Gravitational PE (Near Earth)

Gravitational potential energy U of mass m at distance r from M: U = −GMm/r. Always negative (taking U → 0 at infinity).

Near Earth's surface (h ≪ R): U ≈ mgh + constant. Linear approximation; offset reference is arbitrary.

Work done by gravity = −ΔU. Falling object loses PE, gains KE.

Energy required to ESCAPE from surface = |U_surface| = GMm/R. This gives escape velocity (½mv² = GMm/R).

Orbiting body: total energy E = ½mv² + U = −GMm/(2r) (for circular orbit). Negative total energy = bound.

PE is a SCALAR. PE differences are physically meaningful; absolute PE has an arbitrary zero.

PE of a two-body system depends only on separation, not on individual positions.

Tidal forces arise from gradients in gravitational PE — different parts of an extended body sit at different U.