Summary

DescriptionShell-diag-1.svg	English: A diagram illustrating the derivation of Newton's shell theorem. Shown is a thin shell with a test mass outside the shell (${\displaystyle r>R}$).
Date	25 February 2017
Source	Own work
Author	Xaonon

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue

Source

This image and the others in the same series (2, 3, 4) were generated from the MetaPost code presented below. The code is released under the same license as the images themselves.

% shell-diag.mp
% A diagram illustrating the derivation of Newton's shell theorem.  To be
% processed with MetaPost:
%   mpost --mem=metafun.mem -s 'outputformat="svg"' -s prologues=3 shell-diag.mp

color bandshade, fillshade;
bandshade = 0.7 [blue, white];
fillshade = 0.9 white;

numeric dotsize, deg;
dotsize = 5 bp;
deg = length( fullcircle )/360;

freelabeloffset := 3/4 freelabeloffset;
labeloffset := 2 labeloffset;

def dot( expr P ) =
  fill fullcircle scaled dotsize shifted P withcolor black;
enddef;

def draw_circle( expr R, stroke ) =
  save p;
  pen p;
  p = currentpen;
  pickup p scaled stroke;
  draw fullcircle scaled 2R;
  pickup p;
enddef;

vardef anglebetween( expr a, b, rad, str ) =
  save endofa, endofb, common, curve, where;
  pair endofa, endofb, common;
  path curve;
  numeric where;
  endofa = point length( a ) of a;
  endofb = point length( b ) of b;
  if round point 0 of a = round point 0 of b:
    common = point 0 of a;
  else:
    common = a intersectionpoint b;
  fi;
  where = turningnumber( common--endofa--endofb--cycle );
  curve = (unitvector( endofa - common ){(endofa - common) rotated (90 * where)} ..
           unitvector( endofb - common )) scaled rad shifted common;
  draw thefreelabel( str, point 1/2 of curve, common ) withcolor black;
  curve
enddef;

def draw_angle( expr a, b, rad, str ) =
  begingroup
    save p;
    pen p;
    p = currentpen;
    pickup p scaled 1/2;
    draw anglebetween( a, b, rad, str );
    pickup p;
  endgroup
enddef;

def label_line( expr a, b, disp, str ) =
  begingroup
  save mid, opp;
  pair mid, opp;
  mid = 1/2 [a, b];
  opp = -disp rotated (angle( b - a ) - 90) shifted mid;
  draw thefreelabel( str, mid, opp );
  draw a -- b;
  endgroup
enddef;

def draw_thinshell( expr R, r, theta, dtheta, thetarad, phirad ) =
  begingroup
    save M, m;
    pair M, m;
    M = (0, 0);
    m = (r, 0);

    save circ;
    path circ;
    circ = fullcircle scaled 2R;

    save thetapt, dthetapt;
    pair thetapt, dthetapt;
    thetapt   = point (theta * deg) of circ;
    dthetapt  = point ((theta + dtheta) * deg) of circ;

    save upper, lower, band;
    path upper, lower, band;
    upper = subpath (0, 4) of circ;
    lower = subpath (4, 8) of circ;
    band = buildcycle( upper, (xpart thetapt,  R) -- (xpart thetapt,  -R),
                       lower, (xpart dthetapt, R) -- (xpart dthetapt, -R) );

    % draw figures
    save p;
    pen p;
    p = currentpen;
    pickup p scaled 1/2;
    fill band withcolor bandshade;
    draw band;
    pickup p;

    save near, far;
    pair near, far;
    if theta < 90:
      near = 3/4[ulcorner band, llcorner band];
      far  = right shifted near;
    else:
      near = 3/4[urcorner band, lrcorner band];
      far  = left shifted near;
    fi;
    draw thefreelabel( btex $dM$ etex, near, far );

    dot( M );
    %label.llft( btex $M$ etex, M );

    dot( m );
    label.lrt( btex $m$ etex, m );

    draw M -- thetapt;
    label_line( M, m, right, btex $r$ etex );
    label_line( m, thetapt, right, btex $s$ etex );
    if R <> r:
      label_line( M, dthetapt, left, btex $R$ etex );
    else:
      draw M -- dthetapt;
    fi;

    draw_angle( m -- M, m -- thetapt, phirad, btex $\phi$ etex );
    draw_angle( M -- m, M -- thetapt, thetarad, btex $\theta$ etex );
    draw_angle( M -- thetapt, M -- dthetapt, R, btex $d\theta$ etex );
  endgroup
enddef;

def draw_thickshell( expr Ra, Rb, r ) =
  begingroup
    save m;
    pair m;
    m = (r, 0);

    fill fullcircle scaled 2Rb withcolor fillshade;
    fill fullcircle scaled 2r  withcolor bandshade;
    unfill fullcircle scaled 2Ra;

    dot( origin );
    dot( m );
    label.lrt( btex $m$ etex, m );
    label_line( origin, m, right, btex $r$ etex );

    draw_circle( Rb, 2 );
    if Ra > 0:
      draw_circle( Ra, 2 );
      label_line( origin, dir( 100 ) scaled Rb, left,  btex $R_b$ etex );
      label_line( origin, dir( 80 )  scaled Ra, right, btex $R_a$ etex );
    else:
      label_line( origin, dir( 90 )  scaled Rb, left,  btex $R_b$ etex );
    fi;
  endgroup
enddef;

% Thin shell, r > R
beginfig(1)
  numeric R;
  R = 1 in;
  draw_thinshell( R, 3R, 50, 15, 1/4 in, 3/4 in );
  draw_circle( R, 2 );
  setbounds currentpicture to boundingbox currentpicture enlarged 1pt;
endfig;

% Thin shell, r < R
beginfig(2)
  numeric R;
  R = 1 in;
  draw_thinshell( R, 0.7R, 125, 15, 1/8 in, 1/3 in );
  draw_circle( R, 2 );
  setbounds currentpicture to boundingbox currentpicture enlarged 1pt;
endfig;

% Thick shell
beginfig(3)
  numeric Ra, Rb, r;
  Ra = 0.8 in;
  Rb = 1.3 in;
  r = 1 in;

  draw_thickshell( Ra, Rb, r );
  setbounds currentpicture to boundingbox currentpicture enlarged 1pt;
endfig;

% Solid sphere
beginfig(4)
  numeric Ra, Rb, r;
  Ra = 0;
  Rb = 1.3 in;
  r = 1 in;

  draw_thickshell( Ra, Rb, r );
  setbounds currentpicture to boundingbox currentpicture enlarged 1pt;
endfig;

end