The fastest possible rate of rotation of a planet is that for which the gravitational force on material at the equator just barely provides the centripetal force needed for the rotation. (why? )
(a) show that the corresponding shortest period of rotation is t = √3π/grho, where rho is the uniform density (mass per unit volume) of the spherical planet.
(b) calculate the rotation period assuming a density of 3.0 g/cm³, typical of many planets, satellites, and asteroids. no astronomical object has ever been found to be spinning with a period shorter than that determined by this analysis.

6862.96871 seconds

Explanation:

M = Mass of Planet

G = Gravitational constant

r = Radius

= Density

T = Rotation period

In this system the gravitational force will balance the centripetal force

.

Hence, proved

The rotation period of the astronomical object is 6862.96871 seconds

the answer would be c

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