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review derivation: Newton's Law of Gravitation

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from https://www.youtube.com/watch?v=fJYdFIZlD8k

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
17 substitute LHS of two expressions into expr
  1. 4264859781; locally 8320848:
    \(F \propto m_1\)
    \(F pdg_{5022} propto\)
  2. 4490788873; locally 5440061:
    \(F \propto m_2\)
    \(F pdg_{4851} propto\)
  3. 1571582377; locally 6174613:
    \(F_{gravitational} \propto \frac{1}{r^2}\)
    \(pdg_{2867} = \frac{k}{pdg_{2530}^{2}}\)
  1. 3650814381; locally 1206000:
    \(F_{gravitational} \propto \frac{m_1 m_2}{r^2}\)
    \(\frac{pdg_{2867} pdg_{4851} pdg_{5022} propto}{pdg_{2530}^{2}}\)
Nothing to split 4264859781:
4490788873:
1571582377:
3650814381:
4264859781:
4490788873:
1571582377:
3650814381:
8 substitute LHS of expr 1 into expr 2
  1. 6026694087; locally 3755872:
    \(F_{centripetal} = m \frac{v^2}{r}\)
    \(pdg_{1687} = \frac{pdg_{5156} v^{2}}{pdg_{2530}}\)
  2. 4820320578; locally 5891249:
    \(F_{gravitational} = F_{centripetal}\)
    \(pdg_{2867} = pdg_{1687}\)
  1. 4267808354; locally 2239910:
    \(F_{gravitational} = m \frac{v^2}{r}\)
    \(pdg_{2867} = \frac{pdg_{1357}^{2} pdg_{5156}}{pdg_{2530}}\)
LHS diff is 0 RHS diff is pdg5156*(-pdg1357**2 + v**2)/pdg2530 6026694087:
4820320578:
4267808354:
6026694087:
4820320578:
4267808354:
10 declare initial expr
  1. 6785303857; locally 5154120:
    \(C = 2 \pi r\)
    \(pdg_{3034} = 2 pdg_{2530} pdg_{3141}\)
no validation is available for declarations 6785303857:
6785303857:
9 declare initial expr
  1. 3411994811; locally 9055493:
    \(v_{\rm average} = \frac{d}{t}\)
    \(pdg_{6709} = \frac{pdg_{1943}}{pdg_{1467}}\)
no validation is available for declarations 3411994811:
3411994811:
5 declare assumption
  1. 4820320578; locally 5891249:
    \(F_{gravitational} = F_{centripetal}\)
    \(pdg_{2867} = pdg_{1687}\)
no validation is available for declarations 4820320578:
4820320578:
19 declare final expr
  1. 1292735067; locally 8373934:
    \(F_{gravitational} = G \frac{m_1 m_2}{r^2}\)
    \(pdg_{2867} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
no validation is available for declarations 1292735067:
1292735067:
7 substitute LHS of expr 1 into expr 2
  1. 8361238989; locally 6969192:
    \(a_{centripetal} = \frac{v^2}{r}\)
    \(a_{c*(e*(n*(t*(r*(i*(p*(e*(t*(a*l)))))))))} = \frac{pdg_{1357}^{2}}{pdg_{2530}}\)
  2. 5345738321; locally 2020292:
    \(F = m a\)
    \(pdg_{4202} = pdg_{5156} pdg_{9140}\)
  1. 6026694087; locally 3755872:
    \(F_{centripetal} = m \frac{v^2}{r}\)
    \(pdg_{1687} = \frac{pdg_{5156} v^{2}}{pdg_{2530}}\)
LHS diff is -pdg1687 + pdg4202 RHS diff is pdg5156*(pdg2530*pdg9140 - v**2)/pdg2530 8361238989:
5345738321: dimensions are consistent
6026694087:
8361238989:
5345738321: N/A
6026694087:
15 simplify
  1. 3004158505; locally 4470678:
    \(\frac{T^2}{r} F_{gravitational} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}\)
    \(\frac{pdg_{2867} pdg_{8762}^{2}}{pdg_{2530}} = 4 pdg_{3141}^{2} pdg_{5156}\)
  1. 3650370389; locally 7324555:
    \(\frac{T^2}{r} F_{gravitational} = 4 \pi^2 m\)
    \(\frac{pdg_{2867} pdg_{8762}^{2}}{pdg_{2530}} = 4 pdg_{3141}^{2} pdg_{5156}\)
valid 3004158505:
3650370389:
3004158505:
3650370389:
11 substitute LHS of expr 1 into expr 2
  1. 6785303857; locally 5154120:
    \(C = 2 \pi r\)
    \(pdg_{3034} = 2 pdg_{2530} pdg_{3141}\)
  2. 3411994811; locally 9055493:
    \(v_{\rm average} = \frac{d}{t}\)
    \(pdg_{6709} = \frac{pdg_{1943}}{pdg_{1467}}\)
  1. 5177311762; locally 7653722:
    \(v = \frac{2 \pi r}{T}\)
    \(pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{8762}}\)
LHS diff is -pdg1357 + pdg6709 RHS diff is -2*pdg2530*pdg3141/pdg8762 + pdg1943/pdg1467 6785303857:
3411994811:
5177311762:
6785303857:
3411994811:
5177311762:
3 change variable X to Y
  1. 1848400430; locally 5546471:
    \(F \propto m\)
    \(F pdg_{5156} propto\)
  1. 3876446703:
    \(m\)
    \(pdg_{5156}\)
  2. 7905984866:
    \(m_1\)
    \(pdg_{5022}\)
  1. 4264859781; locally 8320848:
    \(F \propto m_1\)
    \(F pdg_{5022} propto\)
Nothing to split 1848400430: no LHS/RHS split
4264859781:
1848400430: N/A
4264859781:
18 simplify
  1. 3650814381; locally 1206000:
    \(F_{gravitational} \propto \frac{m_1 m_2}{r^2}\)
    \(\frac{pdg_{2867} pdg_{4851} pdg_{5022} propto}{pdg_{2530}^{2}}\)
  1. 1292735067; locally 8373934:
    \(F_{gravitational} = G \frac{m_1 m_2}{r^2}\)
    \(pdg_{2867} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{2530}^{2}}\)
Nothing to split 3650814381:
1292735067:
3650814381:
1292735067:
13 simplify
  1. 6268336290; locally 9170078:
    \(F_{gravitational} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2\)
    \(pdg_{2867} = \frac{4 pdg_{2530} pdg_{3141}^{2} pdg_{4851}}{pdg_{8762}^{2}}\)
  1. 7672365885; locally 5175707:
    \(F_{gravitational} = \frac{4 \pi^2 m r}{T^2}\)
    \(pdg_{2867} = \frac{4 pdg_{2530} pdg_{3141}^{2} pdg_{4851}}{pdg_{8762}^{2}}\)
valid 6268336290:
7672365885:
6268336290:
7672365885:
14 multiply both sides by
  1. 7672365885; locally 5175707:
    \(F_{gravitational} = \frac{4 \pi^2 m r}{T^2}\)
    \(pdg_{2867} = \frac{4 pdg_{2530} pdg_{3141}^{2} pdg_{4851}}{pdg_{8762}^{2}}\)
  1. 3448601530:
    \(\frac{T^2}{r}\)
    \(\frac{pdg_{9491}^{2}}{pdg_{2530}}\)
  1. 3004158505; locally 4470678:
    \(\frac{T^2}{r} F_{gravitational} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}\)
    \(\frac{pdg_{2867} pdg_{8762}^{2}}{pdg_{2530}} = 4 pdg_{3141}^{2} pdg_{5156}\)
LHS diff is pdg2867*(-pdg8762**2 + pdg9491**2)/pdg2530 RHS diff is 4*pdg3141**2*(pdg4851*pdg9491**2 - pdg5156*pdg8762**2)/pdg8762**2 7672365885:
3004158505:
7672365885:
3004158505:
1 declare initial expr
  1. 5345738321; locally 2020292:
    \(F = m a\)
    \(pdg_{4202} = pdg_{5156} pdg_{9140}\)
no validation is available for declarations 5345738321: dimensions are consistent
5345738321: N/A
6 declare initial expr
  1. 8361238989; locally 6969192:
    \(a_{centripetal} = \frac{v^2}{r}\)
    \(a_{c*(e*(n*(t*(r*(i*(p*(e*(t*(a*l)))))))))} = \frac{pdg_{1357}^{2}}{pdg_{2530}}\)
no validation is available for declarations 8361238989:
8361238989:
4 change variable X to Y
  1. 1848400430; locally 5546471:
    \(F \propto m\)
    \(F pdg_{5156} propto\)
  1. 2346952973:
    \(m\)
    \(pdg_{5156}\)
  2. 9594072504:
    \(m_2\)
    \(pdg_{4851}\)
  1. 4490788873; locally 5440061:
    \(F \propto m_2\)
    \(F pdg_{4851} propto\)
Nothing to split 1848400430: no LHS/RHS split
4490788873:
1848400430: N/A
4490788873:
12 substitute LHS of expr 1 into expr 2
  1. 5177311762; locally 7653722:
    \(v = \frac{2 \pi r}{T}\)
    \(pdg_{1357} = \frac{2 pdg_{2530} pdg_{3141}}{pdg_{8762}}\)
  2. 4267808354; locally 2239910:
    \(F_{gravitational} = m \frac{v^2}{r}\)
    \(pdg_{2867} = \frac{pdg_{1357}^{2} pdg_{5156}}{pdg_{2530}}\)
  1. 6268336290; locally 9170078:
    \(F_{gravitational} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2\)
    \(pdg_{2867} = \frac{4 pdg_{2530} pdg_{3141}^{2} pdg_{4851}}{pdg_{8762}^{2}}\)
LHS diff is 0 RHS diff is 4*pdg2530*pdg3141**2*(-pdg4851 + pdg5156)/pdg8762**2 5177311762:
4267808354:
6268336290:
5177311762:
4267808354:
6268336290:
16 declare guess solution
  1. 3650370389; locally 7324555:
    \(\frac{T^2}{r} F_{gravitational} = 4 \pi^2 m\)
    \(\frac{pdg_{2867} pdg_{8762}^{2}}{pdg_{2530}} = 4 pdg_{3141}^{2} pdg_{5156}\)
  1. 1571582377; locally 6174613:
    \(F_{gravitational} \propto \frac{1}{r^2}\)
    \(pdg_{2867} = \frac{k}{pdg_{2530}^{2}}\)
no validation is available for declarations 3650370389:
1571582377:
3650370389:
1571582377:
this is a big leap of logic that is consistent with Kepler's third law of motion
2 simplify
  1. 5345738321; locally 2020292:
    \(F = m a\)
    \(pdg_{4202} = pdg_{5156} pdg_{9140}\)
  1. 1848400430; locally 5546471:
    \(F \propto m\)
    \(F pdg_{5156} propto\)
Nothing to split 5345738321: dimensions are consistent
1848400430: no LHS/RHS split
5345738321: N/A
1848400430: N/A
Physics Derivation Graph: Steps for Newton's Law of Gravitation

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
4202 variable F
\(F\)
['real']
  • length: 1
  • mass: 1
  • time: -2
force 21
5156 variable m
\(m\)
['real']
  • mass: 1
mass 69
3141 constant \pi
\(\pi\)
['real'] dimensionless pi 3.1415   dimensionless
72
4851 variable m_2
\(m_2\)
real
  • mass: 1
mass 31
1357 variable v
\(v\)
['real']
  • length: 1
  • time: -1
velocity 83
2530 variable r
\(r\)
['real']
  • length: 1
radius 60
9491 variable T
\(T\)
['real']
  • time: 1
period 20
5022 variable m_1
\(m_1\)
real
  • mass: 1
mass 35
3034 variable C
\(C\)
['real']
  • length: 1
circumference 5
1943 variable d
\(d\)
['real']
  • length: 1
displacement 25
6709 variable v_{\rm average}
\(v_{\rm average}\)
real
  • length: 1
  • time: -1
velocity average 2
1687 variable F_{\rm centripetal}
\(F_{\rm centripetal}\)
real
  • length: 1
  • mass: 1
  • time: -2
centripetal force 8
9140 variable a
\(a\)
['real']
  • length: 1
  • time: -2
acceleration 31
8762 variable T_{\rm orbit}
\(T_{\rm orbit}\)
real
  • time: 1
orbital period 14
2867 variable F_{\rm gravity}
\(F_{\rm gravity}\)
real
  • length: 1
  • mass: 1
  • time: -2
force due to gravity 12
1467 variable t
\(t\)
['real']
  • time: 1
time 121
6277 constant G
\(G\)
real
  • length: 3
  • mass: -1
  • time: -2
gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
60
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