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review derivation: mass of the Earth

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
6 replace constant with value
  1. 9440616166; locally 9133599:
    \(m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G}\)
    \(pdg_{5458} = \frac{pdg_{3236}^{2} pdg_{7557}}{pdg_{6277}}\)
  1. 2091584724:
    \(g_{\rm Earth}\)
    \(pdg_{7557}\)
  2. 9590696981:
    \(9.80665\)
    \(9.80665\)
  3. 7816982139:
    \(m/s^2\)
    \(\frac{\text{m}^{2}}{\text{s}^{2}}\)
  1. 7846240076; locally 2593741:
    \(m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}\)
    \(pdg_{5458} = \frac{9 pdg_{3236}^{2}}{pdg_{6277}}\)
no check performed 9440616166:
7846240076:
9440616166:
7846240076:
4 divide both sides by
  1. 9407192813; locally 7388891:
    \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth}\)
    \(\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}} = pdg_{5156} pdg_{7557}\)
  1. 3246378279:
    \(m\)
    \(pdg_{5156}\)
  1. 2308660627; locally 9159337:
    \(G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth}\)
    \(\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}} = pdg_{7557}\)
valid 9407192813:
2308660627:
9407192813:
2308660627:
8 replace constant with value
  1. 7112613117; locally 1218257:
    \(m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}\)
    \(pdg_{5458}\)
  1. 7935917166:
    \(r_{\rm Earth}\)
    \(pdg_{3236}\)
  2. 3723096423:
    \(6.3781*10^6\)
    \(6378100.0\)
  3. 7560908617:
    \(m\)
    \(pdg_{5156}\)
  1. 1132941271; locally 9815516:
    \(m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}\)
    \(pdg_{5458}\)
Nothing to split 7112613117:
1132941271:
7112613117:
1132941271:
1 change variable X to Y
  1. 5345738321; locally 3843242:
    \(F = m a\)
    \(pdg_{4202} = pdg_{5156} pdg_{9140}\)
  1. 9881106100:
    \(a\)
    \(pdg_{9140}\)
  2. 5781435087:
    \(g\)
    \(pdg_{1649}\)
  1. 2484824786; locally 6779814:
    \(F = m g\)
    \(pdg_{4202} = pdg_{1649} pdg_{5156}\)
valid 5345738321:
2484824786: dimensions are consistent
 5345738321:
2484824786: N/A
2 change three variables in expr
  1. 6935745841; locally 2737346:
    \(F = G \frac{m_1 m_2}{x^2}\)
    \(pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}\)
  1. 3921072591:
    \(m_1\)
    \(pdg_{5458}\)
  2. 1193980495:
    \(m_{\rm Earth}\)
    \(pdg_{5458}\)
  3. 4651061153:
    \(m_2\)
    \(pdg_{4851}\)
  4. 9903988330:
    \(m\)
    \(pdg_{5156}\)
  5. 3353418803:
    \(x\)
    \(pdg_{4037}\)
  6. 6535639720:
    \(r_{\rm Earth}\)
    \(pdg_{3236}\)
  1. 8661803554; locally 6771172:
    \(F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}\)
    \(pdg_{4202} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}}\)
LHS diff is 0 RHS diff is pdg5156*pdg6277*(pdg5022 - pdg5458)/pdg3236**2 6935745841:
8661803554:
6935745841:
8661803554:
9 simplify
  1. 1132941271; locally 9815516:
    \(m_{\rm Earth} = \frac{(9.80665 m/s^2) (6.3781*10^6 m)^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}\)
    \(pdg_{5458}\)
  1. 3364286646; locally 1635641:
    \(m_{\rm Earth} = 5.972*10^{24} kg\)
    \(pdg_{5458} = 5.972 \cdot 10^{24} kg\)
Nothing to split 1132941271:
3364286646:
1132941271:
3364286646:
7 replace constant with value
  1. 7846240076; locally 2593741:
    \(m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{G}\)
    \(pdg_{5458} = \frac{9 pdg_{3236}^{2}}{pdg_{6277}}\)
  1. 7326066466:
    \(G\)
    \(pdg_{6277}\)
  2. 9956609318:
    \(6.67430*10^{-11}\)
    \(6.6743 \cdot 10^{-11}\)
  3. 2957211007:
    \(m^3 kg^{-1} s^{-2}\)
    \(\frac{\text{m}^{3}}{\text{s}^{2}}\)
  1. 7112613117; locally 1218257:
    \(m_{\rm Earth} = \frac{(9.80665 m/s^2) r_{\rm Earth}^2}{6.67430*10^{-11}m^3 kg^{-1} s^{-2}}\)
    \(pdg_{5458}\)
Nothing to split 7846240076:
7112613117:
7846240076:
7112613117:
0.5 declare initial expr
  1. 5345738321; locally 3843242:
    \(F = m a\)
    \(pdg_{4202} = pdg_{5156} pdg_{9140}\)
no validation is available for declarations 5345738321:
5345738321:
3 LHS of expr 1 equals LHS of expr 2
  1. 8661803554; locally 6771172:
    \(F = G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2}\)
    \(pdg_{4202} = \frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}}\)
  2. 4800170179; locally 6086107:
    \(F = m g_{\rm Earth}\)
    \(pdg_{4202} = pdg_{5156} pdg_{7557}\)
  1. 9407192813; locally 7388891:
    \(G \frac{m_{\rm Earth} m}{r_{\rm Earth}^2} = m g_{\rm Earth}\)
    \(\frac{pdg_{5156} pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}} = pdg_{5156} pdg_{7557}\)
valid 8661803554:
4800170179: dimensions are consistent
9407192813:
8661803554:
4800170179: N/A
9407192813:
1.5 declare initial expr
  1. 6935745841; locally 2737346:
    \(F = G \frac{m_1 m_2}{x^2}\)
    \(pdg_{4202} = \frac{pdg_{4851} pdg_{5022} pdg_{6277}}{pdg_{4037}^{2}}\)
no validation is available for declarations 6935745841:
6935745841:
5 multiply both sides by
  1. 2308660627; locally 9159337:
    \(G \frac{m_{\rm Earth}}{r_{\rm Earth}^2} = g_{\rm Earth}\)
    \(\frac{pdg_{5458} pdg_{6277}}{pdg_{3236}^{2}} = pdg_{7557}\)
  1. 2685587762:
    \(\frac{r_{\rm Earth}^2}{G}\)
    \(\frac{pdg_{3236}^{2}}{pdg_{6277}}\)
  1. 9440616166; locally 9133599:
    \(m_{\rm Earth} = \frac{g_{\rm Earth} r_{\rm Earth}^2}{G}\)
    \(pdg_{5458} = \frac{pdg_{3236}^{2} pdg_{7557}}{pdg_{6277}}\)
valid 2308660627:
9440616166:
2308660627:
9440616166:
1.25 change variable X to Y
  1. 2484824786; locally 6779814:
    \(F = m g\)
    \(pdg_{4202} = pdg_{1649} pdg_{5156}\)
  1. 9355039511:
    \(g\)
    \(pdg_{1649}\)
  2. 2232825726:
    \(g_{\rm Earth}\)
    \(pdg_{7557}\)
  1. 4800170179; locally 6086107:
    \(F = m g_{\rm Earth}\)
    \(pdg_{4202} = pdg_{5156} pdg_{7557}\)
valid 2484824786: dimensions are consistent
4800170179: dimensions are consistent
 2484824786: N/A
4800170179: N/A
Physics Derivation Graph: Steps for mass of the Earth

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
4851 variable m_2
\(m_2\)		 real
  • mass: 1
 mass 31
1649 variable g
\(g\)		 ['real']
  • length: 1
  • time: -2
 acceleration due to gravity 27
4202 variable F
\(F\)		 ['real']
  • length: 1
  • mass: 1
  • time: -2
 force 21
3236 constant r_{\rm Earth}
\(r_{\rm Earth}\)		 real
  • length: 1
 radius of Earth 6.3781*10^{6}   m
 21
5022 variable m_1
\(m_1\)		 real
  • mass: 1
 mass 35
7557 constant g_{\rm Earth}
\(g_{\rm Earth}\)		 real
  • length: 1
  • time: -2
 average acceleration due to gravity on Earth 9.80665   m * s^-2
 6
6277 constant G
\(G\)		 real
  • length: 3
  • mass: -1
  • time: -2
 gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
 60
9140 variable a
\(a\)		 ['real']
  • length: 1
  • time: -2
 acceleration 31
5156 variable m
\(m\)		 ['real']
  • mass: 1
 mass 69
4037 variable x
\(x\)		 ['real']
  • length: 1
 position 53
5458 constant m_{\rm Earth}
\(m_{\rm Earth}\)		 real
  • mass: 2
 mass of Earth 5.97237*10^24   kg
 34
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