Two moons orbit a planet in circular orbits.Moon A has orbital radius R,and moon B has orbital radius 4R.Moon A takes 20 days to complete one orbit.How long does it take moon B to complete an orbit?
A)20 days
B)80 days
C)160 days
D)320 days
E)5 days

Question

Quizplus · Accepted Answer

The orbital period of a moon (or any satellite) is determined by Kepler's third law, which states that the square of the orbital period (T) is directly proportional to the cube of the radius of its orbit (r), i.e., $T^2 \propto r^3$. If Moon A has an orbital radius R and takes 20 days to complete an orbit, and Moon B has an orbital radius of 4R, then the ratio of their periods squared is equal to the ratio of their radii cubed:$\frac{T_B^2}{T_A^2} = \frac{(4R)^3}{R^3} = 64$Given that $T_A = 20$ days, we can solve for $T_B$:$\frac{T_B^2}{20^2} = 64$$T_B^2 = 64 \times 400$$T_B^2 = 25600$$T_B = 160$ days.

Two Moons Orbit a Planet in Circular Orbits