Two moons orbit a planet in nearly 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 one orbit? Multiple Choice 160 days 80 days 20 days 320 days
The correct answer and explanation is:
Let’s analyze the problem step-by-step:
Given:
- Moon A orbits at radius rr with period TA=20T_A = 20 days.
- Moon B orbits at radius 4r4r with unknown period TBT_B.
- Orbits are nearly circular.
- Question: Find TBT_B.
Step 1: Understand the relationship between orbital radius and period
For moons orbiting a planet under gravity, Kepler’s Third Law applies: T2∝r3T^2 \propto r^3
This means the square of the orbital period TT is proportional to the cube of the orbital radius rr.
We can write the ratio for the two moons: TB2TA2=(4r)3r3=43=64\frac{T_B^2}{T_A^2} = \frac{(4r)^3}{r^3} = 4^3 = 64
Step 2: Solve for TBT_B
TB2202=64 ⟹ TB2=64×400=25600\frac{T_B^2}{20^2} = 64 \implies T_B^2 = 64 \times 400 = 25600 TB=25600=160 daysT_B = \sqrt{25600} = 160 \text{ days}
Answer: Moon B takes 160 days to complete one orbit.
Explanation in 300 words:
Two moons orbiting the same planet follow Kepler’s Third Law, which states that the square of the orbital period TT is proportional to the cube of the orbit’s radius rr. Mathematically, this is: T2∝r3T^2 \propto r^3
This relationship means if you know the orbital period and radius of one moon, you can find the orbital period of another moon by comparing their radii.
In this problem, Moon A orbits at radius rr with a period of 20 days. Moon B orbits farther out, at a radius 4r4r. Since the period depends on the radius raised to the power of 32\frac{3}{2} (because T∝r3/2T \propto r^{3/2}), increasing the radius by a factor of 4 increases the period by 43/24^{3/2}.
Calculate this step: 43/2=(43)1/2=64=84^{3/2} = (4^3)^{1/2} = \sqrt{64} = 8
Thus, Moon B’s period is 8 times that of Moon A: TB=8×20=160 daysT_B = 8 \times 20 = 160 \text{ days}
Therefore, Moon B takes 160 days to complete one orbit around the planet. This result illustrates how orbital periods increase significantly with orbital radius due to the gravitational dynamics dictated by Kepler’s laws.
Final answer: 160 days