# Relationship between F and U; between g and φ

To understand how g is related to φ:

- Similarly, compare $$
and $$g = - G M r 2 φ = - in the above table.GM r - If we differentiate $$
φ = - with respect to r, we will getGM r ⅆ ϕ ⅆ r = - , which has the same expression as g.G M - r 2 - Hence, mathematically
ⅆ ϕ ⅆ r = G M r 2 = - g - To understand the meaning of g
= - observe the two graphs carefully, on the right side where r is positive, the gradient of φ vs r graph is positive but the value of g will be negative. And on the left side where r is negative, the gradient of φ vs r graph is negative but the value of g is positive. Thus, gⅆ ϕ ⅆ r = - ⅆ ϕ ⅆ r

thus F

# Summary

symbol | $$ | |

name | Field strength | Potential |

units | N kg^{-1} or m s^{-2} | J kg^{-1} |

meaning | Force per unit mass | Potential energy per unit mass |

quantity | vector | scalar |

equation | ||

relationship to mass | Force, F | Potential energy, U |

graph | ||

computer model if M = 500. | -6.67*500/(abs(r)*r) | -6.67*500/abs(r) |

relationship between g and φ | ||

relationship between F and U |