A small spherical moon of radius r and mass m orbits a planet of mass M, always showing the same face. Weightlessness at the point nearest the planet occurs when orbital radius is r × ³√(xM/m). Determine x.
Physics · Mixed Chapter
JEE Main Practice

Question

A small spherical moon of radius r and mass m orbits a planet of mass M, always showing the same face. Weightlessness at the point nearest the planet occurs when orbital radius is r × ³√(xM/m). Determine x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — The moon rotates synchronously about the planet. At a near-surface point the moon’s self-gravity gₘ = Gm/r². Step 2 — Effective differential (tidal) acceleration between near-side surface and moon centre is approximately 3GMr/R³ in the relevant rotating frame for the source convention. Step 3 — Weightlessness occurs when Gm/r² = 3GMr/R³. Step 4 — Solve R³ = 3Mr³/m ⇒ R = r ³√(3M/m); x = 3.