Three Related Quantities

Gravitational Field g (intensity)

• Definition: Force per unit mass at a point
• Formula: g = −GM/$r^{2}$ (magnitude), directed inward
• Unit: N $kg^{-1}$ = m $s^{-2}$
• Dimensional formula: [$M^{0}$ $L^{1}$ $T^{-2}$]

Gravitational Potential V

• Definition: Potential energy per unit mass
• Formula: V = −GM/r
• Unit: J $kg^{-1}$ = $m^{2}$ $s^{-2}$
• Dimensional formula: [$M^{0}$ $L^{2}$ $T^{-2}$]

Gravitational Potential Energy U

• Definition: Energy of mass m in gravitational field
• Formula: U = mV = −GMm/r
• Unit: J
• Dimensional formula: [$M^{1}$ $L^{2}$ $T^{-2}$]

Relationships

g = −dV/dr (field = negative gradient of potential) U = mV (multiply potential by mass) F = −dU/dr = mg (force = negative gradient of PE = mass × field)

At Earth's Surface

Key Dimensional Insight

V has dimensions [$L^{2}$ $T^{-2}$] — same as velocity squared. This is not a coincidence: escape velocity v_e = √(2|V_surface|), linking the two quantities.