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Review Newton's Law of Gravitation

step inference rule input feed output step validity (as per SymPy)
1
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 5345738321
    \(F = m a\)
 no validation is available for declarations
2
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 5345738321
    \(F = m a\)
  1. 6923310769
    \(F \propto m\)
 LHS diff is -pdg0000004202 + pdg0004202 RHS diff is -pdg0000005156 + pdg0005156*pdg0009140
3
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 6923310769
    \(F \propto m\)
  1. 7905984866
    \(m_1\)
  2. 3876446703
    \(m\)
  1. 4664063894
    \(F \propto m_1\)
 LHS diff is 0 RHS diff is -pdg0000005022 + pdg0000005156
4
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 6923310769
    \(F \propto m\)
  1. 9594072504
    \(m_2\)
  2. 2346952973
    \(m\)
  1. 7222189955
    \(F \propto m_2\)
 LHS diff is 0 RHS diff is -pdg0000004851 + pdg0000005156
5
  • 0000111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an assumption.
  1. 4820320578
    \(F_{\rm gravity} = F_{\rm centripetal}\)
 no validation is available for declarations
6
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 8361238989
    \(a_{\rm centripetal} = \frac{v^2}{r}\)
 no validation is available for declarations
7
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 5345738321
    \(F = m a\)
  2. 8361238989
    \(a_{\rm centripetal} = \frac{v^2}{r}\)
  1. 6026694087
    \(F_{\rm centripetal} = m \frac{v^2}{r}\)
 LHS diff is a_{c*(e*(n*(t*(r*(i*(p*(e*(t*(a*l)))))))))} - pdg0001687 RHS diff is (pdg0001357**2 - pdg0005156*v**2)/pdg0002530
8
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 4820320578
    \(F_{\rm gravity} = F_{\rm centripetal}\)
  2. 6026694087
    \(F_{\rm centripetal} = m \frac{v^2}{r}\)
  1. 4267808354
    \(F_{\rm gravity} = m \frac{v^2}{r}\)
 LHS diff is pdg0001687 - pdg0002867 RHS diff is pdg0005156*(-pdg0001357**2 + v**2)/pdg0002530
9
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 3411994811
    \(v_{\rm average} = \frac{d}{t}\)
 no validation is available for declarations
10
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 6785303857
    \(C = 2 \pi r\)
 no validation is available for declarations
11
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 3411994811
    \(v_{\rm average} = \frac{d}{t}\)
  2. 6785303857
    \(C = 2 \pi r\)
  1. 5177311762
    \(v = \frac{2 \pi r}{T}\)
 LHS diff is -pdg0001357 + pdg0003034 RHS diff is 2*pdg0002530*pdg0003141*(pdg0008762 - 1)/pdg0008762
12
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 4267808354
    \(F_{\rm gravity} = m \frac{v^2}{r}\)
  2. 5177311762
    \(v = \frac{2 \pi r}{T}\)
  1. 6268336290
    \(F_{\rm gravity} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2\)
 LHS diff is pdg0001357 - pdg0002867 RHS diff is 2*pdg0002530*pdg0003141*(-2*pdg0003141*pdg0004851 + pdg0008762)/pdg0008762**2
13
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 6268336290
    \(F_{\rm gravity} = \frac{m}{r}\left(\frac{2\pi r}{T}\right)^2\)
  1. 7672365885
    \(F_{\rm gravity} = \frac{4 \pi^2 m r}{T^2}\)
 valid
14
  • 0000111182: multiply both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 7672365885
    \(F_{\rm gravity} = \frac{4 \pi^2 m r}{T^2}\)
  1. 3448601530
    \(\frac{T^2}{r}\)
  1. 3004158505
    \(\frac{T^2}{r} F_{\rm gravity} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}\)
 LHS arithmetic error. Diff: pdg0002867*(-pdg0008762**2 + pdg0009491**2)/pdg0002530
15
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 3004158505
    \(\frac{T^2}{r} F_{\rm gravity} = \left( \frac{4 \pi^2 m r}{T^2} \right)\frac{T^2}{r}\)
  1. 3650370389
    \(\frac{T^2}{r} F_{\rm gravity} = 4 \pi^2 m\)
 valid
Note about step 16: this is a big leap of logic that is consistent with Kepler's third law of motion
16
  • 0000111237: declare guess solution
  • number of inputs: 1; feeds: 0; outputs: 1
  • Judicious choice as a guessed solution to Eq.~\ref{eq:#1} is Eq.~\ref{eq:#2},
  1. 3650370389
    \(\frac{T^2}{r} F_{\rm gravity} = 4 \pi^2 m\)
  1. 1189963325
    \(F_{\rm gravity} \propto \frac{1}{r^2}\)
 no validation is available for declarations
17
  • 0000111732: substitute LHS of two expressions into expression
  • number of inputs: 3; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} and LHS of Eq.~\ref{eq:#2} into Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 1189963325
    \(F_{\rm gravity} \propto \frac{1}{r^2}\)
  2. 7222189955
    \(F \propto m_2\)
  3. 4664063894
    \(F \propto m_1\)
  1. 8617866819
    \(F_{\rm gravity} \propto \frac{m_1\ m_2}{r^2}\)
 recognized infrule but not yet supported
18
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 8617866819
    \(F_{\rm gravity} \propto \frac{m_1\ m_2}{r^2}\)
  1. 1292735067
    \(F_{\rm gravity} = G \frac{m_1 m_2}{r^2}\)
 LHS diff is -pdg0002867 + pdg4482727458 RHS diff is -pdg0004851*pdg0005022*pdg0006277/pdg0002530**2 + pdg0000004851*pdg0000005022/pdg0000002530**2
19
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 1292735067
    \(F_{\rm gravity} = G \frac{m_1 m_2}{r^2}\)
no validation is available for declarations

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