embedded in a solid uniform sphere, due to the mass of the sphere only, is directly proportional to the distance of the particle from the center of the sphere. If the earth were such a sphere, with a narrow hole drilled through it along a polar diameter, how long would it take a body dropped in the hole to reach the surface at the opposite side of the earth?

—  R. B. Leighton , Feynman Lectures on Physics. Exercises

I want to prove the statement that the force is proportional to radius inside the uniform sphere.
Consider a spherical shell 1-2 PNG.
The potential in point 1 is:
$\Psi _{1}=\Psi _{\text{0-1}}+\Psi _{\text{2-3}}+\Psi _{\text{1-2,at 1}}$
$\textstyle \Psi _{1}=\Psi _{\text{0-1}}+\Psi _{\text{2-3}}+\int _{R_{1}}^{R_{2}}-{\tfrac {Gdm}{x}}$
$\textstyle \Psi _{1}=\Psi _{\text{0-1}}+\Psi _{\text{2-3}}+\int _{R_{1}}^{R_{2}}-{\tfrac {G4\pi x^{2}dx\rho }{x}}$
$\Psi _{1}=\Psi _{\text{0-1}}+\Psi _{\text{2-3}}-G2\pi \rho (R_{2}^{2}-R_{1}^{2})$
The potential in point 2 is:
$\Psi _{2}=\Psi _{\text{0-1}}+\Psi _{\text{2-3}}+\Psi _{\text{1-2,at 2}}$
$\Psi _{2}=\Psi _{\text{0-1}}+\Psi _{\text{2-3}}-{\tfrac {G(M_{2}-M_{1})}{R_{2}}}$
$\Psi _{2}=\Psi _{\text{0-1}}+\Psi _{\text{2-3}}-G{\tfrac {4}{3}}\pi \rho (R_{2}^{2}-{\tfrac {R_{1}^{3}}{R_{2}}})$
Difference in potential:
$\Delta \Psi =\Psi _{2}-\Psi _{1}=-G{\tfrac {4}{3}}\pi \rho (R_{2}^{2}-{\tfrac {R_{1}^{3}}{R_{2}}})+G2\pi \rho (R_{2}^{2}-R_{1}^{2})=G\pi \rho [{\tfrac {2}{3}}{R_{2}}^{2}-{R_{1}}^{2}(2-{\tfrac {4R_{1}}{3R_{2}}})]$
If $R_{1}=R$,
$R_{2}=R+dR$
then:
$d\Psi =G\pi \rho [{\tfrac {2}{3}}{R_{2}}^{2}-{R_{1}}^{2}(2-{\tfrac {4}{3}})]=G\pi \rho {\tfrac {2}{3}}[{R_{2}}^{2}-{R_{1}}^{2}]=G\pi \rho {\tfrac {2}{3}}(R_{2}-R_{1})(R_{2}+R_{1})=G\pi \rho {\tfrac {2}{3}}dR(2R)=G\pi \rho {\tfrac {4}{3}}RdR$

$\textstyle ({\textbf {field}})=-{\tfrac {d\Psi }{dR}}=-G\pi \rho {\tfrac {4}{3}}R$

Is it correct? Username160611000000 (talk) 06:36, 29 April 2018 (UTC)

Your equations assume gravitational force is proportional to the product of the two masses (the Earth and m), and is inversely proportional to the square of the distance between their centers. This confirms Ab initio Newton's law of universal gravitation, see particularly Newton's_law_of_universal_gravitation#Gravitational_field. DroneB (talk) 11:45, 29 April 2018 (UTC)

It is correct for spherical object from outside. But it must be proven for a particle inside a solid sphere. Username160611000000 (talk) 12:36, 29 April 2018 (UTC)

To answer what Feynman actually asks you will use these values:

Polar radius: 6356.8 km

Mass: 5.97237×1024 kg

Surface gravity:9.807 m/s²

See Earth. Defending use of Center of mass as the particle equivalent of an object in Newton's laws of motion is not the exercise. DroneB (talk) 13:27, 29 April 2018 (UTC)

The solutions is not a question. Accepting on faith the proposition that the force is proportional to the distance I can write $g(R)=-{\tfrac {g_{\text{surface}}}{R_{\text{earth}}}}R$. The equation of motion is then ${\tfrac {d^{2}}{dt^{2}}}x=-{\tfrac {g_{\text{surface}}}{R_{\text{earth}}}}x$. It's the harmonic oscillator and the answer is half of the period.

But I want to prove that the force is proportional to the distance.

Username160611000000 (talk) 15:28, 29 April 2018 (UTC)

The first thing is to prove or accept the shell theorem. Once you have that, you know that the gravitational force is equal to the force exerted by all the shells below the falling object only, while those above have no effect. If the falling object is a distance D away from the center and a shell has radius r, each shell exerts a force G*rho*4pi*r^2/D^2, where rho is the density of the shell (the force is proportional to the area, in other words, and they all are concentric). So we integrate r=0 to r=D and get the sum of the forces is (G/3)*rho*4pi*D^3/D^2 - 0 = 4pi*G*rho*D/3, which agrees with your calculation. Wnt (talk) 15:08, 29 April 2018 (UTC)

• • I have checked and found that $[{\tfrac {2}{3}}{R_{2}}^{2}-{R_{1}}^{2}(2-{\tfrac {4R_{1}}{3R_{2}}})]\neq {\tfrac {2}{3}}[{R_{2}}^{2}-{R_{1}}^{2}]$ for $R_{2}=R_{1}+0.000...01$. So my derivation is not good. More careful calculation gives $[{\tfrac {2}{3}}{R_{2}}^{2}-{R_{1}}^{2}(2-{\tfrac {4R_{1}}{3R_{2}}})]={\tfrac {2(R_{1}-R_{2})^{2}(2R_{1}+R_{2})}{3R_{2}}}$.
    So ${\tfrac {d\Psi }{dR}}\sim {\tfrac {2(R_{2}-R_{1})(2R_{1}+R_{2})}{3R_{2}}}\to 0{\text{ when }}(R_{2}-R_{1})\to 0$ .
  • When an object penetrates a solid sphere some shells appear above the object . But these shells generate the potential. Inside one shell the potential is constant, but it is not when number of shells changes.
    Username160611000000 (talk) 15:35, 29 April 2018 (UTC)
  • I presume that mistake is that $\Psi _{\text{0-1}}\neq const$ for point 1 and 2. Username160611000000 (talk) 17:42, 29 April 2018 (UTC)
  • $\Psi _{1}=\Psi _{23}+\underbrace {\Psi _{12,1}} _{-G2\pi \rho ({R_{2}}^{2}-{R_{1}}^{2})}+\underbrace {\Psi _{01}} _{-{\tfrac {GM_{01}}{R_{1}}}}$
    $\Psi _{2}=\Psi _{23}+\underbrace {\Psi _{12,2}+\Psi _{01}} _{-{\tfrac {GM_{02}}{R_{2}}}}$
    $\Psi _{2}-\Psi _{1}=-{\tfrac {GM_{02}}{R_{2}}}+G2\pi \rho ({R_{2}}^{2}-{R_{1}}^{2})+{\tfrac {GM_{01}}{R_{1}}}=-{\tfrac {G(4/3)\pi \rho {R_{2}}^{3}}{R_{2}}}+G2\pi \rho ({R_{2}}^{2}-{R_{1}}^{2})+{\tfrac {G(4/3)\pi \rho {R_{1}}^{3}}{R_{1}}}$
    $=-G(4/3)\pi \rho {R_{2}}^{2}+G2\pi \rho ({R_{2}}^{2}-{R_{1}}^{2})+G(4/3)\pi \rho {R_{1}}^{2}=G(2/3)\pi \rho ({R_{2}}^{2}-{R_{1}}^{2})$
    Username160611000000 (talk) 18:10, 29 April 2018 (UTC)