a) perpendicular. The equilibrium is unstable in the other direction.
b) P = 2 pi sqrt(k/m) for SHM
k is the restoring force per unit perpendicular displacement. m is the satellite's mass
For a small displacement y (<<D), the restoring force is
F = 2*GMm/D^2*[y/(D/2)], so
k = F/y = 4GMm/D^3
P = 4 pi sqrt(GM/D^3)
M is the planet's mass and G is the universal constant of gravity.
Two identical planets, each of mass M, have their centers a distance D apart. Halfway between them, at gravitational equilibrium, lies a satellite of mass m. The satellite is then displaced a small distance and released, after which it undergoes SHM.
a) In what direction must the satellite be displaced: along the line joining the two planets, or perpendicular to that line?
b) What is the period of the satellite’s SHM?
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