Physics Derivation Graph navigation Sign in

review derivation: mass of the Earth

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.


Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Notes for this derivation:

Options
Alternate views of this derivation:
Edit this content:    

To edit a step, click on the number in the "Index" column in the table below

Clicking on the step index will take you to the page where you can edit that step.

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 215 symbols
symbol ID category latex scope dimension name value Used in derivations references
5022 variable m_1
\(m_1\)
real
  • mass: 1
mass 18
3236 constant r_{\rm Earth}
\(r_{\rm Earth}\)
real
  • length: 1
radius of Earth 6.3781*10^{6}   m
21
5458 constant m_{\rm Earth}
\(m_{\rm Earth}\)
real
  • mass: 2
mass of Earth 5.97237*10^24   kg
34
9140 variable a
\(a\)
['real']
  • length: 1
  • time: -2
acceleration 31
4037 variable x
\(x\)
['real']
  • length: 1
position 53
4202 variable F
\(F\)
['real']
  • length: 1
  • mass: 1
  • time: -2
force 21
5156 variable m
\(m\)
['real']
  • mass: 1
mass 57
1649 variable g
\(g\)
['real']
  • length: 1
  • time: -2
acceleration due to gravity 27
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
4851 variable m_2
\(m_2\)
real
  • mass: 1
mass 17
6277 constant G
\(G\)
real
  • length: 3
  • mass: -1
  • time: -2
gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
46
MESSAGE: