## 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

symbol ID category latex scope dimension name value Used in derivations references
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
31
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
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
21
5022 variable m_1
$$m_1$$
real
• mass: 1
mass
35
9140 variable a
$$a$$
['real']
• length: 1
• time: -2
acceleration 31
5156 variable m
$$m$$
['real']
• mass: 1
mass
69
6277 constant G
$$G$$
real
• length: 3
• mass: -1
• time: -2
gravitational constant 6.67430*10^{-11}   m^3 * kg^-1 * s^-2
60
MESSAGE:
• local variable 'all_df' referenced before assignment