## review derivation: Newton's Law of Gravitation

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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

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